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Math Help - Heat diffusion, Gaussian #2

  1. #1
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    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
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  2. #2
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    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
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  3. #3
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    How do you evaluate (-x/2a)*e^(-ax^2) from (-inf, inf)?
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  4. #4
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    Quote Originally Posted by tbyou87 View Post
    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.
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