Expectation of Logistic Distribution/Lemma 2

Lemma for Expectation of Logistic Distribution

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

Proof

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

\(=\)

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

Primitive of Logarithm of x

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

\(\ds -1\)

$\blacksquare$