Definite Integral from 0 to 1 of Logarithm of x by Logarithm of One plus x
Jump to navigation
Jump to search
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 |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definite Integrals involving Logarithmic Functions: $15.95$