Gaussian Integral

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Theorem

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


Proof

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

$\blacksquare$