Answer
$$\frac{1}{4} \cos \theta-\frac{1}{20} \cos 5 \theta+C $$
Work Step by Step
We integrate as follows:
\begin{align*}
\int \sin \theta \cos \theta \cos 3 \theta d \theta&=\frac{1}{2} \int 2 \sin \theta \cos \theta \cos 3 \theta d \theta\\
&=\frac{1}{2} \int \sin 2 \theta \cos 3 \theta d \theta\\
&=\frac{1}{2} \int \frac{1}{2}(\sin (2-3) \theta+\sin (2+3) \theta) d \theta\\
&=\frac{1}{4} \int(\sin (-\theta)+\sin 5 \theta) d \theta\\
&=\frac{1}{4} \int(-\sin \theta+\sin 5 \theta) d \theta\\
&=\frac{1}{4} \cos \theta-\frac{1}{20} \cos 5 \theta+C
\end{align*}
