Definite Integral from 0 to 1 of Logarithm of x over One minus x

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Theorem

$\ds \int_0^1 \frac {\ln x} {1 - x} \rd x = -\frac {\pi^2} 6$


Proof

\(\ds \int_0^1 \frac {\ln x} {1 - x} \rd x\) \(=\) \(\ds \int_0^1 \ln x \paren {\sum_{n \mathop = 0}^\infty x^n} \rd x\) Sum of Infinite Geometric Sequence
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \paren {\int_0^1 x^n \ln x \rd x}\) Fubini's Theorem
\(\ds \) \(=\) \(\ds -\sum_{n \mathop = 0}^\infty \frac {\map \Gamma 2} {\paren {n + 1}^2}\) Definite Integral from $0$ to $1$ of $x^m \paren {\ln x}^n$: $n \gets 1$, $m \gets n$
\(\ds \) \(=\) \(\ds -\sum_{n \mathop = 1}^\infty \frac 1 {n^2}\) shifting the index, Gamma Function Extends Factorial
\(\ds \) \(=\) \(\ds -\frac {\pi^2} 6\) Basel Problem

$\blacksquare$


Also see


Sources

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