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Writing complicated integral in terms of the Gamma function

  1. Apr 12, 2015 #1
    1. The problem statement, all variables and given/known data
    Write ##\int_{0}^{1}x^2(ln\frac{1}{x})^3 dx## in terms of the gamma function

    2. Relevant equation
    ##\Gamma(p+1)=p\Gamma(p)##
    3. The attempt at a solution
    Say ##x=e^{-u}## one would eventually obtain the integral

    ##\int_{-\infty}^{0} u^3 e^{-u} du##

    STEPS:
    ##x=e^{-u}## ##e^{u}=1/x## ##u=ln(1/x)## ##du=xdx##

    ##\int_{0}^{1}x^2(ln\frac{1}{x})^3 dx=\int_{0}^{-\infty} e^{-u} x u^3 \frac{du}{x}=\int_{-\infty}^{0} u^3 e^{-u} du##

    Which wants to look like a gamma function but isnt because of the limits.
     
    Last edited: Apr 12, 2015
  2. jcsd
  3. Apr 12, 2015 #2

    mfb

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    I don't know what went wrong with the substitution (can you show the steps?), but the last integral diverges.
     
  4. Apr 12, 2015 #3
    sure I'll edit to add.
     
  5. Apr 12, 2015 #4
    I've added it now mfb.
     
  6. Apr 13, 2015 #5

    mfb

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    I don't think that is right.
    And your integral limits look wrong, too.
    x=1 corresponds to u=0, but x=0 corresponds to a different value for u.
     
  7. Apr 13, 2015 #6
    Yeah I've corrected it now. Thanks!
     
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