Example 2. Find the integral $\int \operatorname{sh}^{3} x d x$.

Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. 

Example 2. Find the integral $\int \operatorname{sh}^{3} x d x$.
Solution. Representing the integrand in the form

$$
\operatorname{sh}^{3} x=\operatorname{sh}^{2} x \operatorname{sh} x
$$

and taking into account that

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

we obtain

$$
\begin{aligned}
\int \operatorname{sh}^{3} x d x & =\int \operatorname{sh}^{2} x \operatorname{sh} x d x=\int \operatorname{sh}^{2} x d(\operatorname{ch} x)=\int\left(\operatorname{ch}^{2} x-1\right) d(\operatorname{ch} x)= \\
& =\int \operatorname{ch}^{2} x d(\operatorname{ch} x)-\int d(\operatorname{ch} x)=\frac{\operatorname{ch}^{3} x}{3}-\operatorname{ch} x+C
\end{aligned}
$$