What is `int ((1)/(cos^(2)x) - (1)/(sin^(x)x))dx` equal to ?
where c is the constant of integration

Options

A. `2 "cosec" 2x + c`
B. `-2 cot 2x + c`
C. `2 sec 2x + c`
D. `-2 tan 2x + c`

Answer

A

Explanation

Let `I = int ((1)/(cos^(2)x) - (1)/(sin^(2)x)) dx`
`= int (sec^(2) x - "cosec"^(2) x) dx`
`= int sec^(2)x dx - int "cosec"^(2) x dx`
`tan x + cot x + c`
`= tan x + (1)/(tan x) + c = (tan^(2) x+ 1)/(tan x) + c = (sec^(2)x)/(tan x) + c`
`= (2)/(2 sin x cos x) + c = (2)/(sin 2x) + c = 2 "cosec" 2x + c`