JEE Main 2020 (Online) 3rd September Evening ShiftQuestion 1
If the value of the integral $\int_0^{1/2}\frac{x^2}{(1-x^2)^{3/2}}dx$ is $\frac k6$, then $k$ is equal to :
Definite Integration
Solution
Concept. $(1-x^2)^{3/2}$ suggests $x=\sin\theta$. Formulas used. (1) $\tan^2\theta=\sec^2\theta-1$ and $\int\sec^2\theta\,d\theta=\tan\theta$Step 1 - Substitute. $x=\sin\theta$, $dx=\cos\theta\,d\theta$; $x=0\to\theta=0$, $x=\frac12\to\theta=\frac\pi6$: $$\int_0^{\pi/6}\frac{\sin^2\theta\cos\theta}{\cos^3\theta}\,d\theta=\int_0^{\pi/6}\tan^2\theta\,d\theta.$$ Step 2 - Integrate. By (1), $$\big[\tan\theta-\theta\big]_0^{\pi/6}=\frac{1}{\sqrt3}-\frac\pi6=\frac{2\sqrt3-\pi}{6}.$$ Step 3 - Value. $k=2\sqrt3-\pi$. Answer: (B)