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Integration problem to calculate partition function of a gase in a blackbody

  1. Jan 11, 2009 #1
    1. The problem statement, all variables and given/known data
    This is the integration i have to solve
    I=[tex]\int x^{2}In(1-exp(-ax))dx[/tex]
    integration is from zero to infinity

    3. The attempt at a solution
    I know that it should be solved with integration by parts
    du=[a exp(-ax)] / [1-exp(-ax)]
    v=x[tex]^{3}[/tex] /3
    when i put this into the integration formula
    I=u*v-[tex]\int v*du[/tex]
    it becomes more complicated
    I=In(1-exp(-ax))*x[tex]^{3}[/tex]/3 - [tex]\int dx * (x^3/3) * [a exp(-ax)] / [1-exp(-ax)][/tex]
    so what should i do after this , i can't figure it out, am i doing it wrong?
  2. jcsd
  3. Jan 11, 2009 #2
    Use Gamma function and expand integrand using geometric series.
  4. Jan 11, 2009 #3
    if i do that i should calculate the series from zero to infinity. when will i know that i should stop?
  5. Jan 11, 2009 #4
    You should get some good looking series at the end, and you can get a closed form expression. Usually, one ends up with some kind of zeta functions.

    You might get things like:
    [tex]\sum \frac{1}{n^2}=\frac{\pi^2}{6}[/tex]

    [tex]\sum \frac{1}{n^4}=\frac{\pi^4}{90}[/tex]

    which can be simplified by consulting some tables.
  6. Jan 11, 2009 #5
    well thank you for your help. now i will give it a try.
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