# Thread: Need help in Integral

1. ## Need help in Integral

∫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 ?

2. ## Re: Need help in Integral

Originally Posted by nirmal019
∫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

$\int_{-\infty}^{\infty} x^4 e^{-\frac{x^2}{2}} dx$

$=2 \int_0^{\infty} x^4 e^{-\frac{x^2}{2}} dx$

let $u=x^2 \implies du = 2x dx$. also, $x=0 \implies u=0; and x=\infty \implies u=\infty$so you have

$\int_0^{\infty} u^{3/2} e^{-\frac{u}{2}} du$

try solving from here...use gamma function

3. ## Re: Need help in Integral

Thank you so much I have solved the promblem . Actually I never thought of the Gamma function and trying different methods :P

4. ## Re: Need help in Integral

Using Laplace transform:

$\mathcal{L}_x\left[\int_{-\infty }^{\infty } x^4 e^{-\frac{x^2}{2}} \, dx\right](s)$ = $\frac{3 \sqrt{2 \pi }}{s}$

Taking inverse Laplace:

$\mathcal{L}_s^{-1}\left[\frac{3 \sqrt{2 \pi }}{s}\right](x)$ = $3 \sqrt{2 \pi }$