Definite Integral from 0 to 1 of Logarithm of One plus x over x
Jump to navigation
Jump to search
Theorem
- $\ds \int_0^1 \frac {\map \ln {1 + x} } x \rd x = \frac {\pi^2} {12}$
Proof
| \(\ds \int_0^1 \frac {\map \ln {1 + x} } x \rd x\) | \(=\) | \(\ds \int_0^1 \frac 1 x \paren {\sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {x^n} n} \rd x\) | Power Series Expansion for $\map \ln {1 + x}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \paren {\frac {\paren {-1}^{n - 1} } n \int_0^1 x^{n - 1} \rd x}\) | Power Series is Termwise Integrable within Radius of Convergence | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } {n^2}\) | Primitive of Power | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\pi^2} {12}\) | 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: $15.93$