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Calculus Level 5

\[\int _{ 0 }^{ 1 }{ \Gamma \left(1+\frac { x }{ 2 } \right)\Gamma \left(1-\frac { x }{ 2 } \right) \, dx } =\frac { aG }{ b\pi } \]

If the equation holds true for coprime positive integers \(a\) and \(b\), find \(a+b\).

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Notations:

  • \( \Gamma(\cdot) \) denotes the Gamma function.

  • \(G\) denote Catalan's constant, \(\displaystyle G = \sum_{n=0}^\infty \dfrac{ (-1)^n}{(2n+1)^2} \approx 0.916 \).

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