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Two Integrals

  1. Aug 3, 2008 #1
    I have tried many hours on the following integrals and would appreciate any help from you.


    1. Int{x=0 to infinity}(exp(-x)*Ln(x)*Ln(x)dx)

    2. Int{x=0 to Infinity}(exp(-x*x)*Ln(x)dx)



    Any idea guys?
     
  2. jcsd
  3. Aug 4, 2008 #2

    CompuChip

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    Quickly checked it with Mathematica... the indefinite integrals are no fun (they involve hypergeometric functions and error functions) but the definite integrals come out relatively nicely.

    I suppose you will want to somehow use
    [tex]\int_0^\infty e^{-t} \ln(t) \, \mathrm dt = -\gamma[/tex]
    (the Euler gamma). The second one will also involve a Gaussian integral.

    Will think a bit more...
     
  4. Aug 4, 2008 #3

    HallsofIvy

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    This has nothing to do with "number theory". I am moving it to Calculus and Analysis.
     
  5. Aug 5, 2008 #4
    For the first integral ,

    it can be shown to be = Gamma''(1) and

    - Euler's Constant = Gamma'(1)

    where Gamma(x)=Gamma Integral

    I just don't know how to use the above facts.
     
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