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If int e^(sec x)(sec x tan x f(x)+(sec x...

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

A

`sec x + x tan x - (1)/(2)`

B

`x sec x + tan x + (1)/(2)`

C

`sec x - tan x - (1)/(2)`

D

`sec x + tan x + (1)/(2)`

Text Solution

Verified by Experts

The correct Answer is:
D
`int e^(sec x) (sec x tan x f (x) + (sec x tan x + sec^(2) x)) dx = e^(sec x) f (x) + c`
Differentiating both sides w.r.t.x
`implies e^(sec x) sec x tan x f (x) + sec x tan x + sec^(2)) = e^(sec x) f' (x) + f (x) e^(sec x). Sec x tan x`
`implies f'(x) = sec x tan x + sec^(2) x implies f(x) = sec x + tan x + K`
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