integral of cos(θ)cos^5(sin(θ))

Lösung

\int cos (θ) cos^{5} (sin (θ)) d θ

sin (sin (θ)) - \frac{2 sin ^{3} ( sin ( θ ) )}{3} + \frac{sin ^{5} ( sin ( θ ) )}{5} + C

Schritte zur Lösung

\int cos (θ) cos^{5} (sin (θ)) d θ

Wende U-Substitution an

= \int cos^{5} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

(1 - v^{2})^{2} : 1 - 2 v^{2} + v^{4}

= \int 1 - 2 v^{2} + v^{4} d v

Wende die Summenregel an:

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

= \int 1 d v - \int 2 v^{2} d v + \int v^{4} d v

\int 1 d v = v

\int 2 v^{2} d v = \frac{2 v ^{3}}{3}

\int v^{4} d v = \frac{v ^{5}}{5}

= v - \frac{2 v ^{3}}{3} + \frac{v ^{5}}{5}

Ersetze zurück

= sin (sin (θ)) - \frac{2 sin ^{3} ( sin ( θ ) )}{3} + \frac{sin ^{5} ( sin ( θ ) )}{5}

Füge eine Konstante zur Lösung hinzu

= sin (sin (θ)) - \frac{2 sin ^{3} ( sin ( θ ) )}{3} + \frac{sin ^{5} ( sin ( θ ) )}{5} + C

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