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Let
I=∫5π/87π/8∑n=0∞sin2nx dxI = \int_{5\pi/8}^{7\pi/8} \sqrt{\sum_{n=0}^{\infty} \sin^{2n} x} \ dxI=∫5π/87π/8n=0∑∞sin2nx dx
Find ⌊109I⌋\lfloor 10^{9}I \rfloor⌊109I⌋.
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