integral of (sin(θ))^4

Lösung

\int (sin (θ))^{4} d θ

- \frac{1}{4} sin^{3} (θ) cos (θ) + \frac{3}{8} (θ - \frac{1}{2} sin (2 θ)) + C

Schritte zur Lösung

\int (sin (θ))^{4} d θ

Vereinfache

= \int sin^{4} (θ) d θ

Wende integrale Reduktion an

= - \frac{cos ( θ ) sin ^{3} ( θ )}{4} + \frac{3}{4} \cdot \int sin^{2} (θ) d θ

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

= - \frac{cos ( θ ) sin ^{3} ( θ )}{4} + \frac{3}{4} \cdot \frac{1}{2} (θ - \frac{1}{2} sin (2 θ))

Vereinfache

- \frac{cos ( θ ) sin ^{3} ( θ )}{4} + \frac{3}{4} \cdot \frac{1}{2} (θ - \frac{1}{2} sin (2 θ)) : - \frac{1}{4} sin^{3} (θ) cos (θ) + \frac{3}{8} (θ - \frac{1}{2} sin (2 θ))

= - \frac{1}{4} sin^{3} (θ) cos (θ) + \frac{3}{8} (θ - \frac{1}{2} sin (2 θ))

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{4} sin^{3} (θ) cos (θ) + \frac{3}{8} (θ - \frac{1}{2} sin (2 θ)) + C

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