Indefinite Integration — JEE Previous Year Questions

Every Indefinite Integration question asked in JEE Main and JEE Advanced across the last 186 papers — 69 questions, each with its correct answer. Free to read, no account needed.

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All 69 Indefinite Integration questions

Most recent papers first.

Q1·MathematicsSingle correctJEE Main 2026

Let f(x) and g(x) be twice differentiable functions satisfying f''(x) = g''(x) for all x ∈ ℝ, f'(1) = 2g'(1) = 4 and g(2) = 3f(2) = 9. Then f(25) − g(25) is equal to :

(A)20
(B)40
(C)−20
(D)−40

Q2·MathematicsSingle correctJEE Main 2026

Let

f(x) = \int \left(\frac{16x + 24}{x^2 + 2x - 15}\right) dx

. If

f(4) = 14 \log_e(3)

and

f(7) = \log_e(2^{\alpha} \cdot 3^{\beta})

\alpha, \beta \in \mathbb{N}

, then

\alpha + \beta

is equal to:

(A) $31$
(B) $37$
(C) $39$
(D) $41$

Q3·MathematicsSingle correctJEE Main 2026

Let

f(x) = \int \frac{dx}{x^{\left(\frac{2}{3}\right)} + 2x^{\left(\frac{1}{2}\right)}}

be such that

f(0) = -26 + 24 \log_e(2)

. If

f(1) = a + b \log_e(3)

, where a, b

\in

\mathbf{Z}

, then a + b is equal to:

(A)–18
(B)–5
(C)–11
(D)–26

Q4·MathematicsSingle correctJEE Main 2026

\int\left(\dfrac{1 - 5\cos^2 x}{\sin^5 x \cos^2 x}\right)dx = f(x) + C

where C is the constant of integration, then

f\left(\dfrac{\pi}{6}\right) - f\left(\dfrac{\pi}{4}\right)

is equal to

(A) $\dfrac{1}{\sqrt{3}}\left(26 + \sqrt{3}\right)$
(B) $\dfrac{4}{\sqrt{3}}\left(8 - \sqrt{6}\right)$
(C) $\dfrac{1}{\sqrt{3}}\left(26 - \sqrt{3}\right)$
(D) $\dfrac{2}{\sqrt{3}}\left(4 + \sqrt{6}\right)$

Q5·MathematicsSingle correctJEE Main 2026

Let

f(t)=\int\left(\frac{1-\sin(\log_{e}t)}{1-\cos\left(\log_{e}t\right)}\right)dt,\ t>1

. If

f(e^{\pi/2})=-e^{\pi/2}

and

f(e^{\pi/4})=\alpha\ e^{\pi/4}

, then α equals

(A) $-1-\sqrt{2}$
(B) $-1-2\sqrt{2}$
(C) $1+\sqrt{2}$
(D) $-1+\sqrt{2}$

Q6·MathematicsSingle correctJEE Main 2026

Let

I(x) = \int \frac{3dx}{(4x+6)(\sqrt{4x^2+8x+3})}

and

I(0) = \frac{\sqrt{3}}{4}+20

. If

I\left(\frac{1}{2}\right) = \frac{a\sqrt{2}}{b}+c

, where a, b, c ∈ N, gcd(a,b) = 1, then a + b + c is equal to :

(A)29
(B)28
(C)31
(D)30

Q7·MathematicsSingle correctJEE Main 2026

Let

f(x) = \int \frac{(2-x^2).e^x}{\left(\sqrt{1+x}\right)(1-x)^{3/2}} dx

. If

f(0) - 0

, then

f\left(\frac{1}{2}\right)

is equal to :

(A) $\sqrt{3e} - 1$
(B) $\sqrt{2e} + 1$
(C) $\sqrt{2e} - 1$
(D) $\sqrt{3e} + 1$

Q8·MathematicsNumericalJEE Main 2026

\int (\sin x)^{\frac{-11}{2}} (\cos x)^{\frac{-5}{2}} dx = -\frac{p_{1}}{q_{1}}(\cot x)^{\frac{9}{2}} - \frac{p_{2}}{q_{2}}(\cot x)^{\frac{5}{2}} - \frac{p_{3}}{q_{3}}(\cot x)^{\frac{1}{2}} + \frac{p_{4}}{q_{4}}(\cot x)^{\frac{-3}{2}} + C

, where

p_{i}

and

q_{i}

are positive integers with

\gcd(p_{i}, q_{i}) = 1

for

i = 1, 2, 3, 4

and C is the constant of integration, then

\frac{15p_{1}p_{2}p_{3}p_{4}}{q_{1}q_{2}q_{3}q_{4}}

is equal to _______.

Q9·MathematicsIntegerJEE Main 2025

\displaystyle\int\left(\dfrac{1}{x}+\dfrac{1}{x^3}\right)\left(3x^{-24}+x^{-26}\right)^{1/23}dx=-\dfrac{\alpha}{3(\alpha+1)}\left(3x^{\beta}+x^{\gamma}\right)^{\frac{\alpha+1}{\alpha}}+C

x>0

(\alpha,\beta,\gamma\in\mathbb{Z})

, where

C

is the constant of integration, then

\alpha+\beta+\gamma

is equal to __________.

