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  4. The value of the definite integral ∫ limits01 / √2 (x2 d x/√1-x2(1+√1-x2)) is

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

Integrals

Solution:

$I=\int\limits_0^{1 / \sqrt{2}} \frac{x^2 d x}{\sqrt{1-x^2}\left(1+\sqrt{1-x^2}\right)}$
put $x=\sin \theta ; d x=\cos \theta d \theta$
$=\int\limits_0^{\pi / 4} \frac{\sin ^2 \theta \cos \theta d \theta}{\cos \theta(1+\cos \theta)}=\int\limits_0^{\pi / 4}(1-\cos \theta) d \theta==(\theta-\sin \theta)_0^{\pi / 4}=\left(\frac{\pi}{4}-\frac{1}{\sqrt{2}}\right)$