Example 2. Find the integral $\int \cos ^{3} x \sin ^{2} x d x$.
Solution. In this case $m=3, n=2$. We get

$$
\begin{gathered}
\int \cos ^{3} x \sin ^{2} x d x=\int \cos ^{2} x \sin ^{2} x \cos x d x= \\
=\int\left(1-\sin ^{2} x\right) \sin ^{2} x d(\sin x)=\int\left(\sin ^{2} x-\sin ^{4} x\right) d(\sin x)= \\
=\int \sin ^{2} x d(\sin x)-\int \sin ^{4} x d(\sin x)=\frac{\sin ^{3} x}{3}-\frac{\sin ^{5} x}{5}+C
\end{gathered}
$$