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Integrate x^5 exp(x^2)

  1. Oct 15, 2009 #1
    1. The problem statement, all variables and given/known data
    I am losing my first year calculus skills :(
    I don't remember how to integrate x5 exp(x2).
    What is the fastest way?

    2. Relevant equations

    3. The attempt at a solution
    Maybe we need to integrate by parts? But how should I set u and dv?

    Thanks for any help!
  2. jcsd
  3. Oct 15, 2009 #2


    Staff: Mentor

    I would go with integration by parts. There are various possibilities for u and dv, but the one I would try first is u = x4, dv = xex2dx. A good strategy for integration by parts is to choose dv so that it is the most complicated thing that you can actually integrate.

    The goal is to get an integral with x to a power less than 5, and keep applying integration by parts until you get a fairly simple integral, like [tex]\int xe^{x^2}dx[/tex], which can be done by an ordinary substitution.
  4. Oct 15, 2009 #3
    If you are familiar with the trick of tabular integration for integration by parts, you can use the substitution u=x², du=2xdx, to get the solution faster without having to perform multiple integrations by parts.
  5. Oct 15, 2009 #4
    How many times do I have to integrate by parts?
  6. Oct 15, 2009 #5
    Depending on the substitution used, at least 2 times.
  7. Oct 16, 2009 #6
    This integral actually occurs in the middle of a statistics problem. If I know the expectation of a gamma distribution, can I possibly avoid integrating by parts in the above integral? If so, how?
  8. Oct 16, 2009 #7


    Staff: Mentor

    I don't know anything about that. You posed the problem, and you have gotten a couple of strategies for solving it. If you don't know how to do integration by parts, say so, and we'll help you out.
  9. Oct 16, 2009 #8
    Another method that works here is to substitute u=x^2 ...
    This will work for the integral of x^n exp(x^2) when n is odd.
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