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Evaluate $\int_0^1\int_0^{\cos^{-1}y}(\sin x)\sqrt{1+\sin^2x}\,dxdy.$

Can anyone hint me how to start solving this? Or solve the whole thing if you're generous enough. :D

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Since we know that the integrand is independent of $y$, we can perform integration w.r.t. $y$ first. Hence the integration becomes $\int_0^{\pi/2}\int_0^{\cos{x}}(\sin x)\sqrt{1+\sin^2x}\,dydx$ then the integration becomes a lot easier.

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    Very close, @Rose! Check your substitution of $x=\pi/2$ again.2012-12-12