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Definite Integral

  1. Sep 7, 2008 #1
    1. The problem statement, all variables and given/known data

    solve the integral: ∫_(-∞)^∞▒〖x^2 e^(-λ(x-a)^2 ) 〗 dx
    where λ and a are positive real constants

    3. The attempt at a solution

    I tried integration by parts with and without y-substitution but neither worked for me.

    Without substitution, I set up the integral to look like:
    ∫_(-∞)^∞▒〖xe^(-λx^2 )•xe^λa(2x-a) 〗 dx

    u=xe^λa(2x-a) and dv=xe^(-λx^2 ) dx

    after doing this a few times I realized it wouldn't work.

    For y-substitution I used y = x-a. ∫_(-∞)^∞▒〖(y+a)^2 e^(-λ(y)^2 ) 〗
    I then tried to integrate this by parts with u=(y+a)^2 and dv=e^(-λy^2 )
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Sep 7, 2008 #2


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    Homework Helper

    I assume you want to solve
    [tex]\int_{-\infty}^{\infty} x^2 e^{-\lambda (x - a)^2 } \, dx[/tex]

    In that case, try differentiation of an ordinary Gaussian integral w.r.t [itex]\lambda[/itex] (twice).
  4. Sep 7, 2008 #3
    Yea, I didn't have it in the right form. It's for a physics class, so the books says to use a table to help. I think I will try to solve it out anyway. Thanks for the help.
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