Expectation of Logistic Distribution/Lemma 1

Lemma for Expectation of Logistic Distribution

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

Proof

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

\(=\)

\(\ds \bigintlimits {\paren {u - 1} \map \ln {1 - u} - u } 0 1\)

Corollary to Primitive of Logarithm of x

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

\(\ds -1\)

$\blacksquare$