Definite Integral to Infinity of Reciprocal of x Squared plus a Squared/Proof 2
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Theorem
- $\ds \int_0^\infty \dfrac {\d x} {x^2 + a^2} = \frac \pi {2 a}$
for $a \ne 0$.
Proof
| \(\ds \int_0^\infty \dfrac {\d x} {x^2 + a^2}\) | \(=\) | \(\ds \frac \pi { 2 a^{2 - 1} } \csc \left({\frac \pi 2}\right)\) | Definite Integral to Infinity of $\dfrac 1 {1 + x^n}$: Corollary | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac \pi {2 a}\) | Sine of Right Angle |
$\blacksquare$