{"id":94,"date":"2021-12-15T09:53:27","date_gmt":"2021-12-15T09:53:27","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/integrationsmethoden\/"},"modified":"2021-12-15T09:53:27","modified_gmt":"2021-12-15T09:53:27","slug":"integrationsmethoden","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/integrationsmethoden\/","title":{"raw":"Integrationsmethoden","rendered":"Integrationsmethoden"},"content":{"raw":"\n<style>.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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\n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"zeca2210f480f\" class=\"sectionHead\"><span class=\"titlemark\">9.2 <\/span> <a id=\"x1-2630002\"><\/a>Integrationsmethoden<\/h3> <p class=\"noindent\">Wir erinnern daran, dass das unbestimmte Integral einer Funktion <math display=\"inline\"><mi>f<\/mi><\/math> in der Variablen <math display=\"inline\"><mi>x<\/mi><\/math> der Ausdruck <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo> \u222b  <\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">ist, wobei <math display=\"inline\"><mi>F<\/mi><\/math> eine Stammfunktion von <math display=\"inline\"><mi>f<\/mi><\/math> ist. In den Abschnitten <a href=\"..\/..\/chapter\/erste-differentialgleichungen#x1-2520002\">8.5.2<\/a> und <a href=\"..\/..\/chapter\/hyperbolische-funktionen#x1-2460004\">8.4<\/a> haben wir bereits einige Regeln zur Berechnung konkreter unbestimmter Integrale kennengelernt: F\u00fcr <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> (oder sogar&nbsp;<math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math>) ist <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo> \u222b  <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>s<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>s<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"> <mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>s<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo> \u222b  <\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo><mi class=\"qopname\">cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> arcsin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> arctan<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> arsinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/msqrt><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> arcosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Des Weiteren gilt f\u00fcr Funktionen <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> in der Variable <math display=\"inline\"><mi>x<\/mi><\/math> und Zahlen <math display=\"inline\"><msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo> \u222b  <\/mo><msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo> \u222b  <\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo> \u222b  <\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Denn falls <math display=\"inline\"><msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> eine Stammfunktion von <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> ist und <math display=\"inline\"><msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> eine Stammfunktion von <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> ist, so muss die Funktion <math display=\"inline\"><msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> auf Grund der Linearit\u00e4t der Ableitung (Proposition <a href=\"..\/..\/chapter\/die-ableitung#x1-228010r5\">8.5<\/a>) eine Stammfunktion von <math display=\"inline\"><msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> sein. <\/p><p class=\"indent\">Auf \u00e4hnliche Weise lassen sich die anderen Regeln der Differentiation als Identit\u00e4ten f\u00fcr unbestimmte Integrale auffassen, wie wir nun ausf\u00fchren wollen. <a id=\"x1-263001r262\"><\/a> <\/p> <h4 id=\"zd2c0683dd854\" class=\"subsectionHead\"><span class=\"titlemark\">9.2.1 <\/span> <a id=\"x1-2640001\"><\/a>Partielle Integration<\/h4> <p class=\"noindent\">Die Produktregel in Proposition <a href=\"..\/..\/chapter\/die-ableitung#x1-228010r5\">8.5<\/a> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mi>v<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>v<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>u<\/mi><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr zwei differenzierbare Funktionen <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>v<\/mi><\/math> f\u00fchrt ebenso zu einer Integrationsregel, n\u00e4mlich der <span class=\"ecbx-1095\">partiellen Integration<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>u<\/mi><mi>v<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>v<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>u<\/mi><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo><mi>u<\/mi><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>u<\/mi><mi>v<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mo>\u222b  <\/mo><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>v<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-264001r3\" \/><mstyle class=\"maketag\"><mtext>(9.3)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">In der Leibniz-Notation ist <math display=\"inline\"><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>v<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><\/mfrac><\/math> und <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo> <mi>u<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> Deswegen schreibt man die partielle Integration oft auch als <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo> \u222b  <\/mo><mi>u<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>v<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>u<\/mi><mi>v<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mo>\u222b  <\/mo><mi>v<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was formal bloss als Kurzform der Formel in (<a href=\"..\/..\/chapter\/integrationsmethoden#x1-264001r3\">9.3<\/a>) verstanden werden sollte. Die Regel der partiellen Integration ist bereits ein Beispiel, wo eine einfache Regel des Differenzierens eine komplexere Regel des Integrierens als Entsprechung hat. Die Produktregel erlaubt uns, die Ableitung jedes Produkts mittels der Ableitung dessen Faktoren auszudr\u00fccken. Die partielle Integration hingegen erlaubt uns, das unbestimmte Integral eines Produkts mittels dem Integral eines Faktors und eines weiteren Integrals auszudr\u00fccken. Mit etwas Gl\u00fcck (und Geschick) ist das zweite Integral einfacher und kann anschliessend berechnet werden. Wir demonstrieren dies anhand zweier Beispiele. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zeb7203a01703\"> <a id=\"x1-264002r15\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.15 <\/span>(Beispiele partieller Integration)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Wir berechnen das unbestimmte Integral <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><mi>x<\/mi><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Daf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r setzen wir <\/span><math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Eine<\/span> <span class=\"ecti-1095\">Stammfunktion von <\/span><math display=\"inline\"><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">ist <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Damit erhalten wir<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo><mi>x<\/mi><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mo>\u222b  <\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u22c5<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Wir bemerken, dass es gen<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">gt, in solchen Berechnungen immer bloss eine unbekannte Integrationskonstante<\/span> <math display=\"inline\"><mi>C<\/mi><\/math> <span class=\"ecti-1095\">zu<\/span> <span class=\"ecti-1095\">verwenden, da mehrere solche einfach zusammengefasst werden k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen. (Kontrollieren Sie diese<\/span> <span class=\"ecti-1095\">Rechnung durch Ableiten.) Dieselbe Berechnungsmethode f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hrt auch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r unbestimmte Integrale der<\/span> <span class=\"ecti-1095\">Form <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">zum Erfolg.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Wir wollen das unbestimmte Integral <\/span><math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">berechnen. Es mag zuerst etwas <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berraschend sein, dass wir dazu partielle Integration verwenden wollen.<\/span> <span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math><span class=\"ecti-1095\">. Dann<\/span> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Stammfunktion von <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">womit<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>1<\/mn><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mo>\u222b  <\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mo>\u222b  <\/mo><mn>1<\/mn><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Dies kann man wiederum durch Ableiten verifizieren (was nicht notwendig ist, aber einen sehr<\/span> <span class=\"ecti-1095\">einfachen Test darstellt).<\/span><\/p><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zaf2f70ba0f47\"> <a id=\"x1-264005r16\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.16.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Berechnen Sie <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Geben Sie eine rekursive Formel zur Berechnung von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">an.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Berechnen Sie<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Beachten Sie hierbei, dass der Fall<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math> <span class=\"ecti-1095\">getrennt zu behandeln ist.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Berechnen Sie das unbestimmte Integral <\/span><math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><msup><mrow><mi>e<\/mi><\/mrow><mrow><mi>a<\/mi><mi>x<\/mi><\/mrow><\/msup><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> <p class=\"noindent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Bei (iv) ergibt sich nach zweifacher partieller Integration ein Gleichungssystem.<\/span><\/p><\/details><\/dd><\/dl> <\/div> <a id=\"x1-264010r264\"><\/a> <h4 id=\"z707389867a81\" class=\"subsectionHead\"><span class=\"titlemark\">9.2.2 <\/span> <a id=\"x1-2650002\"><\/a>Substitution<\/h4> <p class=\"noindent\">Falls eine Funktion <math display=\"inline\"><mi>g<\/mi><\/math> auf einem Intervall <math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>u<\/mi> <\/mrow> <\/msub> <\/math> das unbestimmte Integral <math display=\"inline\"><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>G<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><\/math> besitzt und <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>x<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>u<\/mi> <\/mrow> <\/msub> <\/math> eine stetig differenzierbare Abbildung auf dem Intervall <math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msub><\/math> ist, dann gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo> \u222b  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>G<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">auf <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>x<\/mi> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Dies folgt unmittelbar aus der Kettenregel in Satz <a href=\"..\/..\/chapter\/die-ableitung#x1-228014r8\">8.8<\/a> und wird oft auch geschrieben als <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo> \u222b  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-265001r4\" \/><mstyle class=\"maketag\"><mtext>(9.4)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr die \u201e neue Variable\u201c <span class=\"maperiod\"><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Alternativ werden wir die obige <span class=\"ecbx-1095\">Substitutionsregel <\/span>gemeinsam mit der Leibniz-Notation auch in folgender informellen Schreibweise                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo> \u222b  <\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>u<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>G<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>u<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>G<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">verwenden, wobei <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo> <mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z9d0eed366239\"> <a id=\"x1-265002r17\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.17.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Es gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><munder class=\"msub\"><mrow><munder accentunder=\"false\"><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/mrow><mo>\ufe38<\/mo><\/munder> <\/mrow><mrow><mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>u<\/mi><\/mrow><\/mfrac> <\/mrow><\/munder><munder class=\"msub\"><mrow><munder accentunder=\"false\"><mrow> <mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><mo>\ufe38<\/mo><\/munder><\/mrow><mrow><mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><\/mrow><\/munder> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo>\u222b  <\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>u<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>u<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">gesetzt<\/span> <span class=\"ecti-1095\">wurde, womit <\/span><span class=\"maperiod\"><math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Es gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><mi class=\"qopname\">sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi class=\"qopname\">tan<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>v<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>v<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>v<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi class=\"qopname\">tan<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><span class=\"ecti-1095\">, und<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>v<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> tan<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>v<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/p><\/dd><\/dl> <\/div> <p class=\"indent\">Wie bereits erw\u00e4hnt, ben\u00f6tigt die Integration mehr \u00dcbung und Vorraussicht als die Differentiation. Obige Substitutionen ben\u00f6tigen zum Beispiel den Blick ob gewisse Faktoren vielleicht die gew\u00fcnschte Ableitung&nbsp;<math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/math> einer inneren Funktionen&nbsp;<math display=\"inline\"><mi>f<\/mi><\/math> darstellen k\u00f6nnte. Manchmal ist dies naheliegend wie in Beispiel <a href=\"..\/..\/chapter\/integrationsmethoden#x1-265002r17\">9.17<\/a>(i), doch manchmal erfordert dies Erfahrung und eine l\u00e4ngere Suche wie in Beispiel&nbsp;<a href=\"..\/..\/chapter\/integrationsmethoden#x1-265002r17\">9.17<\/a>(ii). <a id=\"x1-265005r265\"><\/a> <\/p> <h4 id=\"zfc69cbc62eaa\" class=\"subsectionHead\"><span class=\"titlemark\">9.2.3 <\/span> <a id=\"x1-2660003\"><\/a>Integration rationaler Funktionen<\/h4> <p class=\"noindent\">Wir erinnern daran, dass eine rationale Funktion eine Funktion der Form <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mfrac> <mrow> <mi>p<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>q<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/math> f\u00fcr Polynome <math display=\"inline\"><mi>p<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo> <mi>q<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> und <math display=\"inline\"><mi>q<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> ist, wobei der Definitionsbereich <math display=\"inline\"><mi>\u211d<\/mi><\/math> ohne die Nullstellen von <math display=\"inline\"><mi>q<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> ist. Wir wollen hier ein Verfahren zur Berechnung des unbestimmten Integrals einer rationalen Funktion besprechen. Nach Division mit Rest f\u00fcr Polynome (siehe \u00dcbung <a href=\"..\/..\/chapter\/polynome#x1-82002r17\">3.17<\/a>) k\u00f6nnen wir als erstes ein Polynom abspalten, so dass die verbleibende rationale Funktion von der Form <math display=\"inline\"><mfrac><mrow><msub><mrow><mi>p<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>q<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <\/math> f\u00fcr <math display=\"inline\"><mi class=\"qopname\">deg<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>p<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo><mi class=\"qopname\"> deg<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>q<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> ist.                                                                                                                                                                           <\/p><p class=\"indent\">Da Polynome mittels der Formel <math display=\"inline\"><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><\/math> integriert werden k\u00f6nnen, nehmen wir nun an, dass der Grad von <math display=\"inline\"><mi>p<\/mi><\/math> kleiner als der Grad von <math display=\"inline\"><mi>q<\/mi><\/math> ist. Wir betrachten zuerst einige Spezialf\u00e4lle. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z86cbe1077c0d\"> <a id=\"x1-266001r18\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.18 <\/span>(Integration von elementaren rationalen Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">beliebig und <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">eine nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche Zahl.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Es gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>u<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>u<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/math> <span class=\"ecti-1095\">gesetzt<\/span> <span class=\"ecti-1095\">wurde und <\/span><math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">ist.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei wieder <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/math> <span class=\"ecti-1095\">gesetzt wurde und <\/span><math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">ist.