# Who's up to the challenge? 52

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 $$a+b$$.



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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