Example 7. Find the integral $\int \cos 3 x \cos x d x$.
Solution. Transforming the integrand, we find

$$
\begin{gathered}
\int \cos 3 x \cos x d x=\frac{1}{2} \int(\cos 4 x+\cos 2 x) d x=\frac{1}{2} \int \cos 4 x d x+ \\
+\frac{1}{2} \int \cos 2 x d x=\frac{1}{8} \sin 4 x+\frac{1}{4} \sin 2 x+C
\end{gathered}
$$