Example 1. Find $\int \frac{\operatorname{sh} x d x}{\sqrt{1+\operatorname{ch}^{2} x}}$

Solution It is not hard to notice the possibility of bringing $\operatorname{ch} x$ under the differential sign and applying the tabular formula 13:

$$
\int \frac{\operatorname{sh} x d x}{\sqrt{1+\operatorname{ch}^{2} x}}=\int \frac{d(\operatorname{ch} x)}{\sqrt{1+\operatorname{ch}^{2} x}}=\ln \left(\operatorname{ch} x+\sqrt{1+\operatorname{ch}^{2} x}\right)+C
$$

$$
\text { Example } 2 . \text { Find } \int \frac{d x}{\operatorname{ch} x} \text {. }
$$
Solution. Consider the given integral as an integral of a rational function of the variable $u=e^{x}$ :

$$
\begin{aligned}
& \int \frac{d x}{\operatorname{ch} x}=2 \int \frac{d x}{e^{x}+e^{-x}}=2 \int \frac{e^{x} d x}{e^{2 x}+1}=\left\{\begin{aligned}
e^{x} & =u \\
e^{x} d x & =d u
\end{aligned}\right\}= \\
&=2 \int \frac{d u}{1+u^{2}}=2 \operatorname{arctg} e^{x}+C
\end{aligned}
$$