Q10·MathematicsIntegerJEE Main 2025

\displaystyle\int\dfrac{\left(\sqrt{1+x^2}+x\right)^{10}}{\left(\sqrt{1+x^2}-x\right)^{9}}\,dx=\dfrac{1}{m}\left(\left(\sqrt{1+x^2}+x\right)^{n}\left(n\sqrt{1+x^2}-x\right)\right)+C

, where

C

is the constant of integration and

m,n\in\mathbb{N}

, then

m+n

is equal to ______.

Q11·MathematicsSingle correctJEE Main 2025

Let

f(x)=\displaystyle\int x^3\sqrt{3-x^2}\,dx

. If

5f(\sqrt2)=-4

, then

f(1)

is equal to:

(A) $-\dfrac{2\sqrt2}{5}$
(B) $-\dfrac{8\sqrt2}{5}$
(C) $-\dfrac{4\sqrt2}{5}$
(D) $-\dfrac{6\sqrt2}{5}$

Q12·MathematicsSingle correctJEE Main 2025

f(x)=\int\frac{1}{x^{1/4}(1+x^{1/4})}dx

f(0)=-6

, then

f(1)

is equal to:

(A) $\log_e 2+2$
(B) $4(\log_e 2-2)$
(C) $2-\log_e 2$
(D) $4(\log_e 2+2)$

Q13·MathematicsIntegerJEE Main 2025

\displaystyle\int\dfrac{2x^2+5x+9}{\sqrt{x^2+x+1}}\,dx=x\sqrt{x^2+x+1}+\alpha\sqrt{x^2+x+1}+\beta\log_e\left|x+\dfrac12+\sqrt{x^2+x+1}\right|+C

, where C is the constant of integration, then

\alpha+2\beta

is equal to __________.

Q14·MathematicsIntegerJEE Main 2025

If the area of the larger portion bounded between the curves

x^2+y^2=25

and

y=|x-1|

\dfrac{1}{4}(b\pi+c)

b,c\in\mathbb{N}

, then

b+c

is equal to _______

Q15·MathematicsSingle correctJEE Main 2025

Let

\displaystyle\int x^3\sin x\,dx=g(x)+C

, where

C

is the constant of integration. If

8\left[g\!\left(\tfrac{\pi}{2}\right)+g'\!\left(\tfrac{\pi}{2}\right)\right]=\alpha\pi^3+\beta\pi^2+\gamma,\ \alpha,\beta,\gamma\in\mathbb{Z}

, then

\alpha+\beta-\gamma

equals :

(A)55
(B)47
(C)48
(D)62

Q16·MathematicsSingle correctJEE Main 2025

Let

\displaystyle I(x)=\int\frac{dx}{(x-11)^{11/13}(x+15)^{15/13}}

. If

I(37)-I(24)=\frac{1}{4}\left(\frac{1}{b^{1/13}}-\frac{1}{c^{1/13}}\right)

b,c\in\mathbb{N}

, then

3(b+c)

is equal to

(A) $40$
(B) $39$
(C) $22$
(D) $26$

Q17·MathematicsSingle correctJEE Main 2025

\displaystyle\int e^x\left(\dfrac{x\sin^{-1}x}{\sqrt{1-x^2}}+\dfrac{\sin^{-1}x}{\left(1-x^2\right)^{3/2}}+\dfrac{x}{1-x^2}\right)dx=g(x)+C

, where

C

is the constant of integration, then

g\left(\dfrac{1}{2}\right)

equals:

(A) $\dfrac{\pi}{6}\sqrt{\dfrac{e}{2}}$
(B) $\dfrac{\pi}{4}\sqrt{\dfrac{e}{2}}$
(C) $\dfrac{\pi}{6}\sqrt{\dfrac{e}{3}}$
(D) $\dfrac{\pi}{4}\sqrt{\dfrac{e}{3}}$

Q18·MathematicsSingle correctJEE Main 2024

Let

\int \frac{2 - \tan x}{3 + \tan x}\,dx = \frac{1}{2}\left(\alpha x + \log_e |\beta \sin x + \gamma \cos x|\right) + C

, where C is the constant of integration. Then

\alpha + \frac{\gamma}{\beta}

is equal to:

(A)3
(B)1
(C)4
(D)7

Q19·MathematicsNumericalJEE Main 2024

\displaystyle\int\dfrac{1}{\sqrt[5]{(x-1)^4(x+3)^6}}\,dx=A\left(\dfrac{\alpha x-1}{\beta x+3}\right)^B+C

, where C is the constant of integration, then the value of

\alpha+\beta+20AB

is _______ .

