Use various trigonometric identities to simplify the expression then integrate.
-$$\int \cos ^ { 7 } \theta \sin 2 \theta d \theta$$
A) $\frac { 1 } { 4 } \cos ^ { 7 } \theta + C$
B) $- \frac { 1 } { 5 } \cos ^ { 10 } \theta + C$
C) $\frac { 1 } { 9 } \cos ^ { 8 } \theta + C$
D) $- \frac { 2 } { 9 } \cos ^ { 9 } \theta + C$

Question

Quizplus · Accepted Answer

To simplify and integrate $\int \cos ^ { 7 } \theta \sin 2 \theta d \theta$, we use the trigonometric identity $\sin 2\theta = 2\sin\theta\cos\theta$. This gives us $\int \cos ^ { 7 } \theta (2\sin\theta\cos\theta) d \theta = 2\int \cos^8\theta \sin\theta d\theta$. Let $u = \cos\theta$, so $du = -\sin\theta d\theta$, and the integral becomes $-2\int u^8 du$, which integrates to $-\frac{2}{9}u^9 + C$, or $-\frac{2}{9}\cos^9\theta + C$.

Use Various Trigonometric Identities to Simplify the Expression Then Integrate ∫cos⁡7θsin⁡2θdθ\int \cos ^ { 7 } \theta \sin 2 \theta d \theta∫cos7θsin2θdθ

Use Various Trigonometric Identities to Simplify the Expression Then Integrate $\int \cos ^ { 7 } \theta \sin 2 \theta d \theta$