Example 1.14. Find $\int \sin x \cos 2 x \sin 5 x d x$.
Solution.

$$
\begin{aligned}
& \int \sin x \cos 2 x \sin 5 x d x=\frac{1}{2} \int(\sin 3 x-\sin x) \sin 5 x d x= \\
& =\frac{1}{4} \int \cos 2 x d x-\frac{1}{4} \int \cos 8 x d x- \\
& \quad-\frac{1}{4} \int \cos 4 x d x+\frac{1}{4} \int \cos 6 x d x= \\
& =\frac{1}{8} \sin 2 x-\frac{1}{32} \sin 8 x-\frac{1}{16} \sin 4 x+\frac{1}{24} \sin 6 x+C .
\end{aligned}
$$