Question

\( \int \sec^2 x \cdot \csc^2 x \, dx =\) _____ + C

• A. \( \tan x + \cot x \)
• B. \( \tan x \cdot \cot x \)
• C. \( \tan x - \cot x \)
• D. \( \tan x - \cot 2x \)

Hint:

Always recall standard derivatives of trigonometric functions to solve integrals quickly.

Answer 1

The Correct Option is C

Concept: We use standard derivatives:

• \( \frac{d}{dx}(\tan x) = \sec^2 x \)
• \( \frac{d}{dx}(\cot x) = -\csc^2 x \)

This helps in identifying integrals directly.
Step 1: Rewrite the integrand. \[ \sec^2 x \cdot \csc^2 x = \sec^2 x + \csc^2 x - (\sec^2 x - \csc^2 x) \] But a better approach is to split: \[ = \sec^2 x - (-\csc^2 x) \]
Step 2: Integrate separately. \[ \int \sec^2 x \, dx = \tan x, \quad \int \csc^2 x \, dx = -\cot x \]
Step 3: \[ \int \sec^2 x \cdot \csc^2 x \, dx = \tan x - \cot x + C \]