Definite Integral of Exponential of minus a x squared from 0 to Infinity

Theorem

$\ds \int_0^\infty \map \exp {-a x^2} \rd x = \dfrac 1 2 \sqrt {\dfrac \pi a}$

for $a > 0$.

$\ds \int_0^\infty \map \exp {-x^2} \rd x = \dfrac {\sqrt \pi} 2$

Then:

$\ds \int \map \exp {-a x^2} \rd x$

$=$

$\ds \int \map \exp {-\paren {\sqrt a x}^2} \rd x$

$\ds $

$=$

$\ds \dfrac 1 {\sqrt a} \int \map \exp {-\paren {\sqrt a x}^2} \rd \paren {\sqrt a x}$

$\ds \leadsto \ \ $

$\ds \int_0^\infty \map \exp {-a x^2} \rd x$

$=$

$\ds \dfrac 1 {\sqrt a} \int_0^\infty \map \exp {-\paren {\sqrt a x}^2} \rd \paren {\sqrt a x}$

as the limits of integration stay the same

$\ds $

$=$

$\ds \dfrac 1 {\sqrt a} \dfrac {\sqrt \pi} 2$

$\ds $

$=$

$\ds \dfrac 1 2 \sqrt {\dfrac \pi a}$

$\blacksquare$