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  4. The value of definite integral ∫ limits01 ( dx /√1- x 2(1+√1- x 2)) is equal to

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Q. The value of definite integral $\int\limits_0^1 \frac{ dx }{\sqrt{1- x ^2}\left(1+\sqrt{1- x ^2}\right)}$ is equal to

Integrals

Solution:

Put $x=\sin \theta$
$\therefore I=\int\limits_0^{\pi / 2} \frac{d \theta}{1+\cos \theta}=\left[\tan \frac{\theta}{2}\right]_0^{\pi / 2}=1$