Indefinite Integration
Integration by Substitution
JEE Main 2021
Grade 12
Question:
$\int \frac{\sec^2 x}{(1 + \tan x)^2} \, dx$ equals:
(1) $-\frac{1}{1 + \tan x} + C$
(2) $\frac{1}{1 + \tan x} + C$
(3) $\ln |1 + \tan x| + C$
(4) $-\ln |1 + \tan x| + C$
Step-by-Step Solution
Key Concept: Substitute $t = 1 + \tan x$, noting that $dt = \sec^2 x \, dx$, which converts the integral into a simple power of $t$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)