Example 2.24. $\int_{0}^{\infty} \frac{d x}{4+x^{2}}$.
Solution. We will use formula (2.5)

$$
\int_{0}^{\infty} \frac{d x}{4+x^{2}}=\lim _{b \rightarrow \infty} \int_{0}^{b} \frac{d x}{4+x^{2}}=\left.\lim _{b \rightarrow \infty}(0.5 \operatorname{arctg}(0.5 x))\right|_{0} ^{b}=\frac{\pi}{4}
$$