Definite Integral to Infinity of Exponential of -a x^2

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Theorem

$\ds \int_0^\infty e^{-a x^2} \rd x = \frac 1 2 \sqrt {\frac \pi a}$

where $a$ is a strictly positive real number.


Proof

\(\ds \int_0^\infty e^{-a x^2} \rd x\) \(=\) \(\ds \int_0^\infty e^{-\paren {\sqrt a x}^2} \rd x\)
\(\ds \) \(=\) \(\ds \frac 1 {\sqrt a} \int_0^\infty e^{-t^2} \rd t\) substituting $t = \sqrt a x$
\(\ds \) \(=\) \(\ds \frac 1 2 \sqrt {\frac \pi a}\) Integral to Infinity of $e^{-t^2}$

$\blacksquare$


Sources