What is `int (dx)/(sin^(2) x cos^(2) x)` equal to ?
where c is the constant of integration

Options

A. `tan x + cot x + c`
B. `tan x - cot x + c`
C. `(tan x + cot x)^(2) + c`
D. `(tan x - cot x)^(2) + c`

Answer

B

Explanation

Let `I = int (dx)/(sin^(2) x cos^(2) x) = int ((sin^(2) x + cos^(2) x)dx)/(sin^(2) x cos^(2) x)`
`= int [(sin^(2)x)/(sin^(2) x cos^(2) x) + (cos^(2)x)/(sin^(2) x cos^(2) x)] dx`
`= int [(1)/(cos^(2)x) + (1)/(sin^(2)x)] dx`
`= int (sec^(2) x + "cosec"^(2)x) dx`
`= tan x - cot x + c` where c is constant of Integration