- #1
iceman
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...!
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...!