Gaussian Integrals

$\displaystyle \int \exp\left(\frac{- x^2}{2 \sigma^2}\right) \, dx$	$\textstyle =$	$\displaystyle \sqrt{2\pi \sigma^2}$	(56)
$\displaystyle \int x \exp\left(\frac{- x^2}{2 \sigma^2}\right) \, dx$	$\textstyle =$	$\displaystyle 0$	(57)
$\displaystyle \int x^2 \exp\left(\frac{- x^2}{2 \sigma^2}\right) \, dx$	$\textstyle =$	$\displaystyle \sigma^2 \sqrt{2\pi \sigma^2}$	(58)