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Beliebt Rechnen >

integral of+sin^3(θ)cos^8(θ)

Lösung

\int + sin^{3} (θ) cos^{8} (θ) d θ

Lösung

- \frac{cos ^{9} ( θ )}{9} + \frac{cos ^{11} ( θ )}{11} + C

Schritte zur Lösung

\int sin^{3} (θ) cos^{8} (θ) d θ

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

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

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

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

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

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

Setze in

u = cos (θ)

ein

= - \frac{cos ^{9} ( θ )}{9} + \frac{cos ^{11} ( θ )}{11}

Füge eine Konstante zur Lösung hinzu

= - \frac{cos ^{9} ( θ )}{9} + \frac{cos ^{11} ( θ )}{11} + C

Graph

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