Variance of Logistic Distribution/Lemma 3

Lemma for Variance of Logistic Distribution

 * $\ds \int_{\to 0}^{\to 1} \map \ln u \map \ln {1 - u} \rd u = 2 - \dfrac {\pi^2} 6$

Proof
From Corollary to Power Series Expansion for Logarithm of 1 + x we have:
 * $\ds \ln \paren {1 - x} = -\sum_{n \mathop = 1}^\infty \dfrac {x^n} n$

Therefore: