# Integral proof ..

• Jul 15th 2010, 03:13 PM
BayernMunich
Integral proof ..
Hello,

Problem:

Show that:

$\int_0^1 x^m \, ln^n(x) \, dx = (-1)^n \, \dfrac{n!}{(m+1)^{n+1}}$

where m and n are nonnegative integers..
I want a start, maybe a substitution?
• Jul 15th 2010, 03:20 PM
BayernMunich
Am focus on the gamma function..

I used the substitution $u=-ln(x)$ ..

The integral will be: $\int_0^{\infty} (-1)^n t^n e^{-(m+1)t} \, dt$

I stopped here, any idea?
• Jul 15th 2010, 03:25 PM
BayernMunich
Thanks, I solved it =)

The last step is to substitute $z=(m+1)t$ and recalling that $\Gamma(n+1)=n!$ ..