Answer
$$\int\cos^3\theta\sin2\theta d\theta=-\frac{2}{5}\cos^5\theta+C$$
Work Step by Step
$$A=\int\cos^3\theta\sin2\theta d\theta$$
Use the identity: $$\sin2\theta=2\sin\theta\cos\theta$$
we have $$A=\int\cos^3\theta(2\sin\theta\cos\theta)d\theta$$ $$A=2\int\cos^4\theta\sin\theta d\theta$$ $$A=-2\int\cos^4\theta d(\cos\theta)$$
We set $u=\cos\theta$ $$A=-2\int u^4du=-\frac{2}{5}u^5+C$$ $$A=-\frac{2}{5}\cos^5\theta+C$$
