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Hard Integral

  1. Jun 17, 2014 #1
    1. The problem statement, all variables and given/known data

    Find the Integrals of [tex] \frac{x^2}{e^x+1}\\ \frac{x^3}{e^x+1} [/tex]


    2. Relevant equations

    Integration by parts

    3. The attempt at a solution

    I did IBP twice and it seemed to just get bigger and uglier and now I am stuck. I found the solutions online of the integrals but still do not know how to do them myself.
     
  2. jcsd
  3. Jun 17, 2014 #2

    verty

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    We had a similar thread not too long ago, have a look here.
     
  4. Jun 18, 2014 #3
    As said in the other thread, you need polylogarithms to express the final result. Instead, the definite integral from 0 to infinity can be easily evaluated and you have the following general result:

    $$\int_0^{\infty} \frac{x^{s-1}}{e^x+1}\,dx=\left(1-2^{1-s}\right)\zeta(s)\Gamma(s)$$
     
  5. Jul 1, 2014 #4
    Hey, sorry for the atrociously late reply. I am not familiar with the two functions in the general form you posted. I recognize the one as a gamma function but have never seen the other.
     
  6. Jul 1, 2014 #5

    Orodruin

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  7. Jul 1, 2014 #6
    I have another integral where the term in the exponential is divided by a constant T so it takes the form of [tex] \frac{x^2}{e^\frac{x}{T}+1}\\ \frac{x^3}{e^\frac{x}{T}+1} [/tex]

    Is there some way to get a new variation of this formula that can take these constants into account?
     
  8. Jul 1, 2014 #7

    Orodruin

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    Change variables to ##y = x/T##.
     
  9. Jul 1, 2014 #8
    Then we just use the same integration formula from pranav? Is this because the limits of integration involve an infinity so any constant T in the denominator will have no effect?
     
  10. Jul 1, 2014 #9

    Orodruin

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    It will have an effect, namely multiplying the result by a factor ##T^{n+1}##, where n is the exponent of x, since
    $$
    x^n\,dx = T^{n+1} y^n\,dy
    $$
     
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