Lots of Sines

Calculus Level 3

0π4sinx dx+0π4sin2x dx+0π4sin3x dx+\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 II denote the numerical value of the series of integrals above. Find 1000I.\lfloor1000I\rfloor.

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