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ist, so gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo> \u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>a<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>a<\/mi><\/mrow><\/mfrac><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>a<\/mi><\/mrow><\/mfrac><mi class=\"qopname\">arctan<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>u<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>a<\/mi><\/mrow><\/mfrac><mi class=\"qopname\">arctan<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>a<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>a<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">gesetzt<\/span> <span class=\"ecti-1095\">wurde und <\/span><math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>a<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">ist.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Es gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo>\u222b  <\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>u<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>u<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo> <mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(v)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>u<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><msup><mrow><mi>u<\/mi><\/mrow><mrow><mn>1<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo> <mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/p><\/dd><\/dl> <\/div> <p class=\"indent\">Im Allgemeinen verwenden wir die sogenannte <span class=\"ecbx-1095\">Partialbruchzerlegung <\/span>f\u00fcr die rationale Funktion <math display=\"inline\"><mfrac><mrow><mi>p<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>q<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/math> (nach Division mit Rest so dass <math display=\"inline\"><mi class=\"qopname\"> deg<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>p<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo><mi class=\"qopname\"> deg<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>q<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math>), um die Integration auf obige Beispiele zur\u00fcckzuf\u00fchren. In der Tat l\u00e4sst sich <math display=\"inline\"><mfrac><mrow><mi>p<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>q<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/math> als Linearkombination von einfacheren rationalen Funktionen darstellen. Diese sind von der Form                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mspace class=\"quad\" width=\"1em\" \/><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">oder von der Form <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03bb<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u03bb<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mspace class=\"quad\" width=\"1em\" \/><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/> <mfrac><mrow><msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow> <mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03bb<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u03bb<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wobei <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> eine Nullstelle von <math display=\"inline\"><mi>q<\/mi><\/math> mit Vielfachheit <math display=\"inline\"><mi>k<\/mi><\/math> und <math display=\"inline\"><mi>\u03bb<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mi>\u211d<\/mi><\/math> eine Nullstelle von <math display=\"inline\"><mi>q<\/mi><\/math> mit Vielfachheit <math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> ist und <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z9159d80b648b\"> <a id=\"x1-266007r19\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.19 <\/span>(Integration rationaler Funktionen und die Partialbruchzerlegung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Wir wollen das unbestimmte Integral <\/span><math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo> <mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">bestimmen. Als erstes f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hren wir Division mit Rest<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><\/mtd> <mtd class=\"align-even\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-bin\">\u2212<\/mo><\/mtd> <mtd class=\"align-even\"><munder accentunder=\"false\" class=\"mml-underline\"><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"><mspace class=\"quad\" width=\"1em\" \/> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"><mspace class=\"quad\" width=\"1em\" \/><mspace class=\"thinspace\" width=\"0.17em\" \/><mspace class=\"quad\" width=\"1em\" \/><munder accentunder=\"false\" class=\"mml-underline\"><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"><mspace class=\"qquad\" width=\"2em\" \/><mspace class=\"nbsp\" width=\"0.33em\" \/><mspace class=\"qquad\" width=\"2em\" \/><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mspace class=\"quad\" width=\"1em\" \/><mi class=\"qopname\">Rest<\/mi><mo>  <\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">durch, womit<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><mo> \u222b  <\/mo> <mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Um die Partialbruchzerlegung von <\/span><math display=\"inline\"> <mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">zu erhalten, setzen wir<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>A<\/mi><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>B<\/mi><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>C<\/mi><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r noch unbekannte Zahlen <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>B<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>C<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">multiplizieren mit <\/span><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und erhalten<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><mi>x<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Nun setzen wir in diesem <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">um <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">zu erhalten<\/span> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math><span class=\"ecti-1095\">, um<\/span> <math display=\"inline\"><mi>C<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">zu erhalten.<\/span> <span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">ergibt<\/span> <span class=\"ecti-1095\">sich nun <\/span><math display=\"inline\"><mn>2<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>2<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>2<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">und somit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">(Alternativ kann man auch beide Seiten ausmultiplizieren, die Koeffizienten links<\/span> <span class=\"ecti-1095\">und rechts vergleichen, und auf diese Weise drei Gleichungen in den unbekannten<\/span> <span class=\"ecti-1095\">Variablen<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>B<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>C<\/mi><\/math> <span class=\"ecti-1095\">erhalten.) Daher ist<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mo>\u222b  <\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>D<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Wir berechnen das unbestimmte Integral<\/span> <math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">+<\/mo><mn>2<\/mn><mi>x<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><span class=\"ecti-1095\">. Man beachte dabei,<\/span> <span class=\"ecti-1095\">dass das Polynom <\/span><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">keine reellen Nullstellen hat. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Partialbruchzerlegung machen wir den Ansatz<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>A<\/mi><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>B<\/mi><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Nun multiplizieren wir mit <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und erhalten<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mn>1<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ergibt<\/span> <span class=\"ecti-1095\">sich <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Daher ist<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mn>1<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>C<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Es folgt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo> \u222b  <\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo> \u222b  <\/mo> <mfrac><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo> \u222b  <\/mo> <mfrac><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo> \u222b  <\/mo> <mfrac><mrow><mi>u<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>4<\/mn><\/mrow><\/mfrac><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\"> arctan<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>u<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>D<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>4<\/mn><\/mrow><\/mfrac><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msup><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\"> arctan<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>D<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei wir <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">gesetzt haben und Beispiele <\/span><a href=\"..\/..\/chapter\/integrationsmethoden#x1-266001r18\"><span class=\"ecti-1095\">9.18<\/span><\/a> <span class=\"ecti-1095\">(c) und (d) verwendet haben.<\/span><\/p><\/dd><\/dl> <\/div> <p class=\"indent\">In manchen F\u00e4llen kann obiges Verfahren auch auf das Integral <math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> f\u00fcr ein <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> und <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math> f\u00fchren, was wir mit der trigonometrischen Substitution <math display=\"inline\"><mi class=\"qopname\"> tan<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>a<\/mi><\/mrow><\/mfrac><\/math> (siehe unten) behandeln k\u00f6nnen. Eine andere, allgemeinere Herangehensweise m\u00f6chten wir in folgender Bemerkung f\u00fcr Interessierte behandeln. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z897e2ad8e28b\"> <span class=\"ecti-1095\">Bemerkung <\/span>(Integration rationaler Funktionen mit mehrfachen komplexen Nullstellen)<span class=\"ecti-1095\">.<\/span> <\/h4> <p class=\"indent\">Wie oben schon bemerkt, kann man nach der Partialbruchzerlegung ein Integral einer rationalen Funktion auf die Integration von Ausdr\u00fccken der Form <math display=\"inline\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><\/math> oder von <math display=\"inline\"> <mfrac> <mrow> <mi>A<\/mi><mi>x<\/mi><mo class=\"MathClass-bin\">+<\/mo><mi>B<\/mi><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">+<\/mo><mi>b<\/mi><mi>x<\/mi><mo class=\"MathClass-bin\">+<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><\/math> f\u00fcr <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und f\u00fcr Konstanten <math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>B<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> zur\u00fcckf\u00fchren, wobei die Polynome der Form&nbsp;<math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>b<\/mi><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>c<\/mi><\/math> keine rellen Nullstellen haben. F\u00fcr die Berechnung eines Integrals des zweiten Typs mit <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> m\u00f6chten wir hier einen Algorithmus erl\u00e4utern, wobei wir uns auf den Fall&nbsp;<math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>b<\/mi><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>c<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math> beschr\u00e4nken (auf welchen man den allgemeinen Fall mit quadratischem Erg\u00e4nzen zur\u00fcckf\u00fchren kann). <\/p><p class=\"indent\">Seien also <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> und ein Polynom <math display=\"inline\"><mi>q<\/mi><\/math> von Grad kleiner als <math display=\"inline\"><mn>2<\/mn><mi>k<\/mi><\/math> gegeben. Dann ist das unbestimmte Integral <math display=\"inline\"><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo> <mfrac><mrow><mi>q<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> immer von der Form <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><mi>p<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b1<\/mi><mi class=\"qopname\">arctan<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b2<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-266010r5\" \/><mstyle class=\"maketag\"><mtext>(9.5)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr ein Polynom <math display=\"inline\"><mi>p<\/mi><\/math> von Grad kleiner <math display=\"inline\"><mn>2<\/mn><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><\/math> und Konstanten <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><\/math><\/span><span class=\"period\">.<\/span> Durch Ableiten, auf den gemeinsamen Nenner bringen und Vergleich der Koeffizienten l\u00e4sst sich somit die Stammfunktion ermitteln. <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zc93c121522fd\"> <a id=\"x1-266011r20\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.20.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten in dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung den oben erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">rten Algorithmus genauer erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren und beginnen<\/span> <span class=\"ecti-1095\">mit einem konkreten Beispiel.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Berechnen Sie das Integral <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Sei nun <\/span><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>q<\/mi><\/math> <span class=\"ecti-1095\">ein Polynom von<\/span> <span class=\"ecti-1095\">Grad kleiner als <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>2<\/mn><mi>k<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass die Ableitung von <\/span><math display=\"inline\"> <mfrac><mrow><mi>p<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein beliebiges Polynom <\/span><math display=\"inline\"><mi>p<\/mi><\/math> <span class=\"ecti-1095\">von Grad kleiner als <\/span><math display=\"inline\"><mn>2<\/mn><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">durch<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfrac><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>p<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mi>x<\/mi><mi>p<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">gegeben ist.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Berechnen Sie die Matrixdarstellung <\/span><math display=\"inline\"><mi>M<\/mi><\/math> <span class=\"ecti-1095\">der Abbildung<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03d5<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>p<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>p<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mi>x<\/mi><mi>p<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich der Basis der Monome.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Schliessen Sie auf die Darstellung in<\/span> (<a href=\"..\/..\/chapter\/integrationsmethoden#x1-266010r5\">9.5<\/a>)<span class=\"ecti-1095\">, indem Sie zeigen, dass das Bild von<\/span> <math display=\"inline\"><mi>\u03d5<\/mi><\/math> <span class=\"ecti-1095\">zusammen<\/span> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mn>2<\/mn><mi>x<\/mi><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">den Vektorraum der Polynome von Grad kleiner gleich<\/span> <math display=\"inline\"><mn>2<\/mn><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">aufspannt.<\/span><\/dd><\/dl> <\/div> <a id=\"x1-266016r266\"><\/a> <h4 id=\"ze875100a54a7\" class=\"subsectionHead\"><span class=\"titlemark\">9.2.4 <\/span> <a id=\"x1-2670004\"><\/a>Trigonometrische Substitution<\/h4> <p class=\"noindent\">In allen bisherigen Beispielen der Substitutionsregel in Abschnitt <a href=\"..\/..\/chapter\/integrationsmethoden#x1-2650002\">9.2.2<\/a> hatten wir das Gl\u00fcck, dass das vorhandene Integral (vielleicht nach etwas Arbeit) bereits die richtige Struktur besass. Man verwendet die Substitutionsregel aber oft auch bevor man weiss welches Integral sich eigentlich nach der Substitution ergibt, wobei es gewisse Funktionentypen gibt bei denen eine gewisse Substitution erfahrungsgem\u00e4ss erfolgreich sein k\u00f6nnte. Wir wenden uns nun einem konkreten Beispiel dessen zu. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"ze8b184fee3b8\"> <a id=\"x1-267001r21\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.21 <\/span>(Kreisfl\u00e4che)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">das unbestimmte Integral <\/span><math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">berechnen. Auf Grund der trigonometrischen Identit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ten<\/span> <math display=\"inline\"><msqrt><mrow><msup><mrow> <mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <mi>r<\/mi><mi class=\"qopname\">cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\ud835\udf03<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">bietet<\/span> <span class=\"ecti-1095\">es sich nun an, die Funktion<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>\ud835\udf03<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo><mfrac><mrow> <mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>r<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>\ud835\udf03<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>r<\/mi><mi class=\"qopname\">sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\ud835\udf03<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Substitution zu verwenden. Denn mit dieser Substitution haben wir die Hoffnung, die Wurzel<\/span> <span class=\"ecti-1095\">in einen anderen Ausdruck zu verwandeln.