Use various trigonometric identities to simplify the expression then integrate.
-$$\int \sin \theta \cos \theta \cos 5 \theta d \theta$$
A) $\frac { 1 } { 28 } \cos 4 \theta + C$
B) $\frac { 1 } { 12 } \cos 3 \theta - \frac { 1 } { 28 } \cos 7 \theta + C$
C) $\frac { 1 } { 7 } \cos 7 \theta + C$
D) $\frac { 1 } { 5 } \cos 3 \theta - \frac { 1 } { 10 } \cos 7 \theta + C$

Question

Quizplus · Accepted Answer

To simplify and integrate $\int \sin \theta \cos \theta \cos 5 \theta d \theta$, we use the product-to-sum formulas. The relevant formula here is $\sin A \cos B = \frac{1}{2}[\sin(A-B) + \sin(A+B)]$. Applying this to $\sin \theta \cos 5\theta$, we get $\frac{1}{2}[\sin(\theta-5\theta) + \sin(\theta+5\theta)] = \frac{1}{2}[\sin(-4\theta) + \sin(6\theta)]$. Since $\sin(-4\theta) = -\sin(4\theta)$, the integral becomes $\frac{1}{2}\int (-\sin 4\theta + \sin 6\theta) \cos \theta d\theta$. Further simplification and integration, using trigonometric identities and integration techniques, lead to the result $\frac { 1 } { 12 } \cos 3 \theta - \frac { 1 } { 28 } \cos 7 \theta + C$.

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