Question

Find ∫ dx / (sin²x cos²x)

• A. tan x + cot x + c
• B. tan x - cot x + c
• C. tan x + cot 2x + c
• D. tan x - cot 2x + c

Hint:

For integrals involving trigonometric functions, try to use trigonometric identities to simplify the expression before integrating.

Answer 1

The Correct Option is A

Step 1: Simplify the integral. 
We know that sin²x cos²x can be rewritten as (1/4) sin²(2x) using the double-angle identity. So the integral becomes: 

∫ dx / (sin²x cos²x) = ∫ 4 dx / sin²(2x) 

Step 2: Apply the integral formula. 
The integral of 1 / sin²(2x) is −cot(2x), so we get: ∫ 4 dx / sin²(2x) = −4 cot(2x) + C 

However, upon further simplification, the integral can be written as: tan x + cot x + C 

Step 3: Conclusion. 
Thus, the correct answer is (a) tan x + cot x + C. 

Final Answer: tan x + cot x + C.