# Variance of Logistic Distribution/Lemma 4

$\ds \int_{\to 0}^{\to 1} \map {\ln^2} {\dfrac {1 - u} u} \rd u = \dfrac {\pi^2} 3$
 $\ds \int_{\to 0}^{\to 1} \map {\ln^2} {\dfrac {1 - u} u} \rd u$ $=$ $\ds \int_{\to 0}^{\to 1} \paren {\map {\ln} {1 - u} \rd u - \map {\ln} u}^2 \rd u$ Difference of Logarithms $\ds$ $=$ $\ds \int_{\to 0}^{\to 1} \map {\ln^2} {1 - u} \rd u - 2 \int_{\to 0}^{\to 1} \map \ln {1 - u} \map \ln u \rd u + \int_{\to 0}^{\to 1} \map {\ln^2} u \rd u$ $\ds$ $=$ $\ds 2 - 2 \paren {2 - \dfrac {\pi^2} 6} + 2$ Lemma 1, Lemma 2 and Lemma 3 $\ds$ $=$ $\ds \dfrac {\pi^2} 3$
$\blacksquare$