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Allerdings ist dies umgekehrt zu der Substitution in Abschnitt<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/integrationsmethoden#x1-2650002\"><span class=\"ecti-1095\">9.2.2<\/span><\/a><span class=\"ecti-1095\">, da wir hier die<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">neue<\/span> <span class=\"ecti-1095\">Variable<\/span><span class=\"ecti-1095\">\u201c<\/span> <span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\ud835\udf03<\/mi><\/math> <span class=\"ecti-1095\">verwenden<\/span> <span class=\"ecti-1095\">um die<\/span> <span class=\"ecti-1095\">\u201e<\/span> <span class=\"ecti-1095\">alte Variable<\/span><span class=\"ecti-1095\">\u201c<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>r<\/mi><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">auszudr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">cken. (Anstatt wie in Abschnitt<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/integrationsmethoden#x1-2650002\"><span class=\"ecti-1095\">9.2.2<\/span><\/a> <span class=\"ecti-1095\">wo wir die neue<\/span> <span class=\"ecti-1095\">Variable<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">als Funktion der<\/span> <span class=\"ecti-1095\">alten Variable<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>x<\/mi><\/math> <span class=\"ecti-1095\">gesehen<\/span> <span class=\"ecti-1095\">haben). Da<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">bijektiv ist, ist<\/span> <span class=\"ecti-1095\">dies kein Problem: denn<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>r<\/mi><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>r<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ist zu<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\ud835\udf03<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> arcsin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>r<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent. Weiters<\/span> <span class=\"ecti-1095\">ist die Ableitung von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">gleich<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>r<\/mi><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo>  <mi>\ud835\udf03<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">damit auf ganz<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ungleich<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Gemeinsam mit dem Satz <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber die Ableitung der inversen Funktion (Satz <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-228020r14\"><span class=\"ecti-1095\">8.14<\/span><\/a><span class=\"ecti-1095\">) erhalten wir<\/span> <span class=\"ecti-1095\">daher<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mi>r<\/mi><mi class=\"qopname\">cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\ud835\udf03<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo> \u222b  <\/mo><mfrac><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\ud835\udf03<\/mi> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>2<\/mn><mi>\ud835\udf03<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mi class=\"qopname\"> arcsin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>r<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi>x<\/mi><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei wir eben <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>r<\/mi><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><msqrt><mrow><msup><mrow> <mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <mi>r<\/mi><mi class=\"qopname\">cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\ud835\udf03<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>r<\/mi><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">die trigonometrischen Identit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ten<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msup><mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><msup><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>\ud835\udf03<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">verwendet haben.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Veranschaulichen Sie sich die Substitution und die wichtigsten der obigen Identit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ten<\/span> <span class=\"ecti-1095\">in einem rechtwinkeligen Dreieck. Geben Sie weiters eine geometrische Interpretation<\/span> <span class=\"ecti-1095\">der beiden Terme des unbestimmten Integrals bei der Berechnung des bestimmten<\/span> <span class=\"ecti-1095\">Integrals<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>r<\/mi><\/math> <span class=\"ecti-1095\">an.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Dies zeigt, dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>G<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>r<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mi class=\"qopname\"> arcsin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>r<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi>x<\/mi><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">eine Stammfunktion von <\/span><math display=\"inline\"><msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>r<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/math> <span class=\"ecti-1095\">ist (was wie immer viel einfacher zu <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berpr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">fen ist). Da aber sogar die Funktion<\/span> <math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>r<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/math> <span class=\"ecti-1095\">stetig ist, besitzt<\/span> <math display=\"inline\"><mi>g<\/mi><\/math> <span class=\"ecti-1095\">nach Korollar<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/der-fundamentalsatz-der-integral--und-differentialrechnung#x1-259005r3\"><span class=\"ecti-1095\">9.3<\/span><\/a> <span class=\"ecti-1095\">auch auf ganz<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>r<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Stammfunktion<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>G<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">welche auf<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>r<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">mit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mi>G<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><\/math> <span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">bereinstimmt. Da aber<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>G<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>r<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mi class=\"qopname\"> arcsin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>r<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi>x<\/mi><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">auch eine auf ganz<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>r<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">stetige Funktion<\/span> <span class=\"ecti-1095\">definiert, folgt aus Stetigkeit von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>G<\/mi><\/math> <span class=\"ecti-1095\">und<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mi>G<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/math><span class=\"ecti-1095\">, dass<\/span> <math display=\"inline\"><msub><mrow><mi>G<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>G<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><\/math> <span class=\"ecti-1095\">und damit<\/span> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><mi>G<\/mi><\/math> <span class=\"ecti-1095\">auf ganz<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>r<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">eine Stammfunktion<\/span> <span class=\"ecti-1095\">von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>g<\/mi><\/math><span class=\"ecti-1095\">. Wir bemerken<\/span> <span class=\"ecti-1095\">allerdings, dass<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi class=\"qopname\">arcsin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>r<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">keine<\/span> <span class=\"ecti-1095\">Ableitung in den Punkten<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>r<\/mi><\/math> <span class=\"ecti-1095\">und<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>r<\/mi><\/math> <span class=\"ecti-1095\">besitzt. (Wieso ist dies kein Widerspruch zu obiger Diskussion?)<\/span> <\/p> <\/div> <p class=\"indent\">Substitutionen wie obige nennen sich vielfach <span class=\"ecbx-1095\">trigonometrische Substitutionen<\/span>. Wir werden bei diesen Berechnungen nicht immer so sorgf\u00e4ltig argumentieren und vielmehr der Leibniz Notation vertrauen, doch muss immer Invertierbarkeit der Funktion gegeben sein wenn wir die alte Variable durch die neue Variable ausdr\u00fccken. F\u00fcr die folgende Auflistung der trigonometrischen Substitutionen sei <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">In Ausdr\u00fccken der Form <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mfrac><mrow><mi>n<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup><\/math> f\u00fcr <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> f\u00fchrt wie bereits im obigen Beispiel oft die Substitution <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> mit&nbsp;<math display=\"inline\"><mi>\ud835\udf03<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-close\">)<\/mo><\/math> zum Ziel, wobei sich damit <math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><\/math> und <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi class=\"qopname\">cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\ud835\udf03<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> ergibt. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">In Ausdr\u00fccken der Form <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mfrac><mrow><mi>n<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup><\/math> f\u00fcr <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> f\u00fchrt oft die Substitution <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi class=\"qopname\">tan<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> mit&nbsp;<math display=\"inline\"><mi>\ud835\udf03<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-close\">)<\/mo><\/math> zum Ziel, wobei sich damit <math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>a<\/mi><\/mrow> <mrow><msup><mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><\/math> und <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>a<\/mi><\/mrow> <mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/math> ergibt. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Obwohl dies keine trigonometrische Substitution darstellt, bemerken wir noch Folgendes. Falls ein \u201eeinzelnes\u201c <math display=\"inline\"><mi>x<\/mi><\/math> vor dem Ausdruck <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mfrac><mrow><mi>n<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup><\/math> oder dem Ausdruck <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mfrac><mrow><mi>n<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup><\/math> steht, ist die Substitution <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> respektive <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> teilweise viel einfacher.<\/div><\/div> <p class=\"indent\">Als Merkhilfe kann es helfen f\u00fcr die beiden trigonometrischen Substitution ein rechtwinkeliges Dreieck zu skizzieren und abh\u00e4ngig von der Substitution die Seiten mit Hilfe von Pythagoras entsprechend zu beschriften. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z6cb7e02d7aef\"> <a id=\"x1-267002r22\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.22 <\/span>(Trigonometrische Substitution)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Es gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mfrac><mrow><mn>3<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mfrac><mrow><msup><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mi>a<\/mi> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo> \u222b  <\/mo><mi class=\"qopname\">cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\ud835\udf03<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>d<\/mi><mi>\ud835\udf03<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\ud835\udf03<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><msqrt><mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei wir <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi class=\"qopname\">tan<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><msqrt><mrow><msup><mrow> <mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><\/math> <span class=\"ecti-1095\">verwendet haben. (Veranschaulichen Sie sich die Substitution und obige Identit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ten in einem<\/span> <span class=\"ecti-1095\">Bild.)<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Es ist<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo><mi>x<\/mi><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo> \u222b  <\/mo><msup><mrow><mi>u<\/mi><\/mrow><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/mfrac><mfrac><mrow> <mn>2<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac><msup><mrow><mi>u<\/mi><\/mrow><mrow><mfrac><mrow><mn>3<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/mfrac><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mfrac><mrow><mn>3<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/p><\/dd><\/dl> <\/div> <a id=\"x1-267005r267\"><\/a> <h4 id=\"z8ec10f0caea9\" class=\"subsectionHead\"><span class=\"titlemark\">9.2.5 <\/span> <a id=\"x1-2680005\"><\/a>Weitere Integrationsmethoden<\/h4> <p class=\"noindent\">Es gibt viele weitere Methoden zur Integration; viele davon beruhen auf spezielle Substitutionen. <\/p><p class=\"indent\">Beispielsweise lassen sich gewisse unbestimmte Integrale mit hyperbolischen Substitutionen berechnen. Sei <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> In Ausdr\u00fccken der Form <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mfrac><mrow><mi>n<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup><\/math> f\u00fcr <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> f\u00fchrt oft die Substitution <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi class=\"qopname\">cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> zum Ziel, wobei sich damit <math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>u<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><\/math> und <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> ergibt. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zfc3ca3a17bc7\"> <a id=\"x1-268001r23\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.23.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir berechnen<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo><msqrt><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><msup><mrow><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mo>\u222b  <\/mo><msup><mrow><mi class=\"qopname\">cosh<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mo>\u222b  <\/mo><msup><mrow><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>u<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mo>\u222b  <\/mo><msup><mrow><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Nach Aufl<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sen ergibt sich somit<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo><msqrt><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><msup><mrow><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>u<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi class=\"qopname\">cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>u<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>x<\/mi><msqrt><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/msqrt> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> arcosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <p class=\"indent\">Eine andere Methode, die wir hier kurz erw\u00e4hnen m\u00f6chten, ist die sogenannte Halbwinkelmethode (oder auch Weierstrass-Substitution). Diese ist dann n\u00fctzlich, wenn man das Integral einer rationalen Funktion in <math display=\"inline\"><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> wie zum Beispiel <math display=\"inline\"><mfrac><mrow><msup><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-bin\">+<\/mo><mn>2<\/mn><mn>0<\/mn><mn>1<\/mn><mn>7<\/mn><\/mrow><\/mfrac><\/math> in die Integration einer rationalen Funktion in <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> tan<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> umwandeln m\u00f6chte (siehe auch Beispiel <a href=\"..\/..\/chapter\/integrationsmethoden#x1-265002r17\">9.17<\/a> (b)). <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z3dae513df785\"> <a id=\"x1-268002r24\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.24 <\/span>(Halbwinkelmethode)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten das unbestimmte Integral <\/span><math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo> <mfrac><mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>2<\/mn><mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">mit der Substitution <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> tan<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">berechnen. Zeigen Sie daf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r zuerst die Identit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ten<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>2<\/mn><mi>u<\/mi><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi class=\"qopname\">cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Zeigen Sie anschliessend, dass das obige Integral nach Substitution zu einem Integral einer rationalen<\/span> <span class=\"ecti-1095\">Funktion in <\/span><math display=\"inline\"><mi>u<\/mi><\/math> <span class=\"ecti-1095\">wird und berechnen Sie es.