CUET Mathematics - Calculus - Integrals | Free Questions

Q1:

2026: 31 May Shift 2

Calculus

>

Integrals

Medium

Common

$\displaystyle\int_0^2 x\sqrt{2-x}\,dx$ is equal to :

$\dfrac{8\sqrt{2}}{15}$

$\dfrac{2\sqrt{2}}{15}$

$\dfrac{16\sqrt{2}}{15}$

$\dfrac{\sqrt{2}}{15}$

Correct Answer

Option 3

Correct Answer

Explanation

Q2:

2026: 31 May Shift 2

Calculus

>

Integrals

Medium

Common

$\displaystyle\int \dfrac{x^3}{x+2}dx$ is equal to: (where $C$ is an arbitrary constant and $\log_e x = \log x$ )

$\dfrac{x^3}{3} - x^2 + 4x - 8\log|x+2| + c$

$\dfrac{x^3}{3} + 8\log|x+2| + c$

$x^3 + x^2 - 4x + \log|x+2| + c$

$x^3 + 4x + \log|x+2| + c$

Correct Answer

Option 1

Correct Answer

Explanation

Q3:

2026: 31 May Shift 2

Calculus

>

Integrals

Hard

Core

Match the LIST-I with LIST-II (where $[x]$ denotes greatest integer function)

	LIST-I Integral		LIST-II Values
A.	$\displaystyle\int_{0}^{2} [x]\, dx$	I.	$0$
B.	$\displaystyle\int_{0}^{1.5} [x^2]\, dx$	II.	$1$
C.	$\displaystyle\int_{-\pi/4}^{\pi/4} \sin x \, dx$	III.	$\sqrt{2}$
D.	$\displaystyle\int_{-\pi/4}^{\pi/4} \cos x \, dx$	IV.	$2-\sqrt{2}$

Choose the correct answer from the options given below:

A-II, B-IV, C-I, D-III

A-IV, B-II, C-I, D-III

A-II, B-IV, C-III, D-I

A-IV, B-II, C-III, D-I

Correct Answer

Option 1

Correct Answer

Explanation

Q4:

2026: 31 May Shift 2

Calculus

>

Integrals

Medium

Core

$\displaystyle\int \dfrac{1+\cos\theta}{1-\cos\theta}\,d\theta =$

$\tan\dfrac{\theta}{2} + \sec\dfrac{\theta}{2} + c$ , where $c$ is an arbitrary constant

$\tan\dfrac{\theta}{2} + \theta + c$ , where $c$ is an arbitrary constant

$\cot\left(\dfrac{\theta}{2}\right) + \theta + c$ , where $c$ is an arbitrary constant

$-2\cot\left(\dfrac{\theta}{2}\right) - \theta + c$ , where $c$ is an arbitrary constant

Correct Answer

Option 4

Correct Answer

Explanation

Q5:

2026: 31 May Shift 2

Calculus

>

Integrals

Hard

Core

The value of $\displaystyle\int_{\pi/6}^{\pi/3} \dfrac{\sin x+\cos x}{\sqrt{\sin 2x}}\,dx$ is:

$\sin^{-1}\left(\dfrac{\sqrt{3}+1}{2}\right)$

$2\sin^{-1}\left(\dfrac{\sqrt{3}+1}{2}\right)$

$2\sin^{-1}\left(\dfrac{\sqrt{3}-1}{2}\right)$

$\sin^{-1}\left(\dfrac{\sqrt{3}-1}{2}\right)$

Correct Answer

Option 3

Correct Answer

Explanation

Q6:

2026: 31 May Shift 2

Calculus

>

Integrals

Medium

Applied

In which of the following cases the value of the integral is more than 10?

