Use various trigonometric identities to simplify the expression then integrate.
-$$\int \sin \theta \cos \theta \cos 4 \theta d \theta$$
A) $\frac { 1 } { 6 } \cos 6 \theta + C$
B) $\frac { 1 } { 8 } \cos 2 \theta - \frac { 1 } { 24 } \cos 6 \theta + C$
C) $\frac { 1 } { 24 } \cos 4 \theta + C$
D) $\frac { 1 } { 4 } \cos 2 \theta - \frac { 1 } { 8 } \cos 6 \theta + C$

Question

Quizplus · Accepted Answer

To simplify the integral $\int \sin \theta \cos \theta \cos 4 \theta d \theta$, we can use the trigonometric identity for the product of cosines: $\cos A \cos B = \frac{1}{2}[\cos(A - B) + \cos(A + B)]$. Applying this to $\cos \theta \cos 4\theta$, we get $\frac{1}{2}[\cos(-3\theta) + \cos(5\theta)] = \frac{1}{2}[\cos 3\theta + \cos 5\theta]$. The integral becomes $\frac{1}{2}\int \sin \theta (\cos 3\theta + \cos 5\theta) d\theta$, which can be split into two integrals. Using the trigonometric identity for the product of sine and cosine, $\sin A \cos B = \frac{1}{2}[\sin(A + B) + \sin(A - B)]$, we can further simplify and integrate each part. The result is $\frac{1}{8}\cos 2\theta - \frac{1}{24}\cos 6\theta + C$, matching option B.

Use Various Trigonometric Identities to Simplify the Expression Then Integrate ∫sin⁡θcos⁡θcos⁡4θdθ\int \sin \theta \cos \theta \cos 4 \theta d \theta∫sinθcosθcos4θdθ

Use Various Trigonometric Identities to Simplify the Expression Then Integrate $\int \sin \theta \cos \theta \cos 4 \theta d \theta$