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Two integrals I am trying to solve without closed form antiderivatives

  1. Feb 4, 2007 #1
    How do I solve the integral of the functions x*exp(-a(x-b)^2) and x^2(exp(-a(x-b)^2) where a and b are positive real numbers?

    I tried integration by parts and cannot think of how to find the integral. In addition, while I was able to find exp(-a(x-b)^2) in an integral table, the two functions I listed above, I was not able to find. Any hints, suggestions, or integration tables you can point me towards would be appreciated.
  2. jcsd
  3. Feb 4, 2007 #2


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    Set [tex]u=\sqrt{a}(x-b)\to{dx}=\frac{du}{\sqrt{a}}[/tex]
    Thus, for example, your first integrand transforms as:

    the first term is easily integrated, the other is seen to be the error function integral that is well tabulated.
  4. Feb 5, 2007 #3

    I looked up the second term in an integration table (no problem). The first term I was able to integrate using substitution v = u^2 and dv=2udu. Here is my problem though. Can I solve this term as a definite integral with the bounds positive infinity and negative infinity? If so, how? I know the first term is going to equal -1/2a*exp(-v). But if I set the limit equal to negative infinity, this goes infinity. Does this term have a definite value?
  5. Feb 6, 2007 #4


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    Nothing will go to infinity if you do the calculations correctly.
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