# Math Help - Heat diffusion, Gaussian #2

1. ## Heat diffusion, Gaussian #2

Hi,
I have to evaluate intergral from -inf, inf of (x^2)(e^(-ax^2))dx.
I am supposed to use the fact that integral from -inf, inf of (e^(-ax^2)) = sqrt(pi/a).

I'm pretty sure you need to to it by parts so that you can get x^2, 2x and 2 and then some form of the other integral. I'm just not sure how to evaluate it.

Thanks

2. Here $a>0$:

$\int_{-\infty}^{\infty} \underbrace{x}_{u} \cdot \underbrace{\left( x e^{-ax^2} \right)}_{v'} dx = - \frac{x}{2a}\cdot e^{-ax^2} \big|_{-\infty}^{\infty} + \int_{-\infty}^{\infty} \frac{1}{2a}e^{-ax^2} dx$

3. How do you evaluate (-x/2a)*e^(-ax^2) from (-inf, inf)?

4. Originally Posted by tbyou87
How do you evaluate (-x/2a)*e^(-ax^2) from (-inf, inf)?
One thing you should know about exponentials is that they are faster than polynomials. So $e^{-ax^2}$ goes to zero faster than $x$ to $\infty$. Thus, the overall product is zero.