Gaussian integral
= Gaussian integral
{c}
{title2=$\int_{-\infty}^{\infty}e^{-x^2}\,dx=\sqrt\pi$}
{wiki}
The Gaussian integral is
$$
\int_{-\infty}^{\infty}e^{-ax^2}\,dx=\sqrt{\frac\pi a},
\qquad a>0.
$$
Squaring the integral and changing to polar coordinates proves the formula.