Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 20249 AprilMorning Shift
Exam
JEE Main
Year
2024
Date
9 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Indefinite Integration

JEE Main 2024 · 9 April · Morning Shift · Mathematics · Indefinite Integration

Let ∫ (2- x)/(3+ x) ~d x=(1)/(2)(α x+ _ e |β x+γ x|)+C , where C is...

Let ∫2−tan⁡x3+tan⁡x dx=12(αx+log⁡e∣βsin⁡x+γcos⁡x∣)+C\int \frac{2-\tan x}{3+\tan x} \mathrm{~d} x=\frac{1}{2}\left(\alpha x+\log _{\mathrm{e}}|\beta \sin x+\gamma \cos x|\right)+C, where CC is the constant of integration. Then α+γβ\alpha+\frac{\gamma}{\beta} is equal to :

Options

  1. A7
  2. B4
  3. C1
  4. D3

Correct answer

B. 4

Step-by-step solution

∫2−tan⁡x3+tan⁡xdx=∫2cos⁡x−sin⁡x3cos⁡x+sin⁡xdx2cos⁡x−sin⁡x=A(3cos⁡x+sin⁡x)+B(cos⁡x−3sin⁡x)3A+B=2A−3B=−1\begin{aligned} & \int \frac{2-\tan x}{3+\tan x} d x=\int \frac{2 \cos x-\sin x}{3 \cos x+\sin x} d x \\ & 2 \cos x-\sin x=A(3 \cos x+\sin x)+B(\cos x-3 \sin x) \\ & 3 A+B=2 \\ & A-3 B=-1\end{aligned}⇒A=12,B=12∴∫2cos⁡x−sin⁡x3cos⁡x+sin⁡xdx=x2+12ln⁡∣3cos⁡x+sin⁡x∣+C=12(x+ln⁡∣3cos⁡x+sin⁡x∣)+C=12(αx+ln⁡∣βsin⁡x+γcos⁡x∣)+Cα=1,β=1,γ=3∴α+γβ=1+31=4\begin{aligned} & \Rightarrow A=\frac{1}{2}, B=\frac{1}{2} \\ & \therefore \int \frac{2 \cos x-\sin x}{3 \cos x+\sin x} d x \\ & =\frac{x}{2}+\frac{1}{2} \ln |3 \cos x+\sin x|+C \\ & =\frac{1}{2}(x+\ln |3 \cos x+\sin x|)+C \\ & =\frac{1}{2}(\alpha x+\ln |\beta \sin x+\gamma \cos x|)+C \\ & \alpha=1, \beta=1, \gamma=3 \\ & \therefore \alpha+\frac{\gamma}{\beta}=1+\frac{3}{1}=4\end{aligned}$
Practice Indefinite Integration on Quantrex Academy →

More from Indefinite Integration

Let f(x) = ∫ ((16x + 24)/(x^2 + 2x - 15)) dx . If f(4) = 14 _e(3) and f(7) = _e(2^ α · 3^… JEE Main 2026Let f(x)=∫ d x x^ ((2)/(3)) +2 x^ ((1)/(2)) be such that f(0)=-26+24 _ e (2) . If… JEE Main 2026Let f(t)=∫( 1- ( _ e t) 1- ( _ e t) ) d t, t>1 . If f(e^ π / 2 )=-e^ π / 2 and f(e^ π / 4… JEE Main 2026Let I(x)=∫ 3 d x (4 x+6)( 4 x^ 2 +8 x+3 ) and I(0)= 3 4 +20 . If I((1)/(2))= a 2 b +c… JEE Main 2026Let f(x)=∫ (2-x^ 2 ) · e^ x ( 1+x )(1-x)^ 3 / 2 ~d x . If f(0)=0 , then f((1)/(2)) is… JEE Main 2026If ∫( x)^ (-11)/(2) ( x)^ (-5)/(2) d x= - p_ 1 q_ 1 ( x)^ (9)/(2) - p_ 2 q_ 2 ( x)^… JEE Main 2026If aligned ∫((1)/(x)+(1)/(x^3)) & ( [23] 3 x^ -24 +x^ -26 ) d x & =-(α)/(3(α+1))(3… JEE Main 2025If ∫ ( 1+x^2 +x)^ 10 ( 1+x^2 -x)^9 d x= (1)/(m)(( 1+x^2 +x)^n(n 1+x^2 -x))+C where C is… JEE Main 2025Let f(x)=∫ x^3 3-x^2 d x . If 5 f( 2 )=-4 , then f(1) is equal to JEE Main 2025If f(x)=∫ 1 x^ 1 / 4 (1+x^ 1 / 4 ) d x, f(0)=-6 , then f(1) is equal to : JEE Main 2025If ∫ 2 x^2+5 x+9 x^2+x+1 ~d x=x x^2+x+1 +α x^2+x+1 +β _e|x+(1)/(2)+ x^2+x+1 |+C , where C… JEE Main 2025Let ∫ x^3 x ~d x=g(x)+C , where C is the constant of integration. If 8(g((π)/(2))+g^… JEE Main 2025Let I(x)=∫ d x (x-11)^ (11)/(13) (x+15)^ (15)/(13) . If I(37)-I(24)=(1)/(4)( 1 ~b^… JEE Main 2025If ∫ e^x( x ^ -1 x 1-x^2 + ^ -1 x (1-x^2)^ 3 / 2 +(x)/(1-x^2)) d x=g(x)+C , where C is… JEE Main 2025If ∫ 1 [5] (x-1)^4(x+3)^6 ~d x=A((α x-1)/(β x+3))^B+C , where C is the constant of… JEE Main 2024Let I(x)=∫ (6)/( ^2 x(1- x)^2) d x . If I(0)=3 , then I((π)/(12)) is equal to JEE Main 2024 Full Indefinite Integration list Indefinite Integration JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs