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Beliebt Rechnen >

integral of sin^7(x)*cos^8(x)

Lösung

\int sin^{7} (x) \cdot cos^{8} (x) d x

Lösung

- \frac{cos ^{9} ( x )}{9} + \frac{3 cos ^{11} ( x )}{11} - \frac{3 cos ^{13} ( x )}{13} + \frac{cos ^{15} ( x )}{15} + C

Schritte zur Lösung

\int sin^{7} (x) cos^{8} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int (1 - cos^{2} (x))^{3} sin (x) cos^{8} (x) d x

Wende U-Substitution an

= \int - u^{8} (1 - u^{2})^{3} d u

Multipliziere aus

- u^{8} (1 - u^{2})^{3} : - u^{8} + 3 u^{10} - 3 u^{12} + u^{14}

= \int - u^{8} + 3 u^{10} - 3 u^{12} + u^{14} 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 u^{8} d u + \int 3 u^{10} d u - \int 3 u^{12} d u + \int u^{14} d u

\int u^{8} d u = \frac{u ^{9}}{9}

\int 3 u^{10} d u = \frac{3 u ^{11}}{11}

\int 3 u^{12} d u = \frac{3 u ^{13}}{13}

\int u^{14} d u = \frac{u ^{15}}{15}

= - \frac{u ^{9}}{9} + \frac{3 u ^{11}}{11} - \frac{3 u ^{13}}{13} + \frac{u ^{15}}{15}

Setze in

u = cos (x)

ein

= - \frac{cos ^{9} ( x )}{9} + \frac{3 cos ^{11} ( x )}{11} - \frac{3 cos ^{13} ( x )}{13} + \frac{cos ^{15} ( x )}{15}

Füge eine Konstante zur Lösung hinzu

= - \frac{cos ^{9} ( x )}{9} + \frac{3 cos ^{11} ( x )}{11} - \frac{3 cos ^{13} ( x )}{13} + \frac{cos ^{15} ( x )}{15} + C

Graph

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