integral of (sin^5(x))/(cos^7(x))

Lösung

\int \frac{sin ^{5} ( x )}{cos ^{7} ( x )} d x

\frac{1}{6} sec^{6} (x) - \frac{1}{2} sec^{4} (x) + \frac{1}{2} sec^{2} (x) + C

Schritte zur Lösung

\int \frac{sin ^{5} ( x )}{cos ^{7} ( x )} d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int \frac{( 1 - cos ^{2} ( x ) ) ^{2} sin ( x )}{cos ^{7} ( x )} d x

Wende U-Substitution an

= \int - \frac{( 1 - u ^{2} ) ^{2}}{u ^{7}} d u

Multipliziere aus

- \frac{( 1 - u ^{2} ) ^{2}}{u ^{7}} : - \frac{1}{u ^{7}} + \frac{2}{u ^{5}} - \frac{1}{u ^{3}}

= \int - \frac{1}{u ^{7}} + \frac{2}{u ^{5}} - \frac{1}{u ^{3}} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int \frac{1}{u ^{7}} d u + \int \frac{2}{u ^{5}} d u - \int \frac{1}{u ^{3}} d u

\int \frac{1}{u ^{7}} d u = - \frac{1}{6 u ^{6}}

\int \frac{2}{u ^{5}} d u = - \frac{1}{2 u ^{4}}

\int \frac{1}{u ^{3}} d u = - \frac{1}{2 u ^{2}}

= - (- \frac{1}{6 u ^{6}}) - \frac{1}{2 u ^{4}} - (- \frac{1}{2 u ^{2}})

Setze in

u = cos (x)

ein

= - (- \frac{1}{6 cos ^{6} ( x )}) - \frac{1}{2 cos ^{4} ( x )} - (- \frac{1}{2 cos ^{2} ( x )})

Vereinfache

- (- \frac{1}{6 cos ^{6} ( x )}) - \frac{1}{2 cos ^{4} ( x )} - (- \frac{1}{2 cos ^{2} ( x )}) : \frac{1}{6} sec^{6} (x) - \frac{1}{2} sec^{4} (x) + \frac{1}{2} sec^{2} (x)

= \frac{1}{6} sec^{6} (x) - \frac{1}{2} sec^{4} (x) + \frac{1}{2} sec^{2} (x)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{6} sec^{6} (x) - \frac{1}{2} sec^{4} (x) + \frac{1}{2} sec^{2} (x) + C

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