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