Variance of Logistic Distribution/Lemma 1

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Lemma for Variance of Logistic Distribution

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


Proof

let:

\(\ds x\) \(=\) \(\ds \paren {1 - u}\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d x} {\d u}\) \(=\) \(\ds -1\) Derivative of Power
\(\ds \int_{\to 0}^{\to 1} \map {\ln^2} {1 - u} \rd u\) \(=\) \(\ds -\int_{\to 1}^{\to 0} \map {\ln^2} x \rd x\)
\(\ds \) \(=\) \(\ds \int_{\to 0}^{\to 1} \map {\ln^2} x \rd x\)
\(\ds \) \(=\) \(\ds \bigintlimits {x \ln^2 x } 0 1 - 2 \int_{\to 0}^{\to 1} \map \ln x \rd x\) Primitive of Power of Logarithm of x
\(\ds \) \(=\) \(\ds \paren {\paren {0 - 0} - 2 \paren {-1} }\) Expectation of Logistic Distribution:Lemma 2
\(\ds \) \(=\) \(\ds 2\)

$\blacksquare$