JEE PYQ: Indefinite Integration Question 8
Question 8 - 2020 (03 Sep Shift 2)
If $\int \sin^{-1}\left(\sqrt{\frac{x}{1+x}}\right),dx = A(x)\tan^{-1}(\sqrt{x}) + B(x) + C$, where $C$ is a constant of integration, then the ordered pair $(A(x), B(x))$ can be:
(1) $(x+1, -\sqrt{x})$
(2) $(x+1, \sqrt{x})$
(3) $(x-1, -\sqrt{x})$
(4) $(x-1, \sqrt{x})$
Show Answer
Answer: (1)
Solution
$I = \int \tan^{-1}\sqrt{x},dx$. By parts: $= x\tan^{-1}\sqrt{x} - \int \frac{x}{(1+x)}\cdot\frac{1}{2\sqrt{x}},dx = x\tan^{-1}\sqrt{x} - \sqrt{x} + \tan^{-1}\sqrt{x} + C = (x+1)\tan^{-1}\sqrt{x} - \sqrt{x} + C$. So $A(x) = x+1$, $B(x) = -\sqrt{x}$.
BETA
