Definite Integral from 0 to 1 of Difference of Powers of x over Logarithm of x

Theorem

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

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

Proof
Let:


 * $x = e^{-u}$

We have, by Derivative of Exponential Function:


 * $\dfrac {\d x} {\d u} = -e^{-u}$

By Exponential Tends to Zero and Infinity:


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

and:


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

So: