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Question:

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

sec x + tan x + 1/2

sec x - tan x - 1/2

x sec x + tan x + 1/2

sec x + x tan x - 1/2

Solution:

Correct option is D.
sec x + tan x + 1/2
∫ esec x (sec x tan x f(x) + (sec x tan x + sec2 x)) dx = esec x f(x) + C
Differentiating both sides w.r.t x
esec x (sec x tan x f(x) + (sec x tan x + sec2 x)) = esec x . sec x tan x f(x) + esec x f'(x)
= sec2 x + tan x sec x
⇒ f'(x) = tan x + sec x + c