Definite Integral from 0 to 1 of Power of x by Power of Logarithm of x/Proof 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 have by Exponential of Zero:


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

We also have, by Exponential Tends to Zero and Infinity:


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

So: