Example 3 (to $4^{\circ}$ ). Find $\int \sin ^{4} x \cos ^{3} x d x$.
Solution. We are dealing with case 1); $m=3$ - an odd number. Let $\sin x=u$ and bring $\cos x$ under the differential sign, since $\cos x d x=d(\sin x)$. We have:

$$
\begin{aligned}
\int \sin ^{4} x\left(1-\sin ^{2} x\right) \cos x d x=\int & u^{4}\left(1-u^{2}\right) d u= \\
& =\frac{u^{5}}{5}-\frac{u^{7}}{7}+C=\frac{1}{5} \sin ^{5} x-\frac{1}{7} \sin ^{7} x+C
\end{aligned}
$$