Example 2. Calculate
(1) $(\operatorname{cis} x)^{\prime}$,
(2) $\left(\operatorname{sh}^{-1} x\right)^{\prime}$,
(3) $\int \frac{x}{\sqrt{x^{2}-1}} \, dx$;
(4) $\int \frac{d x}{1-x^{2}}$.
(1) $(\operatorname{ch} x)^{\prime}=(\operatorname{cosi} x)^{\prime}$
$$
=-i \sin i x=\frac{1}{i} \sin i x=\operatorname{sh} x \text {. }
$$
(2) $\left(\operatorname{sh}^{-1} x\right)^{\prime}=\left(\frac{1}{i} \sin ^{-1} i x\right)^{\prime}$
$$
=\frac{1}{i} \frac{i}{\sqrt{1-(i x)^{2}}}=\frac{1}{\sqrt{1+x^{2}}} .
$$
(3) $\int \frac{d x}{\sqrt{x^{2}-1}}=\int \frac{i d x}{\sqrt{(i x)^{2}-i^{2}}}$
$$
\begin{array}{l}
=i \int \frac{d x}{\sqrt{1-x^{2}}}=-i \cos ^{-1} x+C \\
=\frac{1}{i} \cos ^{-1} x+C=\operatorname{ch}^{-1} x+C .
\end{array}
$$
(4) $\int \frac{d x}{1-x^{2}}=\frac{1}{i} \int \frac{d(i x)}{1+(i x)^{2}}$
$$
=\frac{1}{i} \operatorname{tg}^{-1} i x+C=\operatorname{th}^{-1} x+C(|x|<1) .
\end{array}
$$