How to prove $\int_Xexp(-\frac12X'AX)dX=(2\pi)^\frac{n}2{\lvert{A}\rvert}^{-\frac12}$
If $A$ is positive definite, then $\int_{\mathbb{R}^n}\mathrm{e}^{-\langle Ax,x\rangle}\text{d}x=\left|\det\left({\pi}^{-1}A\right)\right|^{-1/2}$