Trig-integral

Calculus Level 5

Let

I=5π/87π/8n=0sin2nx dxI = \int_{5\pi/8}^{7\pi/8} \sqrt{\sum_{n=0}^{\infty} \sin^{2n} x} \ dx

Find 109I\lfloor 10^{9}I \rfloor.

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