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Question Number 5321 by qw last updated on 07/May/16

$$\underset{0}{\stackrel{{\sin }^{2}x}{\int }}{\sin }^{-1}\sqrt{t}dt+\underset{0}{\stackrel{{\cos }^{2}x}{\int }}{\cos }^{-1}\sqrt{t}dt=$$

Commented by Yozzii last updated on 08/May/16

$$I={\int }_0^{{sin}^{2}x}{sin}^{-1}\sqrt{t}dt+{\int }_0^{{cos}^{2}x}{cos}^{-1}\sqrt{t}dt$$$$I=-\frac{1}{2}cos2{xsin}^{-1}\mid sinx\mid +\frac{1+cos2x}{2}{cos}^{-1}\mid cosx\mid +\frac{1}{2}{sin}^{-1}\mid sin(\frac{\pi }{2}-x)\mid$$$$For0<x<\frac{\pi }{2},$$$$I={\int }_0^{{sin}^{2}x}{sin}^{-1}\sqrt{t}dt+{\int }_0^{{cos}^{2}x}{cos}^{-1}\sqrt{t}dt=\frac{\pi }{4}.$$

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