Example 2.38. $I=\int_{-1}^{1} \frac{d x}{\sqrt{1-x^{2}}}$.


Solution. According to formulas (2.16) and (2.17), we have

$$
\begin{gathered}
I=\lim _{\substack{\varepsilon \rightarrow+0 \\
\alpha \rightarrow+0}}\left[\int_{0}^{1-\varepsilon} \frac{d x}{\sqrt{1-x^{2}}}+\int_{-1+\alpha}^{0} \frac{d x}{\sqrt{1-x^{2}}}\right]= \\
=\lim _{\substack{\varepsilon \rightarrow+0 \\
\alpha \rightarrow+0}}[\arcsin (1-\varepsilon)-\arcsin 0+\arcsin 0-\arcsin (-1+\alpha)]= \\
=2 \arcsin 1=\pi .
\end{gathered}
$$