Q20·MathematicsSingle correctJEE Main 2024

Let

I(x)=\displaystyle\int\dfrac{6}{\sin^2 x\cdot(1-\cot x)^2}\,dx

. If

I(0)=3

, then

I\left(\dfrac{\pi}{12}\right)

is equal to:

(A) $\sqrt3$
(B) $3\sqrt3$
(C) $6\sqrt3$
(D) $2\sqrt3$

Q21·MathematicsSingle correctJEE Main 2024

\int\frac{1}{a^{2}\sin^{2}x+b^{2}\cos^{2}x}\,dx=\frac{1}{12}\tan^{-1}(3\tan x)+

constant, then the maximum value of

a\sin x+b\cos x

, is:

(A) $\sqrt{40}$
(B) $\sqrt{39}$
(C) $\sqrt{42}$
(D) $\sqrt{41}$

Q22·MathematicsNumericalJEE Main 2024

\displaystyle\int\operatorname{cosec}^5 x\,dx=\alpha\cot x\,\operatorname{cosec} x\left(\operatorname{cosec}^2 x+\tfrac{3}{2}\right)+\beta\log_e\left|\tan\tfrac{x}{2}\right|+C

, where

\alpha,\beta\in\mathbb{R}

and C is constant of integration, then the value of

8(\alpha+\beta)

equals

Q23·MathematicsSingle correctJEE Main 2024

\displaystyle\int\dfrac{\sin^{\frac{3}{2}}x+\cos^{\frac{3}{2}}x}{\sqrt{\sin^3 x\,\cos^3 x\,\sin(x-\theta)}}\,dx=A\sqrt{\cos\theta\tan x-\sin\theta}+B\sqrt{\cos\theta-\sin\theta\cot x}+C

, where

C

is the integration constant, then

AB

is equal to:

(A) $4\,\mathrm{cosec}(2\theta)$
(B) $4\sec\theta$
(C) $2\sec\theta$
(D) $8\,\mathrm{cosec}(2\theta)$

Q24·MathematicsSingle correctJEE Main 2024

For

x\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)

, if

y(x)=\displaystyle\int\dfrac{\operatorname{cosec}x+\sin x}{\operatorname{cosec}x\sec x+\tan x\sin^2 x}dx

and

\displaystyle\lim_{x\to\left(\frac{\pi}{2}\right)^-}y(x)=0

then

y\left(\dfrac{\pi}{4}\right)

is equal to

(A) $\tan^{-1}\left(\dfrac{1}{\sqrt{2}}\right)$
(B) $\dfrac{1}{2}\tan^{-1}\left(\dfrac{1}{\sqrt{2}}\right)$
(C) $-\dfrac{1}{\sqrt{2}}\tan^{-1}\left(\dfrac{1}{\sqrt{2}}\right)$
(D) $\dfrac{1}{\sqrt{2}}\tan^{-1}\left(-\dfrac{1}{2}\right)$

Q25·MathematicsSingle correctJEE Main 2024

The integral

\displaystyle\int\dfrac{(x^8-x^2)\,dx}{(x^{12}+3x^6+1)\tan^{-1}\left(x^3+\dfrac{1}{x^3}\right)}

is equal to :

(A) $\log_e\left(\left|\tan^{-1}\left(x^3+\dfrac{1}{x^3}\right)\right|^{1/3}\right)+C$
(B) $\log_e\left(\left|\tan^{-1}\left(x^3+\dfrac{1}{x^3}\right)\right|^{1/2}\right)+C$
(C) $\log_e\left(\left|\tan^{-1}\left(x^3+\dfrac{1}{x^3}\right)\right|\right)+C$
(D) $\log_e\left(\left|\tan^{-1}\left(x^3+\dfrac{1}{x^3}\right)\right|^{3}\right)+C$

Q26·MathematicsNumericalJEE Main 2023

Let

f(x) = \int \frac{dx}{\left(3 + 4x^2\right)\sqrt{4 - 3x^2}}

|x| < \frac{2}{\sqrt{3}}

. If

f(0) = 0

and

f(1) = \frac{1}{\alpha\beta}\tan^{-1}\left(\frac{\alpha}{\beta}\right)

\alpha, \beta > 0

, then

\alpha^2 + \beta^2

is equal to____

Q27·MathematicsNumericalJEE Main 2023

Let

I=\int\sqrt{\dfrac{x+7}{x}}\,dx

and

I(1)=\alpha+7\log_e(1+2\sqrt2)

, then

\alpha^4

is equal to _________.

