# Math Help - Another Gaussian integral...

1. ## Another Gaussian integral...

How do you solve this?

$
\int^{\infty}_{0} x^2 e^{-x^2}
$

2. Originally Posted by chutsu
How do you solve this?

$
\int^{\infty}_{0} x^2 e^{-x^2}
$
Integration by parts will reduce leave a known definite integral (splitting the integrand $(x)(xe^{-x^2})$ )

CB

3. What CaptainBlack said but I couldn't resist posting this...

Set $t=x^2$, $x = \sqrt{t}$, dx = $\frac{1}{2}t^{-\tfrac{1}{2}}$.

Integral becomes

$\frac{1}{2} \int_0^{\infty} t^{\tfrac{1}{2}} e^{-t} dt = \tfrac{1}{2}\Gamma(\tfrac{3}{2}) = \tfrac{1}{4}\Gamma(\tfrac{1}{2}) = \tfrac{1}{4}\sqrt{\pi}$

Gamma function - Wikipedia, the free encyclopedia