A. $\displaystyle\int_0^2 |x|\,dx$

B. $\displaystyle\int_0^2 (x+3)^2\,dx$

C. $\displaystyle\int_0^2 e^x\,dx$

D. $\displaystyle\int_0^2 (2x+5)\,dx$

Choose the correct answer from the options given below:

A and C only

C and D only

B and D only

A and D only

Correct Answer

Option 3

Correct Answer

Explanation

Q7:

2026: 31 May Shift 2

Calculus

>

Integrals

Easy

Applied

If $\displaystyle\int_{-1}^{b} x^3\,dx = \dfrac{15}{4}$ , then $b$ can take the value

0

1

2

3

Correct Answer

Option 3

Correct Answer

Explanation

Q8:

2026: 30 May Shift 2

Calculus

>

Integrals

Easy

Common

If $\int_0^1 (4x^3 + 3x^2 + 2x + k)dx = 0$ , then the value of 'k' is

1

-1

3

-3

Correct Answer

Option 4

Correct Answer

Explanation

Q9:

2026: 30 May Shift 2

Calculus

>

Integrals

Medium

Common

$\int e^{x\log_e a}.e^x dx$ is:

[where C is an arbitrary constant]

$\frac{e^x}{1+\log_e(ae)} + C$

$\frac{a^x}{\log_e(ae)} + C$

$(ae)^x + C$

$\frac{a^xe^x}{1+\log_e a} + C$

Correct Answer

Option 4

Correct Answer

Explanation

Q10:

2026: 30 May Shift 2

Calculus

>

Integrals

Medium

Core

The value of integral $\int_{-\pi/2}^{\pi/2} \frac{1}{1+e^{\sin x}}dx$ is

$\pi$

$\frac{\pi}{2}$

$\frac{\pi}{4}$

$2\pi$

Correct Answer

Option 2

Correct Answer

Explanation

Q11:

2026: 30 May Shift 2

Calculus

>

Integrals

Medium

Core

If $0 < x < \frac{\pi}{4}$ , then $\int\sqrt{1-\sin2x}\,dx$ is:

[where C is an arbitrary constant]

$\sin x+\cos x+C$

$\sin x-\cos x+C$

$-\sin x-\cos x+C$

$-\sin x+\cos x+C$

Correct Answer

Option 1

Correct Answer

Explanation

Q12:

2026: 30 May Shift 2

Calculus

>

Integrals

Easy

Core

Match List-I with List-II

List-I	List-II
Definite integral	Value
(A) $\int_2^3 x^2 dx$	(I) $-\frac{1}{2}$
(B) $\int_{-1}^1 (x+1)dx$	(II) $\frac{19}{3}$
(C) $\int_0^1 (x-1)dx$	(III) $\frac{16}{3}$
(D) $\int_0^4 \sqrt{x}\,dx$	(IV) 2

Choose the correct answer from the options given below:

(A) - (II), (B) - (IV), (C) - (III), (D) - (I)

(A) - (I), (B) - (III), (C) - (IV), (D) - (II)

(A) - (IV), (B) - (II), (C) - (III), (D) - (I)

(A) - (II), (B) - (IV), (C) - (I), (D) - (III)

Correct Answer

Option 4

Correct Answer

Explanation

Q13:

2026: 30 May Shift 2

Calculus

>

Integrals

Medium

Applied

Match List-I with List-II

List-I	List-II
(A) The value of $I=\int_0^1 \frac{1}{\sqrt{x}}dx$ is	(I) 1
(B) The value of $I=\int_0^1 (5x^2+1)dx$ is	(II) $\frac{1}{2}$
(C) The value of $I=\int_0^\infty e^{-x}dx$ is	(III) $\frac{8}{3}$
(D) The value of $I=\int_1^e \frac{\log x}{x}dx$ is	(IV) 2

Choose the correct answer from the options given below:

(A) - (I), (B) - (III), (C) - (IV), (D) - (II)

(A) - (IV), (B) - (III), (C) - (I), (D) - (II)

(A) - (IV), (B) - (I), (C) - (II), (D) - (III)

(A) - (II), (B) - (IV), (C) - (III), (D) - (I)

Correct Answer

Option 2

Correct Answer

Explanation

Q14:

2026: 30 May Shift 1

Calculus

>

Integrals

Hard

Common

Which of the following is/are correct?

A. $\int e^x\left[\frac{1}{(x-2)}-\frac{1}{(x-2)^2}\right]dx=\frac{e^x}{(x-2)}+C$ , where C is an arbitrary constant.

B. $\int e^x\left[\frac{1}{(x-2)^3}-\frac{3}{(x-2)^4}\right]dx=\frac{e^x}{(x-2)^2}+C$ , where C is an arbitrary constant.