Q28·MathematicsSingle correctJEE Main 2023

\int\left(\left(\frac{x}{e}\right)^{2x}+\left(\frac{e}{x}\right)^{2x}\right)\log_e x\,dx=\frac{1}{\alpha}\left(\frac{x}{e}\right)^{2x}-\frac{1}{\gamma}\left(\frac{e}{x}\right)^{2x}+C

, where

C

is the constant of integration, then

\alpha+2\beta+3\gamma-4\delta

is equal to:

(A) $1$
(B) $-4$
(C) $-8$
(D) $4$

Q29·MathematicsSingle correctJEE Main 2023

I(x)=\int e^{\sin^2 x}(\cos x\sin 2x-\sin x)\,dx

and

I(0)=1

, then

I\left(\dfrac{\pi}{3}\right)

is equal to

(A) $-\dfrac12 e^{\frac34}$
(B) $e^{\frac34}$
(C) $\dfrac12 e^{\frac34}$
(D) $-e^{\frac34}$

Q30·MathematicsSingle correctJEE Main 2023

The integral

\displaystyle\int\left(\left(\dfrac{x}{2}\right)^{x}+\left(\dfrac{2}{x}\right)^{x}\right)\log_{2}x\,dx

is equal to

(A) $\left(\dfrac{x}{2}\right)^{x}+\left(\dfrac{2}{x}\right)^{x}+C$
(B) $\left(\dfrac{x}{2}\right)^{x}-\left(\dfrac{2}{x}\right)^{x}+C$
(C) $\left(\dfrac{x}{2}\right)^{x}\log_{2}\left(\dfrac{x}{2}\right)+C$
(D) $\left(\dfrac{x}{2}\right)^{x}\log_{2}\left(\dfrac{2}{x}\right)+C$

Q31·MathematicsSingle correctJEE Main 2023

Let

I(x)=\int \frac{(x+1)}{x(1+xe^{x})^{2}}\,dx,\ x>0

. If

\lim_{x\to\infty} I(x)=0

, then

I(1)

is equal to

(A) $\frac{e+1}{e+2}-\log_{e}(e+1)$
(B) $\frac{e+1}{e+2}+\log_{e}(e+1)$
(C) $\frac{e+2}{e+1}+\log_{e}(e+1)$
(D) $\frac{e+2}{e+1}-\log_{e}(e+1)$

Q32·MathematicsSingle correctJEE Main 2023

Let

I(x)=\displaystyle\int\dfrac{x\,\sec^{2}x+\tan x}{(x\tan x+1)^{2}}dx

. If

I(0)=0

, then

I\left(\dfrac{\pi}{4}\right)

is equal to:

(A) $\log_e\dfrac{(\pi+4)^{2}}{16}-\dfrac{\pi^{2}}{4(\pi+4)}$
(B) $\log_e\dfrac{(\pi+4)^{2}}{16}+\dfrac{\pi^{2}}{4(\pi+4)}$
(C) $\log_e\dfrac{(\pi+4)^{2}}{32}-\dfrac{\pi^{2}}{4(\pi+4)}$
(D) $\log_e\dfrac{(\pi+4)^{2}}{32}+\dfrac{\pi^{2}}{4(\pi+4)}$

Q33·MathematicsNumericalJEE Main 2023

\displaystyle\int \sqrt{\sec 2x - 1}\,dx=\alpha \log_e\left|\cos 2x + \beta + \sqrt{\cos 2x\left(1 + \cos\dfrac{1}{\beta}x\right)}\right| + \text{constant}

, then

\beta - \alpha

is equal to _______.

Q34·MathematicsSingle correctJEE Main 2023

Let

f(x)=\displaystyle\int\dfrac{2x}{(x^2+1)(x^2+3)}\,dx

. If

f(3)=\dfrac{1}{2}(\log_e 5-\log_e 6)

, then

f(4)

is equal to:

(A) $\log_e 19-\log_e 20$
(B) $\dfrac{1}{2}(\log_e 17-\log_e 18)$
(C) $\dfrac{1}{2}(\log_e 19-\log_e 17)$
(D) $\dfrac{1}{2}(\log_e 17-\log_e 19)$

Q35·MathematicsSingle correctJEE Main 2022

For

I(x) = \int \frac{\sec^2 x - 2022}{\sin^{2022} x} dx

, if

I\left( \frac{\pi}{4} \right) = 2^{1011}

, then

(A) $3^{1010} I\left( \frac{\pi}{3} \right) - I\left( \frac{\pi}{6} \right) = 0$
(B) $3^{1010} I\left( \frac{\pi}{6} \right) - I\left( \frac{\pi}{3} \right) = 0$
(C) $3^{1011} I\left( \frac{\pi}{3} \right) - I\left( \frac{\pi}{6} \right) = 0$
(D) $3^{1011} I\left( \frac{\pi}{6} \right) - I\left( \frac{\pi}{3} \right) = 0$

Q36·MathematicsSingle correctJEE Main 2022

The integral

\int\frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{\left(1+\frac{2}{\sqrt{3}}\sin 2x\right)}dx

is equal to

(A) $\frac{1}{2}\log_{e}\left|\frac{\tan\left(\frac{x}{2}+\frac{\pi}{12}\right)}{\left(\frac{x}{2}+\frac{\pi}{6}\right)}\right|+C$
(B) $\frac{1}{2}\log_{e}\left|\frac{\tan\left(\frac{x}{2}+\frac{\pi}{6}\right)}{\left(\frac{x}{2}+\frac{\pi}{3}\right)}\right|+C$
(C) $\log_{e}\left|\frac{\tan\left(\frac{x}{2}+\frac{\pi}{6}\right)}{\tan\left(\frac{x}{2}+\frac{\pi}{12}\right)}\right|+C$
(D) $\frac{1}{2}\log_{e}\left|\frac{\tan\left(\frac{x}{2}-\frac{\pi}{12}\right)}{\tan\left(\frac{x}{2}-\frac{\pi}{6}\right)}\right|+C$