<\/span> <\/p> <\/div> <p class=\"indent\">Manchmal f\u00fchrt man auch die eine oder die andere Substitution durch, weil in der zu integrierenden Funktion eine verschachtelte Funktion vorliegt und man einfach keine andere Methode zur Verf\u00fcgung hat. Zum Beispiel bei dem Integral&nbsp;<math display=\"inline\"><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msqrt><mrow><mi>x<\/mi><\/mrow><\/msqrt><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> steht keine der erw\u00e4hnten Methoden zur Verf\u00fcgung, doch ist man versucht&nbsp;<math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><mi>x<\/mi><\/mrow><\/msqrt><\/math> zu setzen um zu sehen was sich daraus ergibt. Dies f\u00fchrt in der Tat zum Erfolg (wieso?). Ebenso in dem Integral der Form&nbsp;<math display=\"inline\"><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> f\u00fchrt der Ansatz&nbsp;<math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> zu einem unbestimmten Integral einer rationalen Funktion (wieso?). <a id=\"x1-268003r268\"><\/a> <\/p> <h4 id=\"zdfc099e7a827\" class=\"subsectionHead\"><span class=\"titlemark\">9.2.6 <\/span> <a id=\"x1-2690006\"><\/a>Das bestimmte Integral<\/h4> <p class=\"noindent\">Alle obigen Regeln zur Berechnung des unbestimmten Integrals lassen sich nach dem Fundamentalsatz der Integral- und Differentialrechnung eins zu eins auch f\u00fcr das Riemann-Integral, welches im Gegensatz zum unbestimmten Integral auch das bestimmte Integral genannt wird, anwenden. Dabei haben wir zwei M\u00f6glichkeiten. <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Eine erste M\u00f6glichkeit ist mit obigen Methoden zuerst das unbestimmte Integral zu berechnen und dann Korollar <a href=\"..\/..\/chapter\/der-fundamentalsatz-der-integral--und-differentialrechnung#x1-259007r4\">9.4<\/a> zur Berechnung des Riemann-Integrals zu verwenden. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Falls wir  aber  nur  an  einem  einzigen  Riemann-Integral  interessiert  sind,  ist  es  oft einfacher, die Ausdr\u00fccke ausserhalb des Integrals so fr\u00fch wie m\u00f6glich zu berechnen. Wir erkl\u00e4ren dies im Folgenden f\u00fcr die partielle Integration und die Substitution.<\/div><\/div> <p class=\"indent\">Sind <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>v<\/mi><\/math> zwei stetig differenzierbare Funktionen auf einem kompakten Intervall                                                                                                                                                                           <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> mit Endpunkten <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>u<\/mi><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>u<\/mi><mi>v<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow> <mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>v<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Denn falls <math display=\"inline\"><mi>F<\/mi><\/math> eine Stammfunktion von <math display=\"inline\"><mi>u<\/mi><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/math> und <math display=\"inline\"><mi>G<\/mi><\/math> eine Stammfunktion von <math display=\"inline\"><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>v<\/mi><\/math> ist, dann gilt f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>C<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mi>u<\/mi><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mo>\u222b  <\/mo><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>v<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>C<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>G<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>C<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr gewisse Integrationskonstanten&nbsp;<span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>C<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>C<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>C<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Somit ist nach Korollar <a href=\"..\/..\/chapter\/der-fundamentalsatz-der-integral--und-differentialrechnung#x1-259007r4\">9.4<\/a> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>u<\/mi><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow> <mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>G<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>G<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>G<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow> <mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow> <mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>v<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Ebenso k\u00f6nnen wir bei einer Substitution in Abschnitt <a href=\"..\/..\/chapter\/integrationsmethoden#x1-2650002\">9.2.2<\/a> die Grenzen f\u00fcr ein Riemann-Integral enstsprechend der Substitution neu berechnen. Sei <math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>x<\/mi> <\/mrow> <\/msub> <\/math> ein Intervall mit Endpunkten <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> sei <math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>u<\/mi> <\/mrow> <\/msub> <\/math> ein weiteres Intervall, sei <math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>u<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> stetig und <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>x<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>u<\/mi> <\/mrow> <\/msub> <\/math> stetig differenzierbar. F\u00fcr ein kompaktes Intervall <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> mit Endpunkten <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> in&nbsp;<math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>x<\/mi> <\/mrow> <\/msub> <\/math> gilt dann <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msubsup><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>u<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">In der Tat, wenn <math display=\"inline\"><mi>G<\/mi><\/math> eine Stammfunktion von <math display=\"inline\"><mi>g<\/mi><\/math> auf&nbsp;<math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>u<\/mi> <\/mrow> <\/msub> <\/math> ist, dann ist nach der Kettenregel <math display=\"inline\"><mi>G<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><\/math> eine Stammfunktion von <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Nach Korollar <a href=\"..\/..\/chapter\/der-fundamentalsatz-der-integral--und-differentialrechnung#x1-259007r4\">9.4<\/a> gilt also                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>G<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow> <mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-rel\">=<\/mo> <mi>G<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>G<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>G<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msubsup> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msubsup><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>u<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Die Annahme der Stetigkeit an <math display=\"inline\"><mi>g<\/mi><\/math> kann abgeschw\u00e4cht werden \u2013 siehe die entsprechende \u00dcbung im Abschnitt <a href=\"..\/..\/chapter\/weitere-lernmaterialien#x1-2930002\">9.8.2<\/a>. <\/p><p class=\"indent\">Wir bemerken an dieser Stelle, dass die in den obigen Abschnitten behandelten Themen oft alles sind, was man f\u00fcr Anwendungen (wie zum Beispiel f\u00fcr die Fl\u00e4chenberechnung unter Graphen) braucht. Nichtsdestotrotz werden wir erst gegen Ende des Kapitels in Abschnitt&nbsp;<a href=\"..\/..\/chapter\/anwendungen#x1-2850007\">9.7<\/a> darauf eingehen. Gewisse Anwendungen wurden schon in Abschnitt&nbsp;<a href=\"..\/..\/chapter\/anwendungen#x1-1150004\">4.4<\/a> diskutiert. <a id=\"x1-269001r269\"><\/a> <\/p> <h4 id=\"z412446ded346\" class=\"subsectionHead\"><span class=\"titlemark\">9.2.7 <\/span> <a id=\"x1-2700007\"><\/a>Leibniz-Notation<\/h4> <p class=\"noindent\">Wir werden die Leibniz-Notation in der Berechnung von unbestimmten und bestimmten Integralen wie bereits oben im Folgenden immer wieder verwenden. Diese Notation verpackt in einem nat\u00fcrlichen Formalismus die partielle Integration <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo> \u222b  <\/mo><mi>u<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>v<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>u<\/mi><mi>v<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mo>\u222b  <\/mo><mi>v<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und die Substitutionsregeln                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo> \u222b  <\/mo><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>u<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>u<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>u<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wobei wir in der zweiten Formulierung der Substitution vorraussetzen, dass <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>x<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>u<\/mi> <\/mrow> <\/msub> <\/math> bijektiv mit nicht verschwindender Ableitung ist und dadurch im linken Integral mit <math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo> <mi>x<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><\/mrow><\/mfrac> <mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><\/mfrac><\/math> multiplizieren konnten und die erste Formulierung der Substitutionsregel anwenden konnten. Wie wir gesehen haben, sind diese Regeln Umformulierungen der Produktregel f\u00fcr die Ableitung und der Kettenregel f\u00fcr die Ableitung (gemeinsam mit der Ableitungsregel f\u00fcr die inverse Abbildung). <\/p><p class=\"indent\">Bei konkreten Integralberechnungen verwenden wir mitunter auch Gleichungen, die <math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> und <math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><\/math> miteinander verbinden. Zum Beispiel bei der trigonometrischen Substitution&nbsp;<math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><\/math> (f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><mi>\ud835\udf03<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-close\">)<\/mo><\/math>) verwenden wir auch die Formel&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><\/math><\/span><span class=\"period\">,<\/span> die formal gesehen keine Bedeutung hat (und deswegen auf keinen Fall in dieser Form in Beweisen auftreten sollte), doch eben im Zuge der Substitution in der Formulierung der Leibniz-Notation einen bequemen Zwischenschritt darstellt. <\/p><p class=\"indent\">Informell taucht in Anwendungen das Symbol <math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> auch oft in Diskussionen auf, die zu einem Riemann-Integral f\u00fchren, wobei <math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> dann f\u00fcr ein (sehr) kleines <math display=\"inline\"><mi>\u0394<\/mi><mi>x<\/mi><\/math> stehen sollte. In Anwendungen werden h\u00e4ufig die Begriffe der Riemann-Summe oder der additiven Intervallfunktion vermieden, wobei es genau diese Begriffe sind, die diese Verwendung von <math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> genau und formal korrekt machen w\u00fcrden (siehe Abschnitte <a href=\"..\/..\/chapter\/anwendungen#x1-1150004\">4.4<\/a> und <a href=\"..\/..\/chapter\/riemann-summen#x1-1800005\">6.5<\/a>). Auf jeden Fall hat in diesem Zusammenhang eine Formel der Gestalt&nbsp;<math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><\/math> auch eine                                                                                                                                                                           Interpretation: Da <math display=\"inline\"><mi>\u0394<\/mi><mi>x<\/mi><\/math> die L\u00e4nge eines kleinen Teilintervalls von&nbsp;<math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> angibt und <math display=\"inline\"><mi>\u0394<\/mi><mi>\ud835\udf03<\/mi><\/math> die L\u00e4nge des entsprechenden Teilintervalls in <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-close\">)<\/mo><\/math> so gibt die Ableitung <math display=\"inline\"><mi>a<\/mi><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><\/math> (bis auf einen kleinen und wie sich herausstellt vernachl\u00e4ssigbaren Fehler) den Gr\u00f6ssenunterschied <math display=\"inline\"><mfrac><mrow><mi>\u0394<\/mi><mi>x<\/mi><\/mrow> <mrow><mi>\u0394<\/mi><mi>\ud835\udf03<\/mi><\/mrow><\/mfrac><\/math> an, der bei Betrachtung von etwaigen Riemann-Summen in der Variable <math display=\"inline\"><mi>x<\/mi><\/math> und der Variable <math display=\"inline\"><mi>\ud835\udf03<\/mi><\/math> als zus\u00e4tzlicher Faktor auftreten w\u00fcrde. Wir m\u00fcssen dies nicht genauer ausf\u00fchren oder die Substitution auf diese Art und Weise beweisen, da wir ja mittels der Kettenregel und dem Fundamentalsatz der Integral- und Differentialrechnung bereits die Substitutionsregel f\u00fcr Riemann-Integrale bewiesen haben und obiger Formalismus diese nur auf eine andere Art pr\u00e4sentiert. Dieser Beweis \u00fcber den Fundamentalsatz verwendet allerdings etwas st\u00e4rkere Annahmen als notwendig (siehe folgende \u00dcbung f\u00fcr den direkten Beweis mit schw\u00e4cheren Annahmen). <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"za2a33f4d2777\"> <a id=\"x1-270001r25\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.25 <\/span>(Substitution f\u00fcr Riemann-integrierbare Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">ein kompaktes<\/span> <span class=\"ecti-1095\">Intervall in<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <span class=\"ecti-1095\">Endpunkten <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">eine stetig differenzierbare<\/span> <span class=\"ecti-1095\">Funktion mit <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><span class=\"ecti-1095\">. Dann ist f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede<\/span> <span class=\"ecti-1095\">Riemann-integrierbare Funktion <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">auch <\/span><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">Riemann-integrierbar und<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msubsup><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <a id=\"x1-270002r270\"><\/a> <h4 id=\"z5023b13c551d\" class=\"subsectionHead\"><span class=\"titlemark\">9.2.8 <\/span> <a id=\"x1-2710008\"><\/a>Neue Funktionen<\/h4> <p class=\"noindent\">Manchmal f\u00fchren obige Methoden zur Bestimmung eines unbestimmten Integrals einer Funktion zu keinem Ergebnis. Dies kann daran liegen, dass die gesuchte Stammfunktion sich nicht mit den bisher bekannten Funktionen ausdr\u00fccken l\u00e4sst. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"za88f3969dea4\"> <a id=\"x1-271001r26\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.26 <\/span>(Integralsinus)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Der <\/span><span class=\"ecbi-1095\">Integralsinus <\/span><span class=\"ecti-1095\">ist die Stammfunktion <\/span><math display=\"inline\"><mi class=\"qopname\">Si<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">der stetigen Funktion<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><mfrac><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mtd><mtd class=\"array\" columnalign=\"center\"> <mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn> <\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"> <mn>1<\/mn> <\/mtd> <mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle> <mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/mtd> <\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">mit der Normalisierung <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\">Si<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Er l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst sich als Potenzreihe schreiben, denn nach Satz<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/integration-von-potenzreihen#x1-216001r85\"><span class=\"ecti-1095\">7.85<\/span><\/a> <span class=\"ecti-1095\">gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">Si<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><mfrac><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>t<\/mi><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zfefaa105cdff\"> <a id=\"x1-271002r27\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.27 <\/span>(Integralkosinus)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Der Integralkosinus <\/span><math display=\"inline\"><mi class=\"qopname\">Ci<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist definiert als die Stammfunktion von <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">mit der Normalisierung <\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi class=\"qopname\">Ci<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Dabei m\u00f6chten wir auf folgende \u00dcbung verweisen, die zeigt, dass der Integralkosinus so wohldefiniert ist. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z4772725f64ee\"> <a id=\"x1-271003r28\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.28.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>F<\/mi><\/math> <span class=\"ecti-1095\">eine Stammfunktion von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass der Grenzwert <\/span><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>F<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">existiert. Dr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">cken Sie <\/span><math display=\"inline\"><mi class=\"qopname\">Ci<\/mi><mo>  <\/mo><\/math> <span class=\"ecti-1095\">als Summe einer Konstanten (der sogenannten Euler-Mascheroni Konstanten), der Logarithmusfunktion<\/span> <span class=\"ecti-1095\">und einer Potenzreihe aus.<\/span> <\/p> <\/div> <p class=\"indent\">Unter Verwendung uneigentlicher Integrale werden wir sp\u00e4ter weitere wichtige Funktionen kennenlernen, die sich nicht in Termen bekannter Funktionen ausdr\u00fccken lassen \u2013 siehe zum Beispiel <a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-273006r34\">9.34<\/a>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               <a id=\"x1-271004r263\"><\/a> <\/p> \n","rendered":"\n<style scoped=\"scoped\">.