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## Homework Statement

For some Gaussian distribution, let's say [itex]e^{-x^2}[/itex] times a constant, I want to find the expectation value of [itex]x^2[/itex]. In other words, I want to evaluate:

[tex]\int^{\infty}_{-\infty} e^{-x^2}x^2dx[/tex]

## Homework Equations

Integration by parts:

[tex]\int udv = uv - \int vdu[/tex]

## The Attempt at a Solution

I tried a few things actually, but none of them got anywhere. One of the routes I attempted was u-substitution followed by integration by parts. First, since the function is even, you set it equal to:

[tex]2\int^{\infty}_{0} e^{-x^2}x^2dx[/tex]

Letting [itex]a = x^2, da = 2xdx[/itex], we get:

[tex]\int^{\infty}_{0} e^{-a}\sqrt{a}da[/tex]

Then, letting [itex]u = \sqrt{a}, dv = e^{-a}da, du = \frac{1}{2\sqrt{a}}da, v = -e^{-a}[/itex]:

[tex]\int e^{-a}\sqrt{a}da = -\sqrt{a}e^{-a} + \int e^{-a}\frac{1}{2\sqrt{a}}da[/tex]

And here I get stuck. Anybody have hints?