Q37·MathematicsSingle correctJEE Main 2022

\int \frac{\left(x^{2}+1\right)e^{x}}{\left(x+1\right)^{2}}dx = f\left(x\right)e^{x} + C

, Where C is a constant, then

\frac{d^{3}f}{dx^{3}}

at x = 1 is equal to :

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

Q38·MathematicsSingle correctJEE Main 2022

\int \frac{1}{x}\sqrt{\frac{1-x}{1+x}}\,dx = g\left(x\right) + c

g\left(1\right) = 0

, then

g\left(\frac{1}{2}\right)

is equal to :

(A) $\log_{e}\left(\frac{\sqrt{3}-1}{\sqrt{3}+1}\right) + \frac{\pi}{3}$
(B) $\log_{e}\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right) + \frac{\pi}{3}$
(C) $\log_{e}\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right) - \frac{\pi}{3}$
(D) $\frac{1}{2}\log_{e}\left(\frac{\sqrt{3}-1}{\sqrt{3}+1}\right) - \frac{\pi}{6}$

Q39·MathematicsSingle correctJEE Main 2022

Let

g : (0, \infty) \to R

be a differentiable function such that

\int\left(\frac{x(\cos x - \sin x)}{e^x + 1} + \frac{g(x)(e^x + 1 - xe^x)}{(e^x + 1)^2}\right)dx = \frac{xg(x)}{e^x + 1} + c

, for all

x > 0

, where c is an arbitrary constant. Then.

(A)g is decreasing in $\left(0, \frac{\pi}{4}\right)$
(B)g' is increasing in $\left(0, \frac{\pi}{4}\right)$
(C)g + g' is increasing in $\left(0, \frac{\pi}{2}\right)$
(D)g − g' is increasing in $\left(0, \frac{\pi}{2}\right)$

Q40·MathematicsSingle correctJEE Main 2021

The integral

\int \frac{1}{\sqrt[4]{(x - 1)^{3} (x + 2)^{5}}} dx

is equal to : (where C is a constant of integration)

(A) $\frac{3}{4}\left(\frac{x + 2}{x - 1}\right)^{\frac{1}{4}} + C$
(B) $\frac{3}{4}\left(\frac{x + 2}{x - 1}\right)^{\frac{5}{4}} + C$
(C) $\frac{4}{3}\left(\frac{x - 1}{x + 2}\right)^{\frac{1}{4}} + C$
(D) $\frac{4}{3}\left(\frac{x - 1}{x + 2}\right)^{\frac{5}{4}} + C$

Q41·MathematicsNumericalJEE Main 2021

\int \frac{\sin x}{\sin^{3} x + \cos^{3} x} dx = \alpha \log_{e} \left| 1 + \tan x \right| + \beta \log_{e} \left| 1 - \tan x + \tan^{2} x \right| + \gamma \tan^{-1} \left( \frac{2 \tan x - 1}{\sqrt{3}} \right) + C

, when C is constant of integration, then the value of

18(\alpha + \beta + \gamma^{2})

is ______.

Q42·MathematicsNumericalJEE Main 2021

\int \frac{2e^{x} + 3e^{-x}}{4e^{x} + 7e^{-x}}\,dx = \frac{1}{14}\left(ux + v\,\log_{e}(4e^{x} + 7e^{-x})\right) + C

, where C is a constant of integration, then u + v is equal to ___________ .

Q43·MathematicsNumericalJEE Main 2021

\int \frac{dx}{\left(x^{2} + x + 1\right)^{2}} = a\tan^{-1}\left(\frac{2x + 1}{\sqrt{3}}\right) + b\left(\frac{2x + 1}{x^{2} + x + 1}\right) + C

x > 0

where

C

is the constant of integration, then the value of

9\left(\sqrt{3}a + b\right)

is equal to ________.

