A calculus problem by chickpeas and tahini

Calculus Level 4

\[ \large 2\displaystyle\int _{ 0 }^{ \pi /2 }{ \ln { \left( 2\sin { x } \cos { x } \right) dx } } =-A\pi \ln { B } \]

The equation above holds true for positive integers \(A\) and \(B\). Find \(A+B\).

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