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Calculus Level 5

\[\int _{ 0 }^{ \pi /2 }{ \sqrt { \sin { x } } \, dx } =\dfrac { \sqrt { A } }{ \sqrt { \pi } } \left(\Gamma \left( \frac { C }{ D } \right)\right)^B \]

The equation above holds for positive integers \(A,B,C\) and \(D\) with \(C,D\) coprime. Find \( A+B+C+D\).

Notation: \( \Gamma(\cdot) \) denotes the Gamma function.

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