Definite Integral from 0 to 1 of Even Powers of Logarithm of 1 - x over x

Theorem
Let $n \in \Z_{> 0}$ be a (strictly) positive integer.


 * $\ds \int_0^1 \map {\ln^{2n} } {\dfrac {1 - x} x} \rd x = \paren {-1}^{n + 1} B_{2 n} \paren {2^{2 n} - 2} \pi^{2 n}$


 * where $B_{2 n}$ is the $2 n$th Bernoulli number.

Proof
let:

and also:

Then:

From Sum of Infinite Geometric Sequence, we have:


 * $\ds \sum_{k \mathop = 0}^\infty \paren {-1}^k \paren {e^{-u} }^k = \dfrac 1 {1 + e^{-u} }$

Taking the derivative of both sides, we have:

Therefore: