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

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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 $a = 1$.

$\blacksquare$


Sources