Interesting Integrals 10

Calculus Level 4

0sin2(x)sin1(t)dt +0cos2(x)cos1(t)dt=πc\large\int_{0}^{\sin^2 (x)}{\sin^{-1} \left( \sqrt t \right)}\,dt\ + \large\int_{0}^{\cos^2 (x)}{\cos^{-1} \left( \sqrt t \right)}\,dt=\dfrac{\pi}{c}

The equation above holds true for some positive integer cc. Find cc.

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