Example 3. Find $\int \frac{x}{\operatorname{sh} \frac{x^{2}}{2}} d x$.
Solution. We have (see formula 4) from point $2^{\circ}$ )

$$
\begin{aligned}
& \int \frac{x d x}{\operatorname{sh} \frac{x^{2}}{2}}=\left\{\begin{array}{c}
\frac{x^{2}}{2}=u \\
x d x=d u
\end{array}\right\}=\int \frac{d u}{\operatorname{sh} u}=\int \frac{d u}{2 \operatorname{sh} \frac{u}{2} \operatorname{ch} \frac{u}{2}}= \\
& \quad=\int \frac{1}{\operatorname{sh} \frac{u}{2} / \operatorname{ch} \frac{u}{2}} \cdot \frac{d u}{2 \operatorname{ch}^{2} \frac{u}{2}}=\int \frac{d\left(\operatorname{th} \frac{u}{2}\right)}{\operatorname{th} \frac{u}{2}}=\ln \left|\operatorname{th} \frac{u}{2}\right|+C=\ln \left|\operatorname{th} \frac{x^{2}}{4}\right|+C
\end{aligned}
$$