Gaussian Integral

Theorem

$\ds \int_{-\infty}^\infty e^{-x^2} \rd x = \sqrt \pi$


Proof

\(\ds \int_{-\infty}^\infty e^{-x^2} \rd x\) \(=\) \(\ds 2 \int_0^\infty e^{-x^2} \rd x\) Definite Integral of Even Function
\(\ds \) \(=\) \(\ds \frac {2 \sqrt \pi} 2\) Integral to Infinity of Exponential of -t^2
\(\ds \) \(=\) \(\ds \sqrt \pi\)

$\blacksquare$

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