Question

Evaluate the integral: \[ I = \int \frac{\sin^2 x - \cos^2 x}{\sin^2 x \cos^2 x} dx \]

• A. \( \tan x + \cot x + C \)
• B. \( \csc x + \sec x + C \)
• C. \( \tan x + \sec x + C \)
• D. \( \tan x + \csc x + C \)

Hint:

Use trigonometric identities to simplify the fraction before integrating.

Answer 1

The Correct Option is A

Step 1: Splitting the fraction.
We rewrite the given integral: \[ I = \int \frac{\sin^2 x}{\sin^2 x \cos^2 x} dx - \int \frac{\cos^2 x}{\sin^2 x \cos^2 x} dx \] Simplifying each term separately: \[ I = \int \frac{1}{\cos^2 x} dx - \int \frac{1}{\sin^2 x} dx \] Step 2: Recognizing standard integral forms.
Using the standard integral formulas: \[ \int \sec^2 x dx = \tan x + C, \quad \int \csc^2 x dx = -\cot x + C \] Thus, \[ I = \tan x + \cot x + C \] Thus, the correct answer is (A) \( \tan x + \cot x + C \).