If ∫ x+1 2 x-1 ~d x=f(x) 2 x-1 +C, where C is a constant of integration, then f(x) is equal to:

Correct answer: D. (1)/(3)(x+4) Let I=∫ x+1 2 x-1 d x Put 2 x-1 =t array l 2 x-1=t^ 2 d x=t d t I=∫ (t^ 2 +3) 2 d t= t^ 3 6 +(3 t)/(2)+C = (2 x-1)^ (3)/(2) 6 +(3)/(2)(2 x-1)^ (1)/(2) +C = 2 x-1 ((x+4)/(3))+C =f(x) · 2 x-1 +C array Hence, f(x)=(x+4)/(3)

JEE Main 201911 JanuaryEvening Shift

Exam: JEE Main
Year: 2019
Date: 11 January
Shift: Evening Shift
Subject: Mathematics
Chapter: Indefinite Integration

JEE Main 2019 · 11 January · Evening Shift · Mathematics · Indefinite Integration

If ∫ x+1 2 x-1 ~d x=f(x) 2 x-1 +C, where C is a constant of...

\int \frac{x+1}{\sqrt{2 x-1}} \mathrm{~d} x=f(x) \sqrt{2 x-1}+\mathrm{C},

where

\mathrm{C}

is a constant of integration, then

f(x)

is equal to:

Options

A $\frac{1}{3}(x+1)$
B $\frac{2}{3}(x+2)$
C $\frac{2}{3}(x-4)$
D $\frac{1}{3}(x+4)$

Correct answer

D. $\frac{1}{3}(x+4)$

Step-by-step solution

Let

I=\int \frac{x+1}{\sqrt{2 x-1}} d x

Put

\sqrt{2 x-1}=t

\begin{array}{l} \therefore \quad 2 x-1=t^{2} \Rightarrow d x=t d t \\ I=\int \frac{\left(t^{2}+3\right)}{2} d t=\frac{t^{3}}{6}+\frac{3 t}{2}+C \\ =\frac{(2 x-1)^{\frac{3}{2}}}{6}+\frac{3}{2}(2 x-1)^{\frac{1}{2}}+C \\ =\sqrt{2 x-1}\left(\frac{x+4}{3}\right)+C \\ =f(x) \cdot \sqrt{2 x-1}+C \end{array}

Hence,

f(x)=\frac{x+4}{3}

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 2026 Let f(x)=∫ d x x^ ((2)/(3)) +2 x^ ((1)/(2)) be such that f(0)=-26+24 _ e (2) . If… JEE Main 2026 Let 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 2026 Let 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 2026 Let 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 2026 If ∫( x)^ (-11)/(2) ( x)^ (-5)/(2) d x= - p_ 1 q_ 1 ( x)^ (9)/(2) - p_ 2 q_ 2 ( x)^… JEE Main 2026 If aligned ∫((1)/(x)+(1)/(x^3)) & ( [23] 3 x^ -24 +x^ -26 ) d x & =-(α)/(3(α+1))(3… JEE Main 2025 If ∫ ( 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 2025 Let f(x)=∫ x^3 3-x^2 d x . If 5 f( 2 )=-4 , then f(1) is equal to JEE Main 2025 If f(x)=∫ 1 x^ 1 / 4 (1+x^ 1 / 4 ) d x, f(0)=-6 , then f(1) is equal to : JEE Main 2025 If ∫ 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 2025 Let ∫ x^3 x ~d x=g(x)+C , where C is the constant of integration. If 8(g((π)/(2))+g^… JEE Main 2025 Let I(x)=∫ d x (x-11)^ (11)/(13) (x+15)^ (15)/(13) . If I(37)-I(24)=(1)/(4)( 1 ~b^… JEE Main 2025 If ∫ 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 2025 Let ∫ (2- x)/(3+ x) ~d x=(1)/(2)(α x+ _ e |β x+γ x|)+C , where C is the constant of… JEE Main 2024 If ∫ 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 2024 Full Indefinite Integration list Indefinite Integration JEE Maths PYQs JEE Mathematics PYQs All JEE Main PYQs