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

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Theorem

$\ds \int_0^1 \frac {\ln x} {1 + x} \rd x = -\frac {\pi^2} {12}$


Proof

\(\ds \int_0^1 \frac {\ln x} {1 + x} \rd x\) \(=\) \(\ds \int_0^1 \frac {\ln x} {1 - \paren {-x} } \rd x\)
\(\ds \) \(=\) \(\ds \int_0^1 \ln x \paren {\sum_{n \mathop = 0}^\infty \paren {-x}^n} \rd x\) Sum of Infinite Geometric Sequence
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \int_0^1 x^n \ln x \rd x\) Fubini's Theorem
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n + 1} \map \Gamma 2} {\paren {n + 1}^2}\) Definite Integral from $0$ to $1$ of $x^m \paren {\ln x}^n$
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n 1!} {n^2}\) Gamma Function Extends Factorial, shifting the index
\(\ds \) \(=\) \(\ds -\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } {n^2}\) writing $-\paren {-1}^{n + 1} = \paren {-1}^{n + 2} = \paren {-1}^2 \paren {-1}^n = \paren {-1}^n$
\(\ds \) \(=\) \(\ds -\frac {\pi^2} {12}\) Sum of Reciprocals of Squares Alternating in Sign

$\blacksquare$


Sources

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