Integration Homework Help: Solving for the Relationship Between Two Integrals

[henryc09]

Homework Statement

Given that:

The integral between infinity and -infinity of

e^{-ax^2} dx = \sqrt{\pi/a}

show that

The integral between 0 and infinity of

x².e^{-ax^2} dx = 1/4\sqrt{\pi/a^3}

Homework Equations

The Attempt at a Solution

I have found the indefinite integrals of each, but cannot see how to show this relationship. Any help would be appreciated.

 

[flatmaster]

Well, I assume you integrate by parts twice. Then anypart of your integral with the Gaussian, you simply substitute in your given value. any part that can you Plug in your actual limits of inf and -inf, you plug in those numbers.