- #1

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...!!