Example 1.10. Find $\int \sin ^{2} x \cos ^{3} x d x$.
Solution. Since the integrand is odd with respect to $\cos x$:

$$
\begin{gathered}
R(\sin x, \cos x)=\sin ^{2} x \cos ^{3} x \\
R(\sin x,-\cos x)=-\sin ^{2} x \cos ^{3} x=-R(\sin x, \cos x)
\end{gathered}
$$

we apply the substitution $t=\sin x$. In this case, transforming the integrand according to the general substitution scheme and considering

$$
d t=d(\sin x)=\cos x d x, \cos ^{2} x=1-\sin ^{2} x
$$

we get:

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