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Beliebt Rechnen >

integral of cos^5(x)sin^6(x)

Lösung

\int cos^{5} (x) sin^{6} (x) d x

Lösung

\frac{sin ^{7} ( x )}{7} - \frac{2 sin ^{9} ( x )}{9} + \frac{sin ^{11} ( x )}{11} + C

Schritte zur Lösung

\int cos^{5} (x) sin^{6} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int (1 - sin^{2} (x))^{2} cos (x) sin^{6} (x) d x

Wende U-Substitution an

= \int u^{6} (1 - u^{2})^{2} d u

Multipliziere aus

u^{6} (1 - u^{2})^{2} : u^{6} - 2 u^{8} + u^{10}

= \int u^{6} - 2 u^{8} + u^{10} 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^{6} d u - \int 2 u^{8} d u + \int u^{10} d u

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

\int 2 u^{8} d u = \frac{2 u ^{9}}{9}

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

= \frac{u ^{7}}{7} - \frac{2 u ^{9}}{9} + \frac{u ^{11}}{11}

Setze in

u = sin (x)

ein

= \frac{sin ^{7} ( x )}{7} - \frac{2 sin ^{9} ( x )}{9} + \frac{sin ^{11} ( x )}{11}

Füge eine Konstante zur Lösung hinzu

= \frac{sin ^{7} ( x )}{7} - \frac{2 sin ^{9} ( x )}{9} + \frac{sin ^{11} ( x )}{11} + C

Graph

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