Question

Evaluate the integral: \[ \int \tan^2 x \, dx \] The correct answer is:

• A. \( \tan x + x + c \)
• B. \( \tan x - x + c \)
• C. \( \cot x + x + c \)
• D. \( \cot x - x + c \)

Hint:

For integrals involving \(\tan^2 x\), use the identity \(\tan^2 x = \sec^2 x - 1\) to simplify the integral.

Answer 1

The Correct Option is B

We can simplify the integral \(\int \tan^2 x \, dx\) using the identity: \[ \tan^2 x = \sec^2 x - 1. \] Thus, the integral becomes: \[ \int \tan^2 x \, dx = \int (\sec^2 x - 1) \, dx. \] Now, integrate each term: \[ \int \sec^2 x \, dx = \tan x, \] \[ \int 1 \, dx = x. \] Thus, the integral is: \[ \tan x - x + c. \] Therefore, the correct answer is \( (B) \).