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1.50 Example of an integral over the real line

The following improper integral converges:

\int_{-\infty}^{\infty}{{dx}\over{1+x^{2}}}=\pi.

Solution. Setting $x={\tan t}$ , we have $\frac{dx}{dt}={\sec^{2}t=x^{2}+1,}$ so

\int_{0}^{\infty}\frac{dx}{1+x^{2}}={\int_{0}^{\frac{\pi}{2}}dt=\frac{\pi}{2}}

and similarly for $\int_{-\infty}^{0}\frac{dx}{1+x^{2}}$ .