Q44·MathematicsNumericalJEE Main 2021

f\left(x\right) = \displaystyle\int \dfrac{5x^{8} + 7x^{6}}{\left(x^{2} + 1 + 2x^{7}\right)^{2}}\,dx, \left(x \geq 0\right), f\left(0\right) = 0

and

f\left(1\right) = \dfrac{1}{K}

, then the value of K is

Q45·MathematicsSingle correctJEE Main 2021

The integral

\displaystyle\int \dfrac{(2x-1)\cos\sqrt{(2x-1)^{2}+5}}{\sqrt{4x^{2}-4x+6}}\,dx

is equal to (where c is a constant of integration)

(A) $\dfrac{1}{2}\sin\sqrt{\left(2x-1\right)^{2}+5} + c$
(B) $\dfrac{1}{2}\cos\sqrt{\left(2x+1\right)^{2}+5} + c$
(C) $\dfrac{1}{2}\cos\sqrt{\left(2x-1\right)^{2}+5} + c$
(D) $\dfrac{1}{2}\sin\sqrt{\left(2x+1\right)^{2}+5} + c$

Q46·MathematicsNumericalJEE Main 2021

For real numbers

\alpha

\beta

\gamma

and

\delta

, if

\int\frac{(x^{2}-1)+\tan^{-1}\left(\frac{x^{2}+1}{x}\right)}{(x^{4}+3x^{2}+1)\tan^{-1}\left(\frac{x^{2}+1}{x}\right)}dx

=\alpha\log_{e}\left(\tan^{-1}\left(\frac{x^{2}+1}{x}\right)\right)+\beta\tan^{-1}\left(\frac{\gamma(x^{2}-1)}{x}\right)+\delta\tan^{-1}\left(\frac{x^{2}+1}{x}\right)+C

where C is an arbitrary constant, then the value of

10(\alpha+\beta\gamma+\delta)

is equal to __________.

Q47·MathematicsSingle correctJEE Main 2021

The integral ∫e

_{4log}

_{e}

^{3log}_{x}

^{e}

^{2x}_{3log}

+5e

_{e}

_{x}^{2log}

–7e

^{e}

^{2x}_{2log}_{e}

_{x}

dx, x > 0, is equal to: (where c is a constant of integration)

(A)loge|x $^{2}$ + 5x – 7| + c
(B)14 loge|x $^{2}$ + 5x – 7| + c
(C)4loge|x $^{2}$ + 5x – 7| + c
(D)logex $^{2}$ +5x –7 + c

Q48·MathematicsSingle correctJEE Main 2021

The value of the integral

\int \frac{\sin\theta \cdot \sin 2\theta (\sin^{6}\theta + \sin^{4}\theta + \sin^{2}\theta)\sqrt{2\sin^{4}\theta + 3\sin^{2}\theta + 6}}{1 - \cos 2\theta}\, d\theta

is (where c is a constant of integration)

(A) $\frac{1}{18}\left[9 - 2\sin^{6}\theta - 3\sin^{4}\theta - 6\sin^{2}\theta\right]^{\frac{3}{2}} + c$
(B) $\frac{1}{18}\left[11 - 18\sin^{2}\theta + 9\sin^{4}\theta - 2\sin^{6}\theta\right]^{\frac{3}{2}} + c$
(C) $\frac{1}{18}\left[11 - 18\cos^{2}\theta + 9\cos^{4}\theta - 2\cos^{6}\theta\right]^{\frac{3}{2}} + c$
(D) $\frac{1}{18}\left[9 - 2\cos^{6}\theta - 3\cos^{4}\theta - 6\cos^{2}\theta\right]^{\frac{3}{2}} + c$

Q49·MathematicsSingle correctJEE Main 2021

\int \frac{\cos x - \sin x}{\sqrt{8 - \sin 2x}} dx = a \sin^{-1}\left(\frac{\sin x + \cos x}{b}\right) + c

, where

c

is a constant of integration, then the ordered pair

(a, b)

is equal to :

(A) $(1, -3)$
(B) $(1, 3)$
(C) $(-1, 3)$
(D) $(3, 1)$

Q50·MathematicsSingle correctJEE Main 2020

\int (e^{2x} + 2e^{x} - e^{-x} - 1)e^{(e^{x}+e^{-x})}dx = g(x)e^{(e^{x}+e^{-x})}+c

where c is a constant of integration, then g(0) is equal to:

(A)e
(B)2
(C) $e^{2}$
(D)1

Q51·MathematicsSingle correctJEE Main 2020

\displaystyle\int\dfrac{\cos\theta}{5+7\sin\theta-2\cos^2\theta}\,d\theta=A\log_e\left|B(\theta)\right|+C

, where C is a constant of integration, then

\dfrac{B(\theta)}{A}

can be:

(A) $\dfrac{5(\sin\theta+3)}{2\sin\theta+1}$
(B) $\dfrac{5(2\sin\theta+1)}{\sin\theta+3}$
(C) $\dfrac{2\sin\theta+1}{5(\sin\theta+3)}$
(D) $\dfrac{2\sin\theta+1}{\sin\theta+3}$

Q52·MathematicsSingle correctJEE Main 2020

The integral

\int \left( \frac{x}{x \sin x + \cos x} \right)^{2} dx

is equal to (where C is constant of integration):

(A) $\tan x + \frac{x \sec x}{x \sin x + \cos x} + C$
(B) $\tan x - \frac{x \sec x}{x \sin x + \cos x} + C$
(C) $\sec x + \frac{x \tan x}{x \sin x + \cos x} + C$
(D) $\sec x - \frac{x \tan x}{x \sin x + \cos x} + C$

