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

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Theorem

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


Proof

\(\ds \int_0^1 \ln x \map \ln {1 + x} \rd x\) \(=\) \(\ds \int_0^1 \ln x \paren {\sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {x^n} n} \rd x\) Power Series Expansion for $\map \ln {1 + x}$
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } n \paren {\int_0^1 x^n \ln x \rd x}\) Fubini's Theorem
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n} n \paren {\frac {\map \Gamma 2} {\paren {n + 1}^2} }\) Definite Integral from $0$ to $1$ of $x^m \paren {\ln x}^n$
\(\ds \) \(=\) \(\ds 1! \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n} {n \paren {n + 1}^2}\) Gamma Function Extends Factorial
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n} n - \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n} {n + 1} - \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n} {\paren {n + 1}^2}\) partial fraction expansion
\(\ds \) \(=\) \(\ds -\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } n - \sum_{n \mathop = 2}^\infty \frac {\paren {-1}^{n - 1} } n - \sum_{n \mathop = 2}^\infty \frac {\paren {-1}^{n - 1} } {n^2}\) shifting indexes
\(\ds \) \(=\) \(\ds -\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } n - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } n - 1} - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } {n^2} - 1}\)
\(\ds \) \(=\) \(\ds 2 - 2 \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } n - \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } {n^2}\)
\(\ds \) \(=\) \(\ds 2 - 2 \ln 2 - \frac {\pi^2} {12}\) Newton-Mercator Series for $\ln 2$, Sum of Reciprocals of Squares Alternating in Sign

$\blacksquare$


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