Answer
$$\int\cos^22\theta\sin\theta d\theta=-\frac{1}{2}\cos\theta+\frac{1}{12}\cos3\theta-\frac{1}{20}\cos5\theta+C$$
Work Step by Step
$$A=\int\cos^22\theta\sin\theta d\theta$$
Use the identity: $$\cos^2\theta=\frac{1+\cos2\theta}{2}$$
Apply the identity here:
$$A=\int\frac{1+\cos4\theta}{2}\sin\theta d\theta$$ $$A=\frac{1}{2}\Big(\int\sin\theta d\theta+\int\sin\theta\cos4\theta d\theta\Big)$$
Now use the identity for the product of sine and cosine functions: $$\sin mx\cos nx=\frac{1}{2}[\sin(m-n)x+\sin(m+n)x]$$
Therefore, $$A=\frac{1}{2}\Big(-\cos\theta+\frac{1}{2}\int[\sin(-3\theta)+\sin5\theta]d\theta\Big)$$ $$A=\frac{1}{2}\Big(-\cos\theta+\frac{1}{2}\int[-\sin3\theta+\sin5\theta]d\theta\Big)$$ $$A=\frac{1}{2}\Big[-\cos\theta+\frac{1}{2}\Big(\frac{1}{3}\cos3\theta-\frac{1}{5}\cos5\theta\Big)\Big]+C$$ $$A=-\frac{1}{2}\cos\theta+\frac{1}{12}\cos3\theta-\frac{1}{20}\cos5\theta+C$$
