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JEE Main · 2024 · 09 Apr 2024 · Shift 1 · Indefinite Integrals

Let

\int \frac{2 - t a n x}{3 + t a n x} d x = \frac{1}{2} (α x + lo g_{e} ∣ β sin x + γ cos x ∣) + C

, where C is the constant of integration.

Then

α + \frac{γ}{β}

is equal to :

(A)3
(B)1
(C)4✓ Correct
(D)7

Answer: (C)

Solution

Write

\frac{2 - tan x}{3 + tan x} = \frac{2 cos x - sin x}{3 cos x + sin x}

and split the numerator as

A (3 cos x + sin x) + B (cos x - 3 sin x)

. Matching gives

A = B = \frac{1}{2}

, so the integral is

\frac{1}{2} x + \frac{1}{2} ln ∣3 cos x + sin x ∣ = \frac{1}{2} (x + ln ∣ sin x + 3 cos x ∣) + C

. Hence

α = 1, β = 1, γ = 3

and

α + \frac{γ}{β} = 1 + 3 = 4

— option (3).

JEE Main · 2024 · 09 Apr 2024 · Shift 1 · Single correct option

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