Help with Gaussian integration problem please

[rdioface]

Help with Gaussian integration problem please :)

Homework Statement

Compute the improper integral
$\int^{\infty}_{-\infty}x^{2}e^{-x^2}dx$
given
$\int^{\infty}_{-\infty}e^{-x^2}dx=\sqrt{\pi}.$

Homework Equations

Just the rule for doubly-improper integrals I guess:
$\int^{\infty}_{-\infty}f(x)dx=\lim_{a\rightarrow-\infty}\lim_{b\rightarrow\infty}\int^{b}_{a}f(x)dx$

$erf(x)$ is beyond the scope of this course and thus cannot be utilized in any way.

The Attempt at a Solution

I can't see any substitutions that would make things easier, and integration by parts doesn't seem useful (choosing to derive $u=e^{-x^2}$ and integrate $dv=x^{2}dx$ will never simplify or isolate $e^{-x^2}$, and you can't choose to integrate $dv=e^{-x^2}dx$ and derive $u=x^2$ because we are only given the special case of $\int^{\infty}_{-\infty}e^{-x^2}dx$ and not a general antiderivative.

 

[Hootenanny]

I'll give you a little hint.

$$\frac{d}{dx}e^{-x^2} = -2xe^{-x^2}$$

Hence,

$$\begin{aligned} \int_{-\infty}^\infty x^2 e^{-x^2} \text{d}x & = -\frac{1}{2}\int_{-\infty}^\infty x (-2x e^{-x^2}) \text{d}x \\ & = - \frac{1}{2}\int_{-\infty}^\infty x \frac{d}{dx}e^{-x^2}dx\end{aligned}$$

Can you now use integration by parts?