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Gamma Function

  1. Aug 13, 2003 #1
    Hello, can anyone please me here?

    I need to prove that

    int(x^a(lnx)^b.dx= (-1)^b/((1+a)^b+1)*Gamma(b+1)

    by making the substitution x=e^-y

    this is what I have done so far:

    x=e^-y -> y=-lnx

    x=0 -> y=-(-00) =+00
    x=1 -> y=0

    dy/dx = -1/x -> dx=-xdy =-e^-ydy

    then the integral becomes

    int[e^(-ay)*(-y)^b*(-e^-y)dy, lower lim->+00, upper lim-> 0
    = (-1)^b*int[e^-(a+1)y*y^bdy.

    I then made a substituion t=(a+1)y
    so integral becomes

    (-1)^b*int[e^-t*y^bdy]

    this is where I get a little bit lost...!!
     
  2. jcsd
  3. Aug 13, 2003 #2

    Hurkyl

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    Staff Emeritus
    Science Advisor
    Gold Member

    Try the substitution with t again.
     
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