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
- 1970: N.G. de Bruijn: Asymptotic Methods in Analysis (3rd ed.) ... (previous) ... (next): $1.1$ What is asymptotics? $(1.1.1)$