An old IIT problem

Calculus Level 3

\[\int_{\pi }^{2\pi} \lfloor 2 \sin x \rfloor \, dx\]

If the value of the integral above is of the form \(\dfrac{A\pi }{B}\), where \(A\) and \(B\) are coprime integers, find \(\left | A \right |+\left | B \right |\).

Notations:

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