C. $\int e^x\left[\frac{x-4}{(x-2)^3}\right]dx=\frac{e^x}{(x-2)^2}+C$ , where C is an arbitrary constant.

D. $\int e^x\left[\frac{1}{x+2}-\frac{3}{(x+2)^2}\right]dx=\frac{e^x}{(x+2)^2}+C$ , where C is an arbitrary constant.

Choose the correct answer from the options given below:

A, B and D only

A, C and D only

B and C only

A only

Correct Answer

Option 4

Correct Answer

Explanation

Q15:

2026: 30 May Shift 1

Calculus

>

Integrals

Medium

Common

$\int e^x\left(\log x+\frac{1}{x^2}\right)dx$ is equal to: (consider $\log x=\log_e x$ )

[C is an arbitrary constant]

$e^x\log x + c$

$e^x\frac{1}{x^2}+c$

$e^x\left(\log x-\frac{1}{x}\right)+c$

$e^x\left(\log x+\frac{1}{x}\right)+c$

Correct Answer

Option 3

Correct Answer

Explanation

Q16:

2026: 30 May Shift 1

Calculus

>

Integrals

Medium

Core

$\int\frac{\tan^5\sqrt{x}.\sec^2\sqrt{x}}{\sqrt{x}}dx$ is equal to:

$\frac{1}{3}\tan^5\sqrt{x}+C$ , where C is an arbitrary constant

$\frac{1}{3}\tan^6\sqrt{x}+C$ , where C is an arbitrary constant

$\frac{1}{5}\tan^5\sqrt{x}+C$ , where C is an arbitrary constant

$\tan^6\sqrt{x}+C$ , where C is an arbitrary constant

Correct Answer

Option 2

Correct Answer

Explanation

Q17:

2026: 30 May Shift 1

Calculus

>

Integrals

Medium

Core

The value of $\int_{-\pi/3}^{\pi/3}\frac{1}{1+e^{\tan x}}dx$ is:

$\pi/3$

$-\pi/3$

$\frac{2\pi}{3}$

$\frac{\pi}{6}$

Correct Answer

Option 1

Correct Answer

Explanation

Q18:

2026: 30 May Shift 1

Calculus

>

Integrals

Easy

Core

The value of $\int_{-\pi/2}^{\pi/2}(\sin|x|+\cos|x|)dx$ equals:

1

2

3

4

Correct Answer

Option 4

Correct Answer

Explanation

Q19:

2026: 30 May Shift 1

Calculus

>

Integrals

Medium

Applied

$\int_{-2}^{1}|x^2-1|dx$ equals:

$17/3$

$8/3$

$16/3$

$2/3$

Correct Answer

Option 2

Correct Answer

Explanation

Q20:

2026: 30 May Shift 1

Calculus

>

Integrals

Medium

Applied

If the manufacturer's marginal cost function is $100\left(\frac{1}{\sqrt{x}}+50\right)$ , then the cost involved to increase production from 25 units to 100 units is:

₹3,00,000

₹2,75,000

₹3,76,000

₹2,80,000

Correct Answer

Option 3

Correct Answer

Explanation

Q21:

2026: 30 May Shift 1

Calculus

>

Integrals

Medium

Applied

$\int\frac{x-1}{x(x-\log x)}dx$ equals:

(consider: $\log x=\log_e x$ )

$\log|x-\log x|+C$ , C is an arbitrary constant.

$\log|x+\log x|+C$ , C is an arbitrary constant.

$(x-1)+\log x+C$ , C is an arbitrary constant.

$2\log|(x-1)+\log x|+C$ , C is an arbitrary constant.

