Example 1. Find the integral $\int \operatorname{sh}^{2} x d x$.

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Solution. Since

$$
\operatorname{sh}^{2} x=\frac{1}{2}(\operatorname{ch} 2 x-1)
$$

then

$$
\begin{gathered}
\int \operatorname{sh}^{2} x d x=\frac{1}{2} \int(\operatorname{ch} 2 x-1) d x=\frac{1}{2} \int \operatorname{ch} 2 x d x-\frac{1}{2} \int d x= \\
=\frac{1}{4} \int \operatorname{ch} 2 x d(2 x)-\frac{1}{2} \int d x=\frac{1}{4} \operatorname{sh} 2 x-\frac{1}{2} x+C
\end{gathered}
$$