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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doesn't work in WP *\/\ndiv.me details {\n\tmargin: 10px 0px 0px 0px;\n}\ndiv.me dd {\n    width: calc(100% - 30px);\n}\t\n\n\n\/* fixing background of pictures *\/\nimg {\n\tbackground: white;\n}\n\n\/* div-container for centered geoapplet *\/\ndiv.geoapplet {\n\tmargin-left: auto;\n\tmargin-right: auto;\n\tmargin-top: 15px;\n\tmax-width: 100%;\n}\ndiv.geoapplet iframe {\n\tborder-style: none;\n\tmax-height: 110vw;\n}\n\n\/* div-container for centered squeezed tables *\/\ndiv.websqueeze {\n\tmargin-left: auto;\n\tmargin-right: auto;\n}\n\n\/* two containers for squeezing text sizes *\/\ndiv.mesmalltext, div.mesmalltext * {\n\tfont-size: 15px;\n}\nspan.metinytext, span.metinytext * {\n\tfont-size: 12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"zeca2210f480f\" class=\"sectionHead\"><span class=\"titlemark\">9.2 <\/span> <a id=\"x1-2630002\"><\/a>Integrationsmethoden<\/h3> <p class=\"noindent\">Wir erinnern daran, dass das unbestimmte Integral einer Funktion <math display=\"inline\"><mi>f<\/mi><\/math> in der Variablen <math display=\"inline\"><mi>x<\/mi><\/math> der Ausdruck <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo> \u222b  <\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">ist, wobei <math display=\"inline\"><mi>F<\/mi><\/math> eine Stammfunktion von <math display=\"inline\"><mi>f<\/mi><\/math> ist. In den Abschnitten <a href=\"..\/..\/chapter\/erste-differentialgleichungen#x1-2520002\">8.5.2<\/a> und <a href=\"..\/..\/chapter\/hyperbolische-funktionen#x1-2460004\">8.4<\/a> haben wir bereits einige Regeln zur Berechnung konkreter unbestimmter Integrale kennengelernt: F\u00fcr <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> (oder sogar&nbsp;<math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math>) ist <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo> \u222b  <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>s<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>s<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"> <mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>s<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo> \u222b  <\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo><mi class=\"qopname\">cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> arcsin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> arctan<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> arsinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/msqrt><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> arcosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Des Weiteren gilt f\u00fcr Funktionen <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> in der Variable <math display=\"inline\"><mi>x<\/mi><\/math> und Zahlen <math display=\"inline\"><msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo> \u222b  <\/mo><msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo> \u222b  <\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo> \u222b  <\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Denn falls <math display=\"inline\"><msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> eine Stammfunktion von <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> ist und <math display=\"inline\"><msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> eine Stammfunktion von <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> ist, so muss die Funktion <math display=\"inline\"><msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>F<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> auf Grund der Linearit\u00e4t der Ableitung (Proposition <a href=\"..\/..\/chapter\/die-ableitung#x1-228010r5\">8.5<\/a>) eine Stammfunktion von <math display=\"inline\"><msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> sein. <\/p><p class=\"indent\">Auf \u00e4hnliche Weise lassen sich die anderen Regeln der Differentiation als Identit\u00e4ten f\u00fcr unbestimmte Integrale auffassen, wie wir nun ausf\u00fchren wollen. <a id=\"x1-263001r262\"><\/a> <\/p> <h4 id=\"zd2c0683dd854\" class=\"subsectionHead\"><span class=\"titlemark\">9.2.1 <\/span> <a id=\"x1-2640001\"><\/a>Partielle Integration<\/h4> <p class=\"noindent\">Die Produktregel in Proposition <a href=\"..\/..\/chapter\/die-ableitung#x1-228010r5\">8.5<\/a> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mi>v<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>v<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>u<\/mi><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr zwei differenzierbare Funktionen <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>v<\/mi><\/math> f\u00fchrt ebenso zu einer Integrationsregel, n\u00e4mlich der <span class=\"ecbx-1095\">partiellen Integration<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>u<\/mi><mi>v<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>v<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>u<\/mi><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo><mi>u<\/mi><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>u<\/mi><mi>v<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mo>\u222b  <\/mo><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>v<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-264001r3\" \/><mstyle class=\"maketag\"><mtext>(9.3)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">In der Leibniz-Notation ist <math display=\"inline\"><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>v<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><\/mfrac><\/math> und <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo> <mi>u<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> Deswegen schreibt man die partielle Integration oft auch als <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo> \u222b  <\/mo><mi>u<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>v<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>u<\/mi><mi>v<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mo>\u222b  <\/mo><mi>v<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was formal bloss als Kurzform der Formel in (<a href=\"..\/..\/chapter\/integrationsmethoden#x1-264001r3\">9.3<\/a>) verstanden werden sollte. Die Regel der partiellen Integration ist bereits ein Beispiel, wo eine einfache Regel des Differenzierens eine komplexere Regel des Integrierens als Entsprechung hat. Die Produktregel erlaubt uns, die Ableitung jedes Produkts mittels der Ableitung dessen Faktoren auszudr\u00fccken. Die partielle Integration hingegen erlaubt uns, das unbestimmte Integral eines Produkts mittels dem Integral eines Faktors und eines weiteren Integrals auszudr\u00fccken. Mit etwas Gl\u00fcck (und Geschick) ist das zweite Integral einfacher und kann anschliessend berechnet werden. Wir demonstrieren dies anhand zweier Beispiele. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zeb7203a01703\"> <a id=\"x1-264002r15\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.15 <\/span>(Beispiele partieller Integration)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Wir berechnen das unbestimmte Integral <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><mi>x<\/mi><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Daf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r setzen wir <\/span><math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Eine<\/span> <span class=\"ecti-1095\">Stammfunktion von <\/span><math display=\"inline\"><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">ist <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Damit erhalten wir<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo><mi>x<\/mi><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mo>\u222b  <\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u22c5<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Wir bemerken, dass es gen<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">gt, in solchen Berechnungen immer bloss eine unbekannte Integrationskonstante<\/span> <math display=\"inline\"><mi>C<\/mi><\/math> <span class=\"ecti-1095\">zu<\/span> <span class=\"ecti-1095\">verwenden, da mehrere solche einfach zusammengefasst werden k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen. (Kontrollieren Sie diese<\/span> <span class=\"ecti-1095\">Rechnung durch Ableiten.) Dieselbe Berechnungsmethode f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hrt auch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r unbestimmte Integrale der<\/span> <span class=\"ecti-1095\">Form <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">zum Erfolg.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Wir wollen das unbestimmte Integral <\/span><math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">berechnen. Es mag zuerst etwas <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berraschend sein, dass wir dazu partielle Integration verwenden wollen.<\/span> <span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math><span class=\"ecti-1095\">. Dann<\/span> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Stammfunktion von <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">womit<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>1<\/mn><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mo>\u222b  <\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mo>\u222b  <\/mo><mn>1<\/mn><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Dies kann man wiederum durch Ableiten verifizieren (was nicht notwendig ist, aber einen sehr<\/span> <span class=\"ecti-1095\">einfachen Test darstellt).<\/span><\/p><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zaf2f70ba0f47\"> <a id=\"x1-264005r16\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.16.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Berechnen Sie <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Geben Sie eine rekursive Formel zur Berechnung von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">an.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Berechnen Sie<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Beachten Sie hierbei, dass der Fall<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math> <span class=\"ecti-1095\">getrennt zu behandeln ist.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Berechnen Sie das unbestimmte Integral <\/span><math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><msup><mrow><mi>e<\/mi><\/mrow><mrow><mi>a<\/mi><mi>x<\/mi><\/mrow><\/msup><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> <div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Bei (iv) ergibt sich nach zweifacher partieller Integration ein Gleichungssystem.<\/span><\/p><\/details><\/dd><\/dl> <\/div> <a id=\"x1-264010r264\"><\/a> <h4 id=\"z707389867a81\" class=\"subsectionHead\"><span class=\"titlemark\">9.2.2 <\/span> <a id=\"x1-2650002\"><\/a>Substitution<\/h4> <p class=\"noindent\">Falls eine Funktion <math display=\"inline\"><mi>g<\/mi><\/math> auf einem Intervall <math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>u<\/mi> <\/mrow> <\/msub> <\/math> das unbestimmte Integral <math display=\"inline\"><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>G<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><\/math> besitzt und <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>x<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>u<\/mi> <\/mrow> <\/msub> <\/math> eine stetig differenzierbare Abbildung auf dem Intervall <math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msub><\/math> ist, dann gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo> \u222b  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>G<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">auf <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>x<\/mi> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Dies folgt unmittelbar aus der Kettenregel in Satz <a href=\"..\/..\/chapter\/die-ableitung#x1-228014r8\">8.8<\/a> und wird oft auch geschrieben als <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo> \u222b  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-265001r4\" \/><mstyle class=\"maketag\"><mtext>(9.4)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr die \u201e neue Variable\u201c <span class=\"maperiod\"><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Alternativ werden wir die obige <span class=\"ecbx-1095\">Substitutionsregel <\/span>gemeinsam mit der Leibniz-Notation auch in folgender informellen Schreibweise                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo> \u222b  <\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>u<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>G<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>u<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>G<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">verwenden, wobei <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo> <mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z9d0eed366239\"> <a id=\"x1-265002r17\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.17.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Es gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><munder class=\"msub\"><mrow><munder accentunder=\"false\"><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/mrow><mo>\ufe38<\/mo><\/munder> <\/mrow><mrow><mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>u<\/mi><\/mrow><\/mfrac> <\/mrow><\/munder><munder class=\"msub\"><mrow><munder accentunder=\"false\"><mrow> <mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><mo>\ufe38<\/mo><\/munder><\/mrow><mrow><mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><\/mrow><\/munder> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo>\u222b  <\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>u<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>u<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">gesetzt<\/span> <span class=\"ecti-1095\">wurde, womit <\/span><span class=\"maperiod\"><math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Es gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><mi class=\"qopname\">sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi class=\"qopname\">tan<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>v<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>v<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>v<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi class=\"qopname\">tan<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><span class=\"ecti-1095\">, und<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>v<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> tan<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>v<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/p><\/dd><\/dl> <\/div> <p class=\"indent\">Wie bereits erw\u00e4hnt, ben\u00f6tigt die Integration mehr \u00dcbung und Vorraussicht als die Differentiation. Obige Substitutionen ben\u00f6tigen zum Beispiel den Blick ob gewisse Faktoren vielleicht die gew\u00fcnschte Ableitung&nbsp;<math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/math> einer inneren Funktionen&nbsp;<math display=\"inline\"><mi>f<\/mi><\/math> darstellen k\u00f6nnte. Manchmal ist dies naheliegend wie in Beispiel <a href=\"..\/..\/chapter\/integrationsmethoden#x1-265002r17\">9.17<\/a>(i), doch manchmal erfordert dies Erfahrung und eine l\u00e4ngere Suche wie in Beispiel&nbsp;<a href=\"..\/..\/chapter\/integrationsmethoden#x1-265002r17\">9.17<\/a>(ii). <a id=\"x1-265005r265\"><\/a> <\/p> <h4 id=\"zfc69cbc62eaa\" class=\"subsectionHead\"><span class=\"titlemark\">9.2.3 <\/span> <a id=\"x1-2660003\"><\/a>Integration rationaler Funktionen<\/h4> <p class=\"noindent\">Wir erinnern daran, dass eine rationale Funktion eine Funktion der Form <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mfrac> <mrow> <mi>p<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>q<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/math> f\u00fcr Polynome <math display=\"inline\"><mi>p<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo> <mi>q<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> und <math display=\"inline\"><mi>q<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> ist, wobei der Definitionsbereich <math display=\"inline\"><mi>\u211d<\/mi><\/math> ohne die Nullstellen von <math display=\"inline\"><mi>q<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> ist. Wir wollen hier ein Verfahren zur Berechnung des unbestimmten Integrals einer rationalen Funktion besprechen. Nach Division mit Rest f\u00fcr Polynome (siehe \u00dcbung <a href=\"..\/..\/chapter\/polynome#x1-82002r17\">3.17<\/a>) k\u00f6nnen wir als erstes ein Polynom abspalten, so dass die verbleibende rationale Funktion von der Form <math display=\"inline\"><mfrac><mrow><msub><mrow><mi>p<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>q<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <\/math> f\u00fcr <math display=\"inline\"><mi class=\"qopname\">deg<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>p<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo><mi class=\"qopname\"> deg<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>q<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> ist.                                                                                                                                                                           <\/p><p class=\"indent\">Da Polynome mittels der Formel <math display=\"inline\"><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><\/math> integriert werden k\u00f6nnen, nehmen wir nun an, dass der Grad von <math display=\"inline\"><mi>p<\/mi><\/math> kleiner als der Grad von <math display=\"inline\"><mi>q<\/mi><\/math> ist. Wir betrachten zuerst einige Spezialf\u00e4lle. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z86cbe1077c0d\"> <a id=\"x1-266001r18\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.18 <\/span>(Integration von elementaren rationalen Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">beliebig und <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">eine nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche Zahl.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Es gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>u<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>u<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/math> <span class=\"ecti-1095\">gesetzt<\/span> <span class=\"ecti-1095\">wurde und <\/span><math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">ist.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei wieder <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/math> <span class=\"ecti-1095\">gesetzt wurde und <\/span><math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">ist.