Correct Answer

Option 1

Correct Answer

Explanation

Q22:

2026: 29 May Shift 2

Calculus

>

Integrals

Medium

Common

$\int_1^2 \frac{dx}{(x+1)(x+2)} =$

$\log_e 8 - \log_e 9$

$\log_e 9 - \log_e 8$

$\log_e 8 + \log_e 9$

$2\log_e 9 - \log_e 8$

Correct Answer

Option 2

Correct Answer

Explanation

Q23:

2026: 29 May Shift 2

Calculus

>

Integrals

Hard

Common

If $\int \frac{1}{\sqrt{(x-1)(x-2)}}\,dx = \log\left|(x-m)+\sqrt{ax^2+bx+n}\right| + C$ where C is an arbitrary constant and $m,n,a$ and $b$ are constant, then $2m+n+a+b$ is equal to:

3

5

10

6

Correct Answer

Option 1

Correct Answer

Explanation

Q24:

2026: 29 May Shift 2

Calculus

>

Integrals

Medium

Core

The value of $\int_0^{\pi/2} \frac{\cos x}{\cos\frac{x}{2}+\sin\frac{x}{2}}\,dx$ is

$\sqrt{2}$

$2\sqrt{2}-2$

$\sqrt{2}-2$

$2\sqrt{2}+2$

Correct Answer

Option 2

Correct Answer

Explanation

Q25:

2026: 29 May Shift 2

Calculus

>

Integrals

Hard

Core

If $\int \frac{\sin^6 x}{\cos^{10} x}\,dx = -2a\tan^9 x + b\tan^7 x + c$ , (where c is an arbitrary constant), then which one of the following is TRUE?

18a=7b

18a+7b=0

7a=18b

7a+18b=0

Correct Answer

Option 2

Correct Answer

Explanation

Q26:

2026: 29 May Shift 2

Calculus

>

Integrals

Medium

Core

$\int \frac{1}{x(x^6+1)}\,dx =$ (consider $\log_e x = \log x$ )

$\frac{1}{6}\log\left|\frac{x^6}{x^6+1}\right|+C$ , where C is an arbitrary constant

$\frac{1}{6}\log\left|\frac{x}{x^6+1}\right|+C$ , where C is an arbitrary constant

$\frac{1}{5}\log\left|\frac{x}{x^5+1}\right|+C$ , where C is an arbitrary constant

$\frac{1}{5}\log\left|\frac{x^5}{x^5+1}\right|+C$ , where C is an arbitrary constant

Correct Answer

Option 1

Correct Answer

Explanation

Q27:

2026: 29 May Shift 2

Calculus

>

Integrals

Medium

Applied

$\int \frac{dx}{x(x^{10}+1)}$ is equal to

$\log\left|\frac{x^{10}}{x^{10}+1}\right|+C$ , where C is an arbitrary constant

$\log\left|\frac{x^{10}+1}{x^{10}}\right|+C$ , where C is an arbitrary constant

$\frac{1}{10}\log\left|\frac{x^{10}+1}{x^{10}}\right|+C$ , where C is an arbitrary constant

$\frac{1}{10}\log\left|\frac{x^{10}}{x^{10}+1}\right|+C$ , where C is an arbitrary constant

Correct Answer

Option 4

Correct Answer

Explanation

Q28:

2026: 29 May Shift 1

Calculus

>

Integrals

Easy

Common

$\displaystyle\int \dfrac{xdx}{x^2+3x+2}$ is equal to:

(where $c$ is an arbitrary constant)

$-\log|x+1|+2\log|x+2|+c$

$\log|x+1|-2\log|x+2|+c$

$-2\log|x+1|+2\log|x+2|+c$

$\log|x+1|+2\log|x+2|+c$

Correct Answer

Option 1

Correct Answer

Explanation

Q29:

2026: 29 May Shift 1

Calculus

>

Integrals

Easy

Common

$\displaystyle\int_1^e x\log_ex\,dx$ is equal to:

$\dfrac{1}{4}(2e^2+1)$

$\dfrac{1}{4}(e^2+1)$

$\dfrac{1}{4}(e^2-1)$

$\dfrac{1}{4}(2e^2-1)$

Correct Answer

Option 2

Correct Answer

Explanation

Q30:

2026: 29 May Shift 1

Calculus

>

Integrals

Medium

Core

$\displaystyle\int \dfrac{(x-3)e^x}{(x-1)^3}dx$ is equal to:

(where C is an arbitrary constant)

$\dfrac{1}{e^x(x-1)^2}+C$

$\dfrac{(x-1)^2}{e^x}+C$

$(x-1)^2e^x+C$

$\dfrac{e^x}{(x-1)^2}+C$

Correct Answer

Option 4

Correct Answer

Explanation

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