Integration of products of the Gauss Error Function

[Unredeemed]

Homework Statement

Given that the integral from negative to positive infinity of e^(-(x^2))dx is equal to sqrt pi. Find the values of the integrals from negative to positive infinity of e^(-u*(x^2))dx and (x^2)*e^(-(x^2))dx.

Homework Equations

The Attempt at a Solution

I did the first one and got that it would be sqrt(pi/u).
But I honestly didn't know where to begin for the second. I drew graphs of y=e^(-(x^2)) and y=(x^2)*e^(-(x^2)), but it didn't help me massively.

I noticed that the graphs acted very similarly if -1/e>x or 1/e<x. But that might have just been how I'd drawn my graphs.

Can anyone help?

NB: I didn't know it was called the "Gauss Error Function" until i googled it, so this question assumes no knowledge of that.

 

[hunt_mat]

There is a trick:
$$\int_{-\infty}^{\infty}e^{-ux^{2}}dx=\sqrt{\frac{\pi}{u}}$$
Differentiate the above expression w.r.t. u to obtain the answer to the second question.