Hello,

Problem:

Show that:

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

where m and n are nonnegative integers..

I want a start, maybe a substitution?

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- Jul 15th 2010, 03:13 PMBayernMunichIntegral proof ..
**Hello,**

**Problem:**

**Show that:**

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

**where m and n are nonnegative integers..**

**I want a start, maybe a substitution?** - Jul 15th 2010, 03:20 PMBayernMunich
**Am focus on the gamma function..**

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

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

**I stopped here, any idea?** - Jul 15th 2010, 03:25 PMBayernMunich
**Thanks, I solved it =)**

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