Gamma Function

  • Thread starter iceman
  • Start date
  • #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...!!
 

Answers and Replies

  • #2
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,967
19
Try the substitution with t again.
 

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