Hello!

Just a quick mathematics problem I hope some of you can help me with.

What is the integral of
x^2 * exp[-(x^2/a^2)]
from -infinity to infinity?

I'm fairly certain we've covered this specific integral in one my maths courses, and there was a certain trick to it, but I've completely forgotton and I'm completely stumped. I know that the Gaussian integral exp[-(x^2)] gives me root pi, and the derivation is fairly complicated, but I remember that integrating x^2 exp[-(x^2)] was much simpler. But I don't remember how. Help please!

something like substitute in u=x^2, then integrate by parts, but i don't have the time to work it out right now

if I typed it in right, mathematica says (1/2)a^3*sqrt(pi)

It's been a while since I've done this, but it does sound like an "integration by parts" problem.

http://en.wikipedia.org/wiki/Integration_by_parts

Try googling around if wikipedia fails. There's some solid math examples around.

I thought I was going somewhere with e[sup]-(x[sup]2[/sup]/a[sup]2[/sup])[/sup] = 1/(e[sup]x[/sup] - e[sup]a[sup]2[/sup][/sup]) but I wasn't

Fake edit: Well maybe. I'm not sure I'm correct in everything but it's an attempt.