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Integration Problem

  1. Sep 17, 2008 #1
    I feel really dumb for asking this, because I know it's something simple I'm just not seeing. Ok, given that

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

    how to I find

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

    I have tried the substitution u=x^2, and integration by parts, but nothing is working. Any help? Thanks
  2. jcsd
  3. Sep 17, 2008 #2


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    The easy way to do this problem is to generalize your first integral. Can you show the integral of exp(-ax^2) is sqrt(pi/a) for a>0? (Use a substitution x=sqrt(a)*u). Now differentiate that with respect to a. Finally put a=1 again.
  4. Sep 17, 2008 #3
    This might help:

    d/dx (x*[e^-x^2]) = ...
    Solve it and then integrate!
  5. Sep 17, 2008 #4


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    Homework Helper

    Integration by parts should work. You have

    \int_{-\infty}^\infty x^2 e^{-x^2} \, dx = \sqrt \pi


    u = x, \quad dv =x e^{-x^2} dx


    \int u \, dv = uv - \int v \, du

    should, with careful attention to the [tex] uv [tex] term at the infinities, work fine.
  6. Sep 18, 2008 #5
    Thanks everyone.
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