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ist, so gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo> \u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>a<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>a<\/mi><\/mrow><\/mfrac><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>a<\/mi><\/mrow><\/mfrac><mi class=\"qopname\">arctan<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>u<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>a<\/mi><\/mrow><\/mfrac><mi class=\"qopname\">arctan<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>a<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>a<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">gesetzt<\/span> <span class=\"ecti-1095\">wurde und <\/span><math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>a<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">ist.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Es gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo>\u222b  <\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>u<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>u<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo> <mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(v)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>u<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><msup><mrow><mi>u<\/mi><\/mrow><mrow><mn>1<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo> <mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/p><\/dd><\/dl> <\/div> <p class=\"indent\">Im Allgemeinen verwenden wir die sogenannte <span class=\"ecbx-1095\">Partialbruchzerlegung <\/span>f\u00fcr die rationale Funktion <math display=\"inline\"><mfrac><mrow><mi>p<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>q<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/math> (nach Division mit Rest so dass <math display=\"inline\"><mi class=\"qopname\"> deg<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>p<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo><mi class=\"qopname\"> deg<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>q<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math>), um die Integration auf obige Beispiele zur\u00fcckzuf\u00fchren. In der Tat l\u00e4sst sich <math display=\"inline\"><mfrac><mrow><mi>p<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>q<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/math> als Linearkombination von einfacheren rationalen Funktionen darstellen. Diese sind von der Form                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mspace class=\"quad\" width=\"1em\" \/><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">oder von der Form <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03bb<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u03bb<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mspace class=\"quad\" width=\"1em\" \/><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/> <mfrac><mrow><msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow> <mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03bb<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u03bb<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wobei <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> eine Nullstelle von <math display=\"inline\"><mi>q<\/mi><\/math> mit Vielfachheit <math display=\"inline\"><mi>k<\/mi><\/math> und <math display=\"inline\"><mi>\u03bb<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mi>\u211d<\/mi><\/math> eine Nullstelle von <math display=\"inline\"><mi>q<\/mi><\/math> mit Vielfachheit <math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> ist und <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z9159d80b648b\"> <a id=\"x1-266007r19\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.19 <\/span>(Integration rationaler Funktionen und die Partialbruchzerlegung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Wir wollen das unbestimmte Integral <\/span><math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo> <mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">bestimmen. Als erstes f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hren wir Division mit Rest<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><\/mtd> <mtd class=\"align-even\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-bin\">\u2212<\/mo><\/mtd> <mtd class=\"align-even\"><munder accentunder=\"false\" class=\"mml-underline\"><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"><mspace class=\"quad\" width=\"1em\" \/> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"><mspace class=\"quad\" width=\"1em\" \/><mspace class=\"thinspace\" width=\"0.17em\" \/><mspace class=\"quad\" width=\"1em\" \/><munder accentunder=\"false\" class=\"mml-underline\"><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"><mspace class=\"qquad\" width=\"2em\" \/><mspace class=\"nbsp\" width=\"0.33em\" \/><mspace class=\"qquad\" width=\"2em\" \/><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mspace class=\"quad\" width=\"1em\" \/><mi class=\"qopname\">Rest<\/mi><mo>  <\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">durch, womit<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><mo> \u222b  <\/mo> <mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Um die Partialbruchzerlegung von <\/span><math display=\"inline\"> <mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">zu erhalten, setzen wir<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>A<\/mi><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>B<\/mi><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>C<\/mi><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r noch unbekannte Zahlen <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>B<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>C<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">multiplizieren mit <\/span><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und erhalten<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><mi>x<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Nun setzen wir in diesem <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">um <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">zu erhalten<\/span> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math><span class=\"ecti-1095\">, um<\/span> <math display=\"inline\"><mi>C<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">zu erhalten.<\/span> <span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">ergibt<\/span> <span class=\"ecti-1095\">sich nun <\/span><math display=\"inline\"><mn>2<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>2<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>2<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">und somit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">(Alternativ kann man auch beide Seiten ausmultiplizieren, die Koeffizienten links<\/span> <span class=\"ecti-1095\">und rechts vergleichen, und auf diese Weise drei Gleichungen in den unbekannten<\/span> <span class=\"ecti-1095\">Variablen<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>B<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>C<\/mi><\/math> <span class=\"ecti-1095\">erhalten.) Daher ist<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mo>\u222b  <\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>D<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Wir berechnen das unbestimmte Integral<\/span> <math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">+<\/mo><mn>2<\/mn><mi>x<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><span class=\"ecti-1095\">. Man beachte dabei,<\/span> <span class=\"ecti-1095\">dass das Polynom <\/span><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">keine reellen Nullstellen hat. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Partialbruchzerlegung machen wir den Ansatz<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>A<\/mi><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>B<\/mi><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Nun multiplizieren wir mit <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und erhalten<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mn>1<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ergibt<\/span> <span class=\"ecti-1095\">sich <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Daher ist<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mn>1<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>C<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Es folgt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo> \u222b  <\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo> \u222b  <\/mo> <mfrac><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo> \u222b  <\/mo> <mfrac><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo> \u222b  <\/mo> <mfrac><mrow><mi>u<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>4<\/mn><\/mrow><\/mfrac><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\"> arctan<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>u<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>D<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>4<\/mn><\/mrow><\/mfrac><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msup><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\"> arctan<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>D<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei wir <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">gesetzt haben und Beispiele <\/span><a href=\"..\/..\/chapter\/integrationsmethoden#x1-266001r18\"><span class=\"ecti-1095\">9.18<\/span><\/a> <span class=\"ecti-1095\">(c) und (d) verwendet haben.<\/span><\/p><\/dd><\/dl> <\/div> <p class=\"indent\">In manchen F\u00e4llen kann obiges Verfahren auch auf das Integral <math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> f\u00fcr ein <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> und <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math> f\u00fchren, was wir mit der trigonometrischen Substitution <math display=\"inline\"><mi class=\"qopname\"> tan<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>a<\/mi><\/mrow><\/mfrac><\/math> (siehe unten) behandeln k\u00f6nnen. Eine andere, allgemeinere Herangehensweise m\u00f6chten wir in folgender Bemerkung f\u00fcr Interessierte behandeln. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z897e2ad8e28b\"> <span class=\"ecti-1095\">Bemerkung <\/span>(Integration rationaler Funktionen mit mehrfachen komplexen Nullstellen)<span class=\"ecti-1095\">.<\/span> <\/h4> <p class=\"indent\">Wie oben schon bemerkt, kann man nach der Partialbruchzerlegung ein Integral einer rationalen Funktion auf die Integration von Ausdr\u00fccken der Form <math display=\"inline\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><\/math> oder von <math display=\"inline\"> <mfrac> <mrow> <mi>A<\/mi><mi>x<\/mi><mo class=\"MathClass-bin\">+<\/mo><mi>B<\/mi><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">+<\/mo><mi>b<\/mi><mi>x<\/mi><mo class=\"MathClass-bin\">+<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><\/math> f\u00fcr <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und f\u00fcr Konstanten <math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>B<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> zur\u00fcckf\u00fchren, wobei die Polynome der Form&nbsp;<math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>b<\/mi><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>c<\/mi><\/math> keine rellen Nullstellen haben. F\u00fcr die Berechnung eines Integrals des zweiten Typs mit <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> m\u00f6chten wir hier einen Algorithmus erl\u00e4utern, wobei wir uns auf den Fall&nbsp;<math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>b<\/mi><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>c<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math> beschr\u00e4nken (auf welchen man den allgemeinen Fall mit quadratischem Erg\u00e4nzen zur\u00fcckf\u00fchren kann). <\/p><p class=\"indent\">Seien also <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> und ein Polynom <math display=\"inline\"><mi>q<\/mi><\/math> von Grad kleiner als <math display=\"inline\"><mn>2<\/mn><mi>k<\/mi><\/math> gegeben. Dann ist das unbestimmte Integral <math display=\"inline\"><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo> <mfrac><mrow><mi>q<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> immer von der Form <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><mi>p<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b1<\/mi><mi class=\"qopname\">arctan<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b2<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-266010r5\" \/><mstyle class=\"maketag\"><mtext>(9.5)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr ein Polynom <math display=\"inline\"><mi>p<\/mi><\/math> von Grad kleiner <math display=\"inline\"><mn>2<\/mn><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><\/math> und Konstanten <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><\/math><\/span><span class=\"period\">.<\/span> Durch Ableiten, auf den gemeinsamen Nenner bringen und Vergleich der Koeffizienten l\u00e4sst sich somit die Stammfunktion ermitteln. <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zc93c121522fd\"> <a id=\"x1-266011r20\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.20.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten in dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung den oben erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">rten Algorithmus genauer erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren und beginnen<\/span> <span class=\"ecti-1095\">mit einem konkreten Beispiel.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Berechnen Sie das Integral <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Sei nun <\/span><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>q<\/mi><\/math> <span class=\"ecti-1095\">ein Polynom von<\/span> <span class=\"ecti-1095\">Grad kleiner als <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>2<\/mn><mi>k<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass die Ableitung von <\/span><math display=\"inline\"> <mfrac><mrow><mi>p<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein beliebiges Polynom <\/span><math display=\"inline\"><mi>p<\/mi><\/math> <span class=\"ecti-1095\">von Grad kleiner als <\/span><math display=\"inline\"><mn>2<\/mn><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">durch<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfrac><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>p<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mi>x<\/mi><mi>p<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">gegeben ist.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Berechnen Sie die Matrixdarstellung <\/span><math display=\"inline\"><mi>M<\/mi><\/math> <span class=\"ecti-1095\">der Abbildung<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03d5<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>p<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>p<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mi>x<\/mi><mi>p<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich der Basis der Monome.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Schliessen Sie auf die Darstellung in<\/span> (<a href=\"..\/..\/chapter\/integrationsmethoden#x1-266010r5\">9.5<\/a>)<span class=\"ecti-1095\">, indem Sie zeigen, dass das Bild von<\/span> <math display=\"inline\"><mi>\u03d5<\/mi><\/math> <span class=\"ecti-1095\">zusammen<\/span> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mn>2<\/mn><mi>x<\/mi><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">den Vektorraum der Polynome von Grad kleiner gleich<\/span> <math display=\"inline\"><mn>2<\/mn><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">aufspannt.<\/span><\/dd><\/dl> <\/div> <a id=\"x1-266016r266\"><\/a> <h4 id=\"ze875100a54a7\" class=\"subsectionHead\"><span class=\"titlemark\">9.2.4 <\/span> <a id=\"x1-2670004\"><\/a>Trigonometrische Substitution<\/h4> <p class=\"noindent\">In allen bisherigen Beispielen der Substitutionsregel in Abschnitt <a href=\"..\/..\/chapter\/integrationsmethoden#x1-2650002\">9.2.2<\/a> hatten wir das Gl\u00fcck, dass das vorhandene Integral (vielleicht nach etwas Arbeit) bereits die richtige Struktur besass. Man verwendet die Substitutionsregel aber oft auch bevor man weiss welches Integral sich eigentlich nach der Substitution ergibt, wobei es gewisse Funktionentypen gibt bei denen eine gewisse Substitution erfahrungsgem\u00e4ss erfolgreich sein k\u00f6nnte. Wir wenden uns nun einem konkreten Beispiel dessen zu. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"ze8b184fee3b8\"> <a id=\"x1-267001r21\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.21 <\/span>(Kreisfl\u00e4che)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">das unbestimmte Integral <\/span><math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">berechnen. Auf Grund der trigonometrischen Identit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ten<\/span> <math display=\"inline\"><msqrt><mrow><msup><mrow> <mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <mi>r<\/mi><mi class=\"qopname\">cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\ud835\udf03<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">bietet<\/span> <span class=\"ecti-1095\">es sich nun an, die Funktion<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>\ud835\udf03<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo><mfrac><mrow> <mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>r<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>\ud835\udf03<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>r<\/mi><mi class=\"qopname\">sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\ud835\udf03<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Substitution zu verwenden. Denn mit dieser Substitution haben wir die Hoffnung, die Wurzel<\/span> <span class=\"ecti-1095\">in einen anderen Ausdruck zu verwandeln.