Q53·MathematicsSingle correctJEE Main 2020

\int \sin^{-1} \left( \sqrt{\frac{x}{1+x}} \right) dx = A(x) \tan^{-1} \left( \sqrt{x} \right) + B(x) + C,

where C is a constant of integration, then the ordered pair

\left( A(x), B(x) \right)

can be:

(A) $\left( x - 1, -\sqrt{x} \right)$
(B) $\left( x - 1, \sqrt{x} \right)$
(C) $\left( x + 1, -\sqrt{x} \right)$
(D) $\left( x + 1, \sqrt{x} \right)$

Q54·MathematicsSingle correctJEE Main 2020

The integral

\int \dfrac{dx}{(x+4)^{8/7}(x-3)^{6/7}}

is equal to: (where C is a constant of integration)

(A) $-\dfrac{1}{13}\left(\dfrac{x-3}{x+4}\right)^{-13/7}+C$
(B) $-\left(\dfrac{x-3}{x+4}\right)^{-1/7}+C$
(C) $\left(\dfrac{x-3}{x+4}\right)^{1/7}+C$
(D) $\dfrac{1}{2}\left(\dfrac{x-3}{x+4}\right)^{3/7}+C$

Q55·MathematicsSingle correctJEE Main 2020

\int \dfrac{d\theta}{\cos^{2}\theta\,(\tan 2\theta + \sec 2\theta)} = \lambda \tan\theta + 2\log_{e}\left|f(\theta)\right| + C

where C is a constant of integration, then the ordered pair

\left(\lambda,\, f(\theta)\right)

is equal to:

(A) $(-1, 1 + \tan\theta)$
(B) $(1, 1 - \tan\theta)$
(C) $(1, 1 + \tan\theta)$
(D) $(-1, 1 - \tan\theta)$

Q56·MathematicsSingle correctJEE Main 2020

f'(x)=\tan^{-1}(\sec x+\tan x), -\dfrac{\pi}{2}<x<\dfrac{\pi}{2}

, and

f(0)=0

, then

f(1)

is equal to:

(A) $\dfrac{\pi-1}{4}$
(B) $\dfrac{\pi+1}{4}$
(C) $\dfrac{\pi+2}{4}$
(D) $\dfrac{1}{4}$

Q57·MathematicsSingle correctJEE Main 2020

\displaystyle\int \dfrac{\cos x\,dx}{\sin^{3}x\left(1+\sin^{6}x\right)^{2/3}}=f(x)\left(1+\sin^{6}x\right)^{1/\lambda}+c

where c is a constant of integration, then

\lambda f\left(\dfrac{\pi}{3}\right)

is equal to:

(A)-2
(B) $-\dfrac{9}{8}$
(C) $\dfrac{9}{8}$
(D)2

Q58·MathematicsSingle correctJEE Main 2019

Let

\alpha \in (0, \pi/2)

be fixed. If the integral

\displaystyle\int \frac{\tan x + \tan \alpha}{\tan x - \tan \alpha}dx = A(x)\cos 2\alpha + B(x)\sin 2\alpha + C

, where C is a constant of integration, then the functions A(x) and B(x) are respectively:

(A) $x+\alpha$ and $\log_{e}\left|\sin(x-\alpha)\right|$
(B) $x-\alpha$ and $\log_{e}\left|\cos(x-\alpha)\right|$
(C) $x-\alpha$ and $\log_{e}\left|\sin(x-\alpha)\right|$
(D) $x+\alpha$ and $\log_{e}\left|\sin(x+\alpha)\right|$

Q59·MathematicsSingle correctJEE Main 2019

The integral

\int \frac{2x^{3}-1}{x^{4}+x}\, dx

is equal to : (Here C is a constant of integration)

(A) $\frac{1}{2}\log_{e}\left|\frac{x^{3}+1}{x^{2}}\right|+C$
(B) $\frac{1}{2}\log_{e}\frac{|x^{3}+1|^{2}}{|x^{3}|}+C$
(C) $\log_{e}\left|\frac{|x^{3}+1|}{x}\right|+C$
(D) $\log_{e}\frac{|x^{3}+1|}{x^{2}}+C$

Q60·MathematicsSingle correctJEE Main 2019

\displaystyle\int \dfrac{dx}{(x^{2}-2x+10)^{2}}=A\left(\tan^{-1}\left(\dfrac{x-1}{3}\right)+\dfrac{f(x)}{x^{2}-2x+10}\right)+C

where C is a constant of integration, then:

(A) $A=\dfrac{1}{27}$ and $f(x)=-(x-1)$
(B) $A=\dfrac{1}{54}$ and $f(x)=9(x-1)^{2}$
(C) $A=\dfrac{1}{54}$ and $f(x)=3(x-1)$
(D) $A=\dfrac{1}{81}$ and $f(x)=3(x-1)$

