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Question Number 31504 by abdo imad last updated on 09/Mar/18

$$calculate{\int }_0^1\frac{dt}{t+\sqrt{1-t^2}}.$$

Commented by abdo imad last updated on 15/Mar/18

$$letputt=sinx\Rightarrow I={\int }_0^{\frac{\pi }{2}}\frac{cosxdx}{sinx+cosx}$$$$={\int }_0^{\frac{\pi }{2}}\frac{cosx+sinx-sinx}{sinx+cosx}dx=\frac{\pi }{2}-{\int }_0^{\frac{\pi }{2}}\frac{sinx}{sinx+csx}dx$$$$ch.x=\frac{\pi }{2}-tgive{\int }_0^{\frac{\pi }{2}}\frac{sinx}{sinx+cosx}dx={\int }_0^{\frac{\pi }{2}}\frac{cosx}{cosx+sinx}dx$$$$=I\Rightarrow 2I=\frac{\pi }{2}\Rightarrow I=\frac{\pi }{4}.$$

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