Use various trigonometric identities to simplify the expression then integrate.
-$\int \sin ^ { 2 } \theta \cos 5 \theta d \theta$
A) $\frac { 1 } { 10 } \sin 5 \theta - \frac { 1 } { 4 } \sin 3 \theta - \frac { 1 } { 28 } \sin 7 \theta + C$
B) $\frac { 1 } { 2 } \sin 5 \theta - \frac { 1 } { 4 } \sin 3 \theta - \frac { 1 } { 7 } \sin 7 \theta + C$
C) $\frac { 1 } { 10 } \sin 3 \theta - \frac { 1 } { 4 } \sin 5 \theta - \frac { 1 } { 28 } \sin 7 \theta + C$
D) $\frac { 1 } { 10 } \sin 5 \theta - \frac { 1 } { 4 } \sin 7 \theta - \frac { 1 } { 28 } \sin 3 \theta + C$

Question

Quizplus · Accepted Answer

To simplify and integrate $\int \sin^2 \theta \cos 5 \theta d \theta$, use the identity $\sin^2 \theta = \frac{1 - \cos 2\theta}{2}$. Substituting and expanding using trigonometric product-to-sum identities, the integral simplifies to terms involving $\sin 3\theta$, $\sin 5\theta$, and $\sin 7\theta$, leading to the correct answer after integration.

Use Various Trigonometric Identities to Simplify the Expression Then Integrate ∫sin⁡2θcos⁡5θdθ\int \sin ^ { 2 } \theta \cos 5 \theta d \theta∫sin2θcos5θdθ

Use Various Trigonometric Identities to Simplify the Expression Then Integrate $\int \sin ^ { 2 } \theta \cos 5 \theta d \theta$