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Calculus Level 5

01Γ(1+x2)Γ(1x2)dx=aGbπ\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 aa and bb, find a+ba+b.

Notations:

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

  • GG denote Catalan's constant, G=n=0(1)n(2n+1)20.916\displaystyle G = \sum_{n=0}^\infty \dfrac{ (-1)^n}{(2n+1)^2} \approx 0.916 .

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