<p>Evaluate \(\displaystyle\int_0^1\sqrt{\frac{x}{1+x}}\,dx\) [JEE Main 2020]</p>
Step-by-Step Solution
Key Concept: Let x = tan^2\theta: dx = 2tan\theta \cdot sec^2\theta d\theta, \sqrt{x/(1+x}) = |sin \theta|. Or let t^2=x/(1+x).
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<p>Let $x=\tan^2\theta$, $\sqrt{x/(1+x)}=\sin\theta$, $dx=2\tan\theta\sec^2\theta\,d\theta$. Limits: $\theta:0\to\pi/4$.</p>
<p>$$I=\int_0^{\pi/4}\sin\theta\cdot 2\tan\theta\sec^2\theta\,d\theta=2\int_0^{\pi/4}\frac{\sin^2\theta}{\cos^3\theta}\,d\theta$$</p>
<p>Let $t=\sin\theta$: $I=2\int_0^{1/\sqrt{2}}\frac{t^2}{(1-t^2)^{3/2}}dt$. This reduces to $2-\pi/2$.</p>
Correct Answer: A