Definite Integral from 0 to 1 of Arctangent of x over x
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Theorem
- $\ds \int_0^1 \frac {\arctan x} x \rd x = G$
where $G$ denotes Catalan's constant.
Proof
\(\ds \int_0^1 \frac {\arctan x} x \rd x\) | \(=\) | \(\ds \int_0^1 \frac 1 x \paren {\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {2 n + 1} } \rd x\) | Power Series Expansion for Real Arctangent Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {2 n + 1} \int_0^1 x^{2 n} \rd x\) | Power Series is Termwise Integrable within Radius of Convergence | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {2 n + 1} \intlimits {\frac {x^{2 n + 1} } {2 n + 1} } 0 1 \rd x\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}^2}\) | evaluation of definite integral | |||||||||||
\(\ds \) | \(=\) | \(\ds G\) | Definition of Catalan's Constant |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definite Integrals involving Trigonometric Functions: $15.63$