Indefinite Integration
Trigonometric Integrals
IIT-JEE 2014
Grade 12
Question:
$\int (\sqrt{\tan x} + \sqrt{\cot x}) \, dx$ equals:
(1) $\sqrt{2} \sin^{-1}(\sin x - \cos x) + C$
(2) $\sqrt{2} \sin^{-1}(\sin x + \cos x) + C$
(3) $2 \sin^{-1}(\sin x - \cos x) + C$
(4) $\sqrt{2} \cos^{-1}(\sin x - \cos x) + C$
Step-by-Step Solution
Key Concept: Combine the two terms over $\sqrt{\sin x \cos x}$, rewrite $\sin x \cos x$ via $\sin 2x$, and substitute $t = \sin x - \cos x$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)