Example 2.10. $I=\int_{0}^{\pi / 2} \sin ^{3} x \sin 2 x d x$.


Solution. Let's make the substitution $z=\sin x$ and note that $\sin 2 x=2 \sin x \cos x$. We have

$$
\begin{aligned}
& I=\int_{0}^{\pi / 2} \sin ^{3} x \sin 2 x d x=2 \int_{0}^{\pi / 2} \sin ^{4} x \sin x d x= \\
& =2 \int_{0}^{\pi / 2} \sin ^{3} x d \sin x=2 \int_{0}^{1} z^{4} d z=\left.2 \frac{z^{5}}{5}\right|_{0} ^{1}= \\
& \quad=2\left(\frac{1^{5}}{5}-\frac{0^{5}}{5}\right)=\frac{2}{5}=0.4
\end{aligned}
$$