Definite Integral to Infinity of Reciprocal of x Squared plus a Squared/Corollary

Theorem

 * $\ds \int_0^\infty \dfrac {\d x} {1 + x^2} = \frac \pi 2$

for $a \ne 0$.

Proof
From Definite Integral to Infinity of Reciprocal of x Squared plus a Squared:


 * $\ds \int_0^\infty \dfrac {\d x} {x^2 + a^2} = \frac \pi {2 a}$

which holds for for $a \ne 0$.

The result follows by setting $a = 1$.