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Allerdings ist dies umgekehrt zu der Substitution in Abschnitt<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/integrationsmethoden#x1-2650002\"><span class=\"ecti-1095\">9.2.2<\/span><\/a><span class=\"ecti-1095\">, da wir hier die<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">neue<\/span> <span class=\"ecti-1095\">Variable<\/span><span class=\"ecti-1095\">\u201c<\/span> <span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\ud835\udf03<\/mi><\/math> <span class=\"ecti-1095\">verwenden<\/span> <span class=\"ecti-1095\">um die<\/span> <span class=\"ecti-1095\">\u201e<\/span> <span class=\"ecti-1095\">alte Variable<\/span><span class=\"ecti-1095\">\u201c<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>r<\/mi><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">auszudr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">cken. (Anstatt wie in Abschnitt<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/integrationsmethoden#x1-2650002\"><span class=\"ecti-1095\">9.2.2<\/span><\/a> <span class=\"ecti-1095\">wo wir die neue<\/span> <span class=\"ecti-1095\">Variable<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">als Funktion der<\/span> <span class=\"ecti-1095\">alten Variable<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>x<\/mi><\/math> <span class=\"ecti-1095\">gesehen<\/span> <span class=\"ecti-1095\">haben). Da<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">bijektiv ist, ist<\/span> <span class=\"ecti-1095\">dies kein Problem: denn<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>r<\/mi><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>r<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ist zu<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\ud835\udf03<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> arcsin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>r<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent. Weiters<\/span> <span class=\"ecti-1095\">ist die Ableitung von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">gleich<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>r<\/mi><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo>  <mi>\ud835\udf03<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">damit auf ganz<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ungleich<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Gemeinsam mit dem Satz <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber die Ableitung der inversen Funktion (Satz <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-228020r14\"><span class=\"ecti-1095\">8.14<\/span><\/a><span class=\"ecti-1095\">) erhalten wir<\/span> <span class=\"ecti-1095\">daher<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mi>r<\/mi><mi class=\"qopname\">cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\ud835\udf03<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo> \u222b  <\/mo><mfrac><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\ud835\udf03<\/mi> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>2<\/mn><mi>\ud835\udf03<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mi class=\"qopname\"> arcsin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>r<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi>x<\/mi><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei wir eben <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>r<\/mi><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><msqrt><mrow><msup><mrow> <mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <mi>r<\/mi><mi class=\"qopname\">cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\ud835\udf03<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>r<\/mi><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">die trigonometrischen Identit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ten<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msup><mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><msup><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>\ud835\udf03<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">verwendet haben.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Veranschaulichen Sie sich die Substitution und die wichtigsten der obigen Identit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ten<\/span> <span class=\"ecti-1095\">in einem rechtwinkeligen Dreieck. Geben Sie weiters eine geometrische Interpretation<\/span> <span class=\"ecti-1095\">der beiden Terme des unbestimmten Integrals bei der Berechnung des bestimmten<\/span> <span class=\"ecti-1095\">Integrals<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>r<\/mi><\/math> <span class=\"ecti-1095\">an.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Dies zeigt, dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>G<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>r<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mi class=\"qopname\"> arcsin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>r<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi>x<\/mi><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">eine Stammfunktion von <\/span><math display=\"inline\"><msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>r<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/math> <span class=\"ecti-1095\">ist (was wie immer viel einfacher zu <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berpr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">fen ist). Da aber sogar die Funktion<\/span> <math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>r<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/math> <span class=\"ecti-1095\">stetig ist, besitzt<\/span> <math display=\"inline\"><mi>g<\/mi><\/math> <span class=\"ecti-1095\">nach Korollar<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/der-fundamentalsatz-der-integral--und-differentialrechnung#x1-259005r3\"><span class=\"ecti-1095\">9.3<\/span><\/a> <span class=\"ecti-1095\">auch auf ganz<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>r<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Stammfunktion<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>G<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">welche auf<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>r<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">mit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mi>G<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><\/math> <span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">bereinstimmt. Da aber<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>G<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>r<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mi class=\"qopname\"> arcsin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>r<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi>x<\/mi><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">auch eine auf ganz<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>r<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">stetige Funktion<\/span> <span class=\"ecti-1095\">definiert, folgt aus Stetigkeit von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>G<\/mi><\/math> <span class=\"ecti-1095\">und<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mi>G<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/math><span class=\"ecti-1095\">, dass<\/span> <math display=\"inline\"><msub><mrow><mi>G<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>G<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><\/math> <span class=\"ecti-1095\">und damit<\/span> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><mi>G<\/mi><\/math> <span class=\"ecti-1095\">auf ganz<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>r<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">eine Stammfunktion<\/span> <span class=\"ecti-1095\">von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>g<\/mi><\/math><span class=\"ecti-1095\">. Wir bemerken<\/span> <span class=\"ecti-1095\">allerdings, dass<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi class=\"qopname\">arcsin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>r<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">keine<\/span> <span class=\"ecti-1095\">Ableitung in den Punkten<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>r<\/mi><\/math> <span class=\"ecti-1095\">und<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>r<\/mi><\/math> <span class=\"ecti-1095\">besitzt. (Wieso ist dies kein Widerspruch zu obiger Diskussion?)<\/span> <\/p> <\/div> <p class=\"indent\">Substitutionen wie obige nennen sich vielfach <span class=\"ecbx-1095\">trigonometrische Substitutionen<\/span>. Wir werden bei diesen Berechnungen nicht immer so sorgf\u00e4ltig argumentieren und vielmehr der Leibniz Notation vertrauen, doch muss immer Invertierbarkeit der Funktion gegeben sein wenn wir die alte Variable durch die neue Variable ausdr\u00fccken. F\u00fcr die folgende Auflistung der trigonometrischen Substitutionen sei <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">In Ausdr\u00fccken der Form <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mfrac><mrow><mi>n<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup><\/math> f\u00fcr <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> f\u00fchrt wie bereits im obigen Beispiel oft die Substitution <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> mit&nbsp;<math display=\"inline\"><mi>\ud835\udf03<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-close\">)<\/mo><\/math> zum Ziel, wobei sich damit <math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><\/math> und <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi class=\"qopname\">cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\ud835\udf03<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> ergibt. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">In Ausdr\u00fccken der Form <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mfrac><mrow><mi>n<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup><\/math> f\u00fcr <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> f\u00fchrt oft die Substitution <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi class=\"qopname\">tan<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> mit&nbsp;<math display=\"inline\"><mi>\ud835\udf03<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-close\">)<\/mo><\/math> zum Ziel, wobei sich damit <math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>a<\/mi><\/mrow> <mrow><msup><mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><\/math> und <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>a<\/mi><\/mrow> <mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/math> ergibt. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Obwohl dies keine trigonometrische Substitution darstellt, bemerken wir noch Folgendes. Falls ein \u201eeinzelnes\u201c <math display=\"inline\"><mi>x<\/mi><\/math> vor dem Ausdruck <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mfrac><mrow><mi>n<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup><\/math> oder dem Ausdruck <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mfrac><mrow><mi>n<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup><\/math> steht, ist die Substitution <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> respektive <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> teilweise viel einfacher.<\/div><\/div> <p class=\"indent\">Als Merkhilfe kann es helfen f\u00fcr die beiden trigonometrischen Substitution ein rechtwinkeliges Dreieck zu skizzieren und abh\u00e4ngig von der Substitution die Seiten mit Hilfe von Pythagoras entsprechend zu beschriften. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z6cb7e02d7aef\"> <a id=\"x1-267002r22\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.22 <\/span>(Trigonometrische Substitution)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Es gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mfrac><mrow><mn>3<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mfrac><mrow><msup><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mi>a<\/mi> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo> \u222b  <\/mo><mi class=\"qopname\">cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\ud835\udf03<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>d<\/mi><mi>\ud835\udf03<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\ud835\udf03<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><msqrt><mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei wir <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi class=\"qopname\">tan<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><msqrt><mrow><msup><mrow> <mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><\/math> <span class=\"ecti-1095\">verwendet haben. (Veranschaulichen Sie sich die Substitution und obige Identit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ten in einem<\/span> <span class=\"ecti-1095\">Bild.)<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Es ist<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo><mi>x<\/mi><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo> \u222b  <\/mo><msup><mrow><mi>u<\/mi><\/mrow><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/mfrac><mfrac><mrow> <mn>2<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac><msup><mrow><mi>u<\/mi><\/mrow><mrow><mfrac><mrow><mn>3<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/mfrac><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mfrac><mrow><mn>3<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/p><\/dd><\/dl> <\/div> <a id=\"x1-267005r267\"><\/a> <h4 id=\"z8ec10f0caea9\" class=\"subsectionHead\"><span class=\"titlemark\">9.2.5 <\/span> <a id=\"x1-2680005\"><\/a>Weitere Integrationsmethoden<\/h4> <p class=\"noindent\">Es gibt viele weitere Methoden zur Integration; viele davon beruhen auf spezielle Substitutionen. <\/p><p class=\"indent\">Beispielsweise lassen sich gewisse unbestimmte Integrale mit hyperbolischen Substitutionen berechnen. Sei <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> In Ausdr\u00fccken der Form <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mfrac><mrow><mi>n<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup><\/math> f\u00fcr <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> f\u00fchrt oft die Substitution <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi class=\"qopname\">cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> zum Ziel, wobei sich damit <math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>u<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><\/math> und <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> ergibt. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zfc3ca3a17bc7\"> <a id=\"x1-268001r23\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.23.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir berechnen<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo><msqrt><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><msup><mrow><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mo>\u222b  <\/mo><msup><mrow><mi class=\"qopname\">cosh<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mo>\u222b  <\/mo><msup><mrow><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>u<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mo>\u222b  <\/mo><msup><mrow><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Nach Aufl<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sen ergibt sich somit<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo><msqrt><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><msup><mrow><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>u<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi class=\"qopname\">cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>u<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>x<\/mi><msqrt><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/msqrt> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> arcosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <p class=\"indent\">Eine andere Methode, die wir hier kurz erw\u00e4hnen m\u00f6chten, ist die sogenannte Halbwinkelmethode (oder auch Weierstrass-Substitution). Diese ist dann n\u00fctzlich, wenn man das Integral einer rationalen Funktion in <math display=\"inline\"><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> wie zum Beispiel <math display=\"inline\"><mfrac><mrow><msup><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-bin\">+<\/mo><mn>2<\/mn><mn>0<\/mn><mn>1<\/mn><mn>7<\/mn><\/mrow><\/mfrac><\/math> in die Integration einer rationalen Funktion in <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> tan<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> umwandeln m\u00f6chte (siehe auch Beispiel <a href=\"..\/..\/chapter\/integrationsmethoden#x1-265002r17\">9.17<\/a> (b)). <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z3dae513df785\"> <a id=\"x1-268002r24\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.24 <\/span>(Halbwinkelmethode)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten das unbestimmte Integral <\/span><math display=\"inline\"><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo> <mfrac><mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>2<\/mn><mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">mit der Substitution <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> tan<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">berechnen. Zeigen Sie daf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r zuerst die Identit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ten<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>2<\/mn><mi>u<\/mi><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi class=\"qopname\">cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Zeigen Sie anschliessend, dass das obige Integral nach Substitution zu einem Integral einer rationalen<\/span> <span class=\"ecti-1095\">Funktion in <\/span><math display=\"inline\"><mi>u<\/mi><\/math> <span class=\"ecti-1095\">wird und berechnen Sie es.<\/span> <\/p> <\/div> <p class=\"indent\">Manchmal f\u00fchrt man auch die eine oder die andere Substitution durch, weil in der zu integrierenden Funktion eine verschachtelte Funktion vorliegt und man einfach keine andere Methode zur Verf\u00fcgung hat. Zum Beispiel bei dem Integral&nbsp;<math display=\"inline\"><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msqrt><mrow><mi>x<\/mi><\/mrow><\/msqrt><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> steht keine der erw\u00e4hnten Methoden zur Verf\u00fcgung, doch ist man versucht&nbsp;<math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><mi>x<\/mi><\/mrow><\/msqrt><\/math> zu setzen um zu sehen was sich daraus ergibt. Dies f\u00fchrt in der Tat zum Erfolg (wieso?). Ebenso in dem Integral der Form&nbsp;<math display=\"inline\"><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> f\u00fchrt der Ansatz&nbsp;<math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> zu einem unbestimmten Integral einer rationalen Funktion (wieso?). <a id=\"x1-268003r268\"><\/a> <\/p> <h4 id=\"zdfc099e7a827\" class=\"subsectionHead\"><span class=\"titlemark\">9.2.6 <\/span> <a id=\"x1-2690006\"><\/a>Das bestimmte Integral<\/h4> <p class=\"noindent\">Alle obigen Regeln zur Berechnung des unbestimmten Integrals lassen sich nach dem Fundamentalsatz der Integral- und Differentialrechnung eins zu eins auch f\u00fcr das Riemann-Integral, welches im Gegensatz zum unbestimmten Integral auch das bestimmte Integral genannt wird, anwenden. Dabei haben wir zwei M\u00f6glichkeiten. <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Eine erste M\u00f6glichkeit ist mit obigen Methoden zuerst das unbestimmte Integral zu berechnen und dann Korollar <a href=\"..