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JEE Main 2019 · 9 Apr · Shift II

JEE Main 2019 Mathematics question on Indefinite Integration

If ∫esec⁡x(sec⁡xtan⁡xf(x)+(sec⁡xtan⁡x+sec⁡2x))dx=esec⁡xf(x)+C\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)f(x) is
  1. A.
    xsec⁡x+tan⁡x+12x \sec x + \tan x + \dfrac{1}{2}
  2. B.
    sec⁡x+tan⁡x+12\sec x + \tan x + \dfrac{1}{2}
  3. C.
    sec⁡x+xtan⁡x−12\sec x + x\tan x - \dfrac{1}{2}
  4. D.
    sec⁡x−tan⁡x−12\sec x - \tan x - \dfrac{1}{2}
Show answer and solution

Correct option: B

Solution

Differentiating the right side e^(sec x) f(x) and setting it equal to the integrand gives:

e^(sec x) [sec x tan x · f(x) + f'(x)] = e^(sec x) [sec x tan x · f(x) + sec x tan x + sec²x]

So f'(x) = sec x tan x + sec²x, which integrates to f(x) = sec x + tan x + C.

Choosing C = 1/2 gives f(x) = sec x + tan x + 1/2, which matches option 2.

[JEE Main 2019 · 9 Apr · Shift II]

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