Q61·MathematicsSingle correctJEE Main 2019

\int x^{5} e^{-x^{2}} dx = g(x) e^{-x^{2}} + c

, where c is a constant of integration, then

g(-1)

is equal to

(A)-1
(B)1
(C) $-\dfrac{5}{2}$
(D) $-\dfrac{1}{2}$

Q62·MathematicsSingle correctJEE Main 2019

The integral

\displaystyle\int \sec^{2/3}x\,\text{cosec}^{4/3}x\,dx

is equal to: (Here C is a constant of integration)

(A) $3\tan^{-1/3}x+C$
(B) $-\dfrac{3}{4}\tan^{-4/3}x+C$
(C) $-3\cot^{-1/3}x+C$
(D) $-3\tan^{-1/3}x+C$

Q63·MathematicsSingle correctJEE Main 2019

\int \frac{\sin\frac{5x}{2}}{\sin\frac{x}{2}}dx

is equal to: (where c is a constant of integration).

(A) $x+2\sin x+2\sin 2x+c$
(B) $2x+\sin x+2\sin 2x+c$
(C) $x+2\sin x+\sin 2x+c$
(D) $2x+\sin x+\sin 2x+c$

Q64·MathematicsSingle correctJEE Main 2019

The integral

\int \cos(\log_{e}x)dx

is equal to: (where C is a constant of integration)

(A) $\frac{x}{2}\left[\sin(\log_{e}x)-\cos(\log_{e}x)\right]+C$
(B) $x\left[\cos(\log_{e}x)+\sin(\log_{e}x)\right]+C$
(C) $\frac{x}{2}\left[\cos(\log_{e}x)+\sin(\log_{e}x)\right]+C$
(D) $x\left[\cos(\log_{e}x)-\sin(\log_{e}x)\right]+C$

Q65·MathematicsSingle correctJEE Main 2019

The integral

\displaystyle\int \dfrac{3x^{13} + 2x^{11}}{\left(2x^{4} + 3x^{2} + 1\right)^{4}} \, dx

is equal to (where C is a constant of integration)

(A) $\dfrac{x^{4}}{6\left(2x^{4}+3x^{2}+1\right)^{3}} + C$
(B) $\dfrac{x^{12}}{6\left(2x^{4}+3x^{2}+1\right)^{3}} + C$
(C) $\dfrac{x^{4}}{\left(2x^{4}+3x^{2}+1\right)^{3}} + C$
(D) $\dfrac{x^{12}}{\left(2x^{4}+3x^{2}+1\right)^{3}} + C$

Q66·MathematicsSingle correctJEE Main 2019

\int\frac{\sqrt{1-x^{2}}}{x^{4}}dx=A(x)(\sqrt{1-x^{2}})^{m}+C

, for a suitable chosen integer m and a function A(x), where C is a constant of integration, then

(A(x))^{m}

equals:

(A) $\frac{-1}{27x^{9}}$
(B) $\frac{-1}{3x^{3}}$
(C) $\frac{1}{27x^{6}}$
(D) $\frac{1}{9x^{4}}$

Q67·MathematicsSingle correctJEE Main 2019

\int \frac{x+1}{\sqrt{2x-1}}dx=f(x)\sqrt{2x-1}+C

, where C is a constant of integration of integration, then f(x) is equal to:

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

Q68·MathematicsSingle correctJEE Main 2019

f(x) = \int \frac{5x^{8} + 7x^{6}}{\left(x^{2} + 1 + 2x^{7}\right)^{2}} dx, (x \ge 0)

and

f(0) = 0

, then the value of

f(1)

is:

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

Q69·MathematicsSingle correctJEE Main 2019

For

x^2 \neq n\pi + 1

n \in N

(the set of natural numbers), the integral

\int x.\sqrt{\dfrac{2\sin(x^2-1) - \sin 2(x^2-1)}{2\sin(x^2-1) + \sin 2(x^2-1)}} \, dx

(A) $\log_e \left| \dfrac{1}{2}\sec^2(x^2-1) \right| + c$
(B) $\dfrac{1}{2}\log_e \left| \sec(x^2-1) \right| + c$
(C) $\dfrac{1}{2}\log_e \left| \sec^2\left(\dfrac{x^2-1}{2}\right) \right| + c$
(D) $\log_e \left| \sec\left(\dfrac{x^2-1}{2}\right) \right| + c$

Indefinite Integration — frequently asked

How many questions from Indefinite Integration appear in JEE?

Indefinite Integration has appeared in 66 of the last 186 JEE Main and JEE Advanced papers — about 35% of them — contributing 69 questions in total across those papers.

Is Indefinite Integration an important chapter for JEE?

Judged by how often it is actually tested, it appears in roughly 35% of papers. Chapters above about 50% are effectively guaranteed to show up every session, so they repay thorough preparation; lower-frequency chapters are better treated as targeted revision.

Where do these Indefinite Integration questions come from?

Every question is from an official JEE Main or JEE Advanced paper, transcribed from the original paper and tagged to this chapter. Answers follow the official answer key.

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