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Express the following integral in terms of the gamma function

  1. Apr 14, 2010 #1
    1. The problem statement, all variables and given/known data
    This is actually part of a probability problem I'm thinking about. I'm trying to find the nth moment of a certain random variable in terms of the gamma function, which is basically equivalent to solving the following integral or expressing it in terms of the gamma function. Here is the integral:

    2. Relevant equations

    Mathematica:
    Integrate[(a/b^a) x^(n + a - 1) Exp[-(x/b)^a], {x, 0, Infinity}]

    Plain text:
    integral_0^infinity(a x^(n+a-1) e^(-(x/b)^a))/b^a dx

    a, b, and n are constants.

    wolfram alpha:

    http://www.wolframalpha.com/input/?i=integrate+%28%28a%2Fb^a%29*x^%28n%2Ba-1%29*e^%28-1%28x%2Fb%29^a%29%2Cx%2C0%2Cinf%29

    3. The attempt at a solution

    I tried integrating by parts, taking u = -x^n and dv/dx = ... everything else in the expression (since that can be integrated nicely using derivative is present substitution), I was left with -1 + an even nastier integral, so I'm not convinced this is the right method...
     
  2. jcsd
  3. Apr 14, 2010 #2

    vela

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    Don't actually try to integrate it. Use substitutions that turn the integral into the form of the gamma function.
     
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