Question:
\(\int e^{\sec x}\left(\sec x\tan x\, f(x) + \sec x\tan x + \sec^2 x\right)dx = e^{\sec x} f(x) + C\) then a possible choice of f(x) is :-
\(\int e^{\sec x}\left(\sec x\tan x\, f(x) + \sec x\tan x + \sec^2 x\right)dx = e^{\sec x} f(x) + C\) then a possible choice of f(x) is :-
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Differentiate both sides; the e^{sec x} factors cancel and leave an equation for f'(x).
Updated On: Nov 5, 2026
- x sec x + tan x + 1/2
- sec x + x tan x - 1/2
- sec x - tan x - 1/2
- sec x + tan x + 1/2
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The Correct Option is D
Solution and Explanation
Differentiate both sides w.r.t. x: \(e^{\sec x}\left(\sec x\tan x\,f + \sec x\tan x + \sec^2x\right) = e^{\sec x}\left(\sec x\tan x\,f + f'\right)\).
So \(f'(x) = \sec x\tan x + \sec^2 x\).
Integrate: \(f(x) = \sec x + \tan x + c\).
With \(c = \frac12\) this is option (4).
Answer: (4) sec x + tan x + 1/2
So \(f'(x) = \sec x\tan x + \sec^2 x\).
Integrate: \(f(x) = \sec x + \tan x + c\).
With \(c = \frac12\) this is option (4).
Answer: (4) sec x + tan x + 1/2
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