Lots of Sines

Calculus Level 4

\[\large \displaystyle\int_0^\frac{\pi}{4}\sin x\text{ } \mathrm dx+\displaystyle\int_0^\frac{\pi}{4}\sin^2 x\text{ } \mathrm dx+\displaystyle\int_0^\frac{\pi}{4}\sin^3 x\text{ } \mathrm dx+\ldots\]

Let \(I \) denote the numerical value of the series of integrals above. Find \(\lfloor1000I\rfloor.\)

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