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

Corollary to Definite Integral from 0 to 1 of Even Powers of Logarithm of 1 - x over x
Let $n \in \Z_{> 0}$ be a (strictly) positive integer.


 * $\ds \int_0^1 \map {\ln^{2n} } {\dfrac {1 - x} x} \rd x = 2 \map \zeta {2 n} \paren {2 n}! \paren {1 - \dfrac 1 {2^{2n-1} } }$