\/..\/chapter\/der-fundamentalsatz-der-integral--und-differentialrechnung#x1-259007r4\">9.4<\/a> zur Berechnung des Riemann-Integrals zu verwenden. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Falls wir  aber  nur  an  einem  einzigen  Riemann-Integral  interessiert  sind,  ist  es  oft einfacher, die Ausdr\u00fccke ausserhalb des Integrals so fr\u00fch wie m\u00f6glich zu berechnen. Wir erkl\u00e4ren dies im Folgenden f\u00fcr die partielle Integration und die Substitution.<\/div><\/div> <p class=\"indent\">Sind <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>v<\/mi><\/math> zwei stetig differenzierbare Funktionen auf einem kompakten Intervall                                                                                                                                                                           <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> mit Endpunkten <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>u<\/mi><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>u<\/mi><mi>v<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow> <mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>v<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Denn falls <math display=\"inline\"><mi>F<\/mi><\/math> eine Stammfunktion von <math display=\"inline\"><mi>u<\/mi><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/math> und <math display=\"inline\"><mi>G<\/mi><\/math> eine Stammfunktion von <math display=\"inline\"><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>v<\/mi><\/math> ist, dann gilt f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>C<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mi>u<\/mi><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mo>\u222b  <\/mo><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>v<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>C<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>G<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>C<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr gewisse Integrationskonstanten&nbsp;<span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>C<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>C<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>C<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Somit ist nach Korollar <a href=\"..\/..\/chapter\/der-fundamentalsatz-der-integral--und-differentialrechnung#x1-259007r4\">9.4<\/a> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>u<\/mi><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow> <mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>G<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>G<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>G<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow> <mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow> <mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>v<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Ebenso k\u00f6nnen wir bei einer Substitution in Abschnitt <a href=\"..\/..\/chapter\/integrationsmethoden#x1-2650002\">9.2.2<\/a> die Grenzen f\u00fcr ein Riemann-Integral enstsprechend der Substitution neu berechnen. Sei <math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>x<\/mi> <\/mrow> <\/msub> <\/math> ein Intervall mit Endpunkten <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> sei <math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>u<\/mi> <\/mrow> <\/msub> <\/math> ein weiteres Intervall, sei <math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>u<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> stetig und <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>x<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>u<\/mi> <\/mrow> <\/msub> <\/math> stetig differenzierbar. F\u00fcr ein kompaktes Intervall <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> mit Endpunkten <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> in&nbsp;<math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>x<\/mi> <\/mrow> <\/msub> <\/math> gilt dann <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msubsup><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>u<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">In der Tat, wenn <math display=\"inline\"><mi>G<\/mi><\/math> eine Stammfunktion von <math display=\"inline\"><mi>g<\/mi><\/math> auf&nbsp;<math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>u<\/mi> <\/mrow> <\/msub> <\/math> ist, dann ist nach der Kettenregel <math display=\"inline\"><mi>G<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><\/math> eine Stammfunktion von <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Nach Korollar <a href=\"..\/..\/chapter\/der-fundamentalsatz-der-integral--und-differentialrechnung#x1-259007r4\">9.4<\/a> gilt also                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>G<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow> <mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-rel\">=<\/mo> <mi>G<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>G<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>G<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msubsup> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msubsup><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>u<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Die Annahme der Stetigkeit an <math display=\"inline\"><mi>g<\/mi><\/math> kann abgeschw\u00e4cht werden \u2013 siehe die entsprechende \u00dcbung im Abschnitt <a href=\"..\/..\/chapter\/weitere-lernmaterialien#x1-2930002\">9.8.2<\/a>. <\/p><p class=\"indent\">Wir bemerken an dieser Stelle, dass die in den obigen Abschnitten behandelten Themen oft alles sind, was man f\u00fcr Anwendungen (wie zum Beispiel f\u00fcr die Fl\u00e4chenberechnung unter Graphen) braucht. Nichtsdestotrotz werden wir erst gegen Ende des Kapitels in Abschnitt&nbsp;<a href=\"..\/..\/chapter\/anwendungen#x1-2850007\">9.7<\/a> darauf eingehen. Gewisse Anwendungen wurden schon in Abschnitt&nbsp;<a href=\"..\/..\/chapter\/anwendungen#x1-1150004\">4.4<\/a> diskutiert. <a id=\"x1-269001r269\"><\/a> <\/p> <h4 id=\"z412446ded346\" class=\"subsectionHead\"><span class=\"titlemark\">9.2.7 <\/span> <a id=\"x1-2700007\"><\/a>Leibniz-Notation<\/h4> <p class=\"noindent\">Wir werden die Leibniz-Notation in der Berechnung von unbestimmten und bestimmten Integralen wie bereits oben im Folgenden immer wieder verwenden. Diese Notation verpackt in einem nat\u00fcrlichen Formalismus die partielle Integration <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo> \u222b  <\/mo><mi>u<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>v<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>u<\/mi><mi>v<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mo>\u222b  <\/mo><mi>v<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>C<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und die Substitutionsregeln                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo> \u222b  <\/mo><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>u<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo>\u222b  <\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mo> \u222b  <\/mo><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>u<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>u<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wobei wir in der zweiten Formulierung der Substitution vorraussetzen, dass <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>x<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>u<\/mi> <\/mrow> <\/msub> <\/math> bijektiv mit nicht verschwindender Ableitung ist und dadurch im linken Integral mit <math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo> <mi>x<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><\/mrow><\/mfrac> <mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><\/mfrac><\/math> multiplizieren konnten und die erste Formulierung der Substitutionsregel anwenden konnten. Wie wir gesehen haben, sind diese Regeln Umformulierungen der Produktregel f\u00fcr die Ableitung und der Kettenregel f\u00fcr die Ableitung (gemeinsam mit der Ableitungsregel f\u00fcr die inverse Abbildung). <\/p><p class=\"indent\">Bei konkreten Integralberechnungen verwenden wir mitunter auch Gleichungen, die <math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> und <math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>u<\/mi><\/math> miteinander verbinden. Zum Beispiel bei der trigonometrischen Substitution&nbsp;<math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><\/math> (f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><mi>\ud835\udf03<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-close\">)<\/mo><\/math>) verwenden wir auch die Formel&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><\/math><\/span><span class=\"period\">,<\/span> die formal gesehen keine Bedeutung hat (und deswegen auf keinen Fall in dieser Form in Beweisen auftreten sollte), doch eben im Zuge der Substitution in der Formulierung der Leibniz-Notation einen bequemen Zwischenschritt darstellt. <\/p><p class=\"indent\">Informell taucht in Anwendungen das Symbol <math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> auch oft in Diskussionen auf, die zu einem Riemann-Integral f\u00fchren, wobei <math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> dann f\u00fcr ein (sehr) kleines <math display=\"inline\"><mi>\u0394<\/mi><mi>x<\/mi><\/math> stehen sollte. In Anwendungen werden h\u00e4ufig die Begriffe der Riemann-Summe oder der additiven Intervallfunktion vermieden, wobei es genau diese Begriffe sind, die diese Verwendung von <math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> genau und formal korrekt machen w\u00fcrden (siehe Abschnitte <a href=\"..\/..\/chapter\/anwendungen#x1-1150004\">4.4<\/a> und <a href=\"..\/..\/chapter\/riemann-summen#x1-1800005\">6.5<\/a>). Auf jeden Fall hat in diesem Zusammenhang eine Formel der Gestalt&nbsp;<math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><\/math> auch eine                                                                                                                                                                           Interpretation: Da <math display=\"inline\"><mi>\u0394<\/mi><mi>x<\/mi><\/math> die L\u00e4nge eines kleinen Teilintervalls von&nbsp;<math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> angibt und <math display=\"inline\"><mi>\u0394<\/mi><mi>\ud835\udf03<\/mi><\/math> die L\u00e4nge des entsprechenden Teilintervalls in <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>\u03c0<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-close\">)<\/mo><\/math> so gibt die Ableitung <math display=\"inline\"><mi>a<\/mi><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mi>\ud835\udf03<\/mi><\/math> (bis auf einen kleinen und wie sich herausstellt vernachl\u00e4ssigbaren Fehler) den Gr\u00f6ssenunterschied <math display=\"inline\"><mfrac><mrow><mi>\u0394<\/mi><mi>x<\/mi><\/mrow> <mrow><mi>\u0394<\/mi><mi>\ud835\udf03<\/mi><\/mrow><\/mfrac><\/math> an, der bei Betrachtung von etwaigen Riemann-Summen in der Variable <math display=\"inline\"><mi>x<\/mi><\/math> und der Variable <math display=\"inline\"><mi>\ud835\udf03<\/mi><\/math> als zus\u00e4tzlicher Faktor auftreten w\u00fcrde. Wir m\u00fcssen dies nicht genauer ausf\u00fchren oder die Substitution auf diese Art und Weise beweisen, da wir ja mittels der Kettenregel und dem Fundamentalsatz der Integral- und Differentialrechnung bereits die Substitutionsregel f\u00fcr Riemann-Integrale bewiesen haben und obiger Formalismus diese nur auf eine andere Art pr\u00e4sentiert. Dieser Beweis \u00fcber den Fundamentalsatz verwendet allerdings etwas st\u00e4rkere Annahmen als notwendig (siehe folgende \u00dcbung f\u00fcr den direkten Beweis mit schw\u00e4cheren Annahmen). <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"za2a33f4d2777\"> <a id=\"x1-270001r25\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.25 <\/span>(Substitution f\u00fcr Riemann-integrierbare Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">ein kompaktes<\/span> <span class=\"ecti-1095\">Intervall in<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <span class=\"ecti-1095\">Endpunkten <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">eine stetig differenzierbare<\/span> <span class=\"ecti-1095\">Funktion mit <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><span class=\"ecti-1095\">. Dann ist f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede<\/span> <span class=\"ecti-1095\">Riemann-integrierbare Funktion <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">auch <\/span><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">Riemann-integrierbar und<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msubsup><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <a id=\"x1-270002r270\"><\/a> <h4 id=\"z5023b13c551d\" class=\"subsectionHead\"><span class=\"titlemark\">9.2.8 <\/span> <a id=\"x1-2710008\"><\/a>Neue Funktionen<\/h4> <p class=\"noindent\">Manchmal f\u00fchren obige Methoden zur Bestimmung eines unbestimmten Integrals einer Funktion zu keinem Ergebnis. Dies kann daran liegen, dass die gesuchte Stammfunktion sich nicht mit den bisher bekannten Funktionen ausdr\u00fccken l\u00e4sst. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"za88f3969dea4\"> <a id=\"x1-271001r26\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.26 <\/span>(Integralsinus)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Der <\/span><span class=\"ecbi-1095\">Integralsinus <\/span><span class=\"ecti-1095\">ist die Stammfunktion <\/span><math display=\"inline\"><mi class=\"qopname\">Si<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">der stetigen Funktion<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><mfrac><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mtd><mtd class=\"array\" columnalign=\"center\"> <mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn> <\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"> <mn>1<\/mn> <\/mtd> <mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle> <mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/mtd> <\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">mit der Normalisierung <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\">Si<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Er l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst sich als Potenzreihe schreiben, denn nach Satz<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/integration-von-potenzreihen#x1-216001r85\"><span class=\"ecti-1095\">7.85<\/span><\/a> <span class=\"ecti-1095\">gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">Si<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><mfrac><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>t<\/mi><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zfefaa105cdff\"> <a id=\"x1-271002r27\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.27 <\/span>(Integralkosinus)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Der Integralkosinus <\/span><math display=\"inline\"><mi class=\"qopname\">Ci<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist definiert als die Stammfunktion von <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">mit der Normalisierung <\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi class=\"qopname\">Ci<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Dabei m\u00f6chten wir auf folgende \u00dcbung verweisen, die zeigt, dass der Integralkosinus so wohldefiniert ist. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z4772725f64ee\"> <a id=\"x1-271003r28\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.28.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>F<\/mi><\/math> <span class=\"ecti-1095\">eine Stammfunktion von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass der Grenzwert <\/span><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>F<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">existiert. Dr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">cken Sie <\/span><math display=\"inline\"><mi class=\"qopname\">Ci<\/mi><mo>  <\/mo><\/math> <span class=\"ecti-1095\">als Summe einer Konstanten (der sogenannten Euler-Mascheroni Konstanten), der Logarithmusfunktion<\/span> <span class=\"ecti-1095\">und einer Potenzreihe aus.<\/span> <\/p> <\/div> <p class=\"indent\">Unter Verwendung uneigentlicher Integrale werden wir sp\u00e4ter weitere wichtige Funktionen kennenlernen, die sich nicht in Termen bekannter Funktionen ausdr\u00fccken lassen \u2013 siehe zum Beispiel <a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-273006r34\">9.34<\/a>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               <a id=\"x1-271004r263\"><\/a> <\/p> \n","protected":false},"author":1089,"menu_order":2,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-94","chapter","type-chapter","status-publish","hentry"],"part":92,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/94","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/users\/1089"}],"version-history":[{"count":0,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/94\/revisions"}],"part":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/parts\/92"}],"metadata":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/94\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=94"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=94"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=94"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=94"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}