∫x^(4)e^(-x^(2)/2)dx in the limit [-∞,∞]
I tried to solve this problem many times but i just couldn't . Can someone help me ?
write down the even function as
$\displaystyle \int_{-\infty}^{\infty} x^4 e^{-\frac{x^2}{2}} dx$
$\displaystyle =2 \int_0^{\infty} x^4 e^{-\frac{x^2}{2}} dx$
let $\displaystyle u=x^2 \implies du = 2x dx$. also, $\displaystyle x=0 \implies u=0; and x=\infty \implies u=\infty$so you have
$\displaystyle \int_0^{\infty} u^{3/2} e^{-\frac{u}{2}} du$
try solving from here...use gamma function
Using Laplace transform:
$\displaystyle \mathcal{L}_x\left[\int_{-\infty }^{\infty } x^4 e^{-\frac{x^2}{2}} \, dx\right](s) $ = $\displaystyle \frac{3 \sqrt{2 \pi }}{s}$
Taking inverse Laplace:
$\displaystyle \mathcal{L}_s^{-1}\left[\frac{3 \sqrt{2 \pi }}{s}\right](x)$ = $\displaystyle 3 \sqrt{2 \pi }$