If `int e^(sec x)(sec x tan x f(x)+(sec x tan x + sec^(2) x))dx = e^(sec x)f(x) + C`, then a possible choice of f(x) is

Options

A. `x sec x + tan x + (1)/(2)`
B. `sec x + tan x + (1)/(2)`
C. `sec x + x tan x -(1)/(2)`
D. `sec x - tan x -(1)/(2)`

Answer

B

Explanation

Given, `int e^(sec x)[(sec x tan x)f(x)+(sec x tan x + sec^(2)x)]dx=e^(sec x)* f(x)+C`
On differentiating both sides w.r.t.x, we get `e^(sec x)[(sec x tan x)f(x)+(sec x tan x + sec^(2) x)]`
`=e^(sec x)f'(x) + e^(sec x)(sec x tan x)f(x)`
`rArr e^(sec x)(sec x tan x + sec^(2)x)=e^(sec x)f'(x)`
`rArr f'(x)=sec x tan x + sec^(2)x`
So, `f(x)=int f'(x)dx=int (sec x tan x + sec^(2)x)dx`
`= sec x + tan x + C`
So, possible value of f(x) from options, is `f(x) = sec x + tan x + (1)/(2)`.