Variance of Logistic Distribution/Lemma 2

Lemma for Variance of Logistic Distribution

$\ds \int_{\to 0}^{\to 1} \map {\ln^2} u \rd u = 2$

Proof

\(\ds \int_{\to 0}^{\to 1} \map {\ln^2} u \rd u\)

\(=\)

\(\ds \bigintlimits {u \ln^2 u } 0 1 - 2 \int_{\to 0}^{\to 1} \map \ln u \rd u\)

Primitive of Power of Logarithm of x

\(\ds \)

\(=\)

\(\ds \paren {\paren {0 - 0} - 2 \paren {-1} }\)

Expectation of Logistic Distribution:Lemma 2

\(\ds \)

\(=\)

\(\ds 2\)

$\blacksquare$