Definite Integral from 0 to 1 of Power of x by Power of Logarithm of x

Theorem

 * $\displaystyle \int_0^1 x^m \paren {\ln x}^n \rd x = \frac {\paren {-1}^n \map \Gamma {n + 1} } {\paren {m + 1}^{n + 1} }$

where $n$ and $m$ are real numbers with $n, m > -1$.

Proof
Let:


 * $\displaystyle x = \map \exp {-\frac u {m + 1} }$

Then, by Derivative of Exponential Function:


 * $\displaystyle \frac {\d x} {\d u} = -\frac 1 {m + 1} \map \exp {-\frac u {m + 1} }$

We also have by Exponential Tends to Zero and Infinity:


 * as $x \to 1$, $u \to 0$

and:


 * as $x \to 0$, $u \to \infty$

So: