Answer
$$\frac{2}{5} \cos ^{5} \theta-\cos ^{3} \theta+\cos \theta+C $$
Work Step by Step
We integrate as follows:
\begin{align*}
\int \sin ^{3} \theta \cos 2 \theta d \theta&=\int \sin ^{2} \theta \cos 2 \theta \sin \theta d \theta\\
&=\int\left(1-\cos ^{2} \theta\right)\left(2 \cos ^{2} \theta-1\right) \sin \theta d \theta\\
&=\int\left(-2 \cos ^{4} \theta+3 \cos ^{2} \theta-1\right) \sin \theta d \theta\\
&=-2 \int \cos ^{4} \theta \sin \theta d \theta+3 \int \cos ^{2} \theta \sin \theta d \theta-\int \sin \theta d \theta\\
&=\frac{2}{5} \cos ^{5} \theta-\cos ^{3} \theta+\cos \theta+C
\end{align*}
