Integration problem

Calculus Level 4

\[I=\displaystyle\int \limits^{\frac{\pi }{4} }_{0}\sin \left( x\right) dx +\displaystyle\int \limits^{\frac{\pi }{4} }_{0}\sin ^{2}\left( x\right) dx + \displaystyle\int \limits^{\frac{\pi }{4} }_{0}\sin ^{3}\left( x\right) dx + \displaystyle\int \limits^{\frac{\pi }{4} }_{0}\sin ^{4}\left( x\right) dx+\cdots \] Given that \[I=\sqrt{A} - \frac{\pi}{B} .\] Find \(A+B\).

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