CUET Mathematics - Calculus - Integrals | Free Questions - Page 10

Q1:

2022: 6 Aug Shift 2

Calculus

>

Integrals

Easy

Core

The value of $\int_0^1 e^x (x + 1) \, dx$ is equal to

$e + 1$

$3e$

$e$

$3e - 2$

Correct Answer

Option 3

Correct Answer

Explanation

Q2:

2022: 6 Aug Shift 2

Calculus

>

Integrals

Easy

Core

If $f'(x) = 4x^5 - 6x$ and $f(0) = 3$ , then $f(3)$ is equal to

$462$

$219$

$138$

$957$

Correct Answer

Option 1

Correct Answer

Explanation

Q3:

2022: 6 Aug Shift 2

Calculus

>

Integrals

Easy

Core

The value of $\int_{-4}^{4} \log_e \left( \frac{1-x}{1+x} \right) dx$ is equal to

$2$

$1$

$0$

$8 \log_e(2)$

Correct Answer

Option 3

Correct Answer

Explanation

Q4:

2022: 4 Aug Shift 1

Calculus

>

Integrals

Medium

Core

$\int \frac{x}{(x^2+3)(x^2+4)} dx =$

$\frac{1}{2}\log\left(\frac{x^2+4}{x^2+3}\right) + C$

$\log\left(\frac{x^2+3}{x^2+4}\right) + C$

$\frac{1}{2}\log\left(\frac{x^2+3}{x^2+4}\right) + C$

$\log\left(\frac{x^2+4}{x^2+3}\right) + C$

Correct Answer

Option 3

Correct Answer

Explanation

Q5:

2022: 4 Aug Shift 1

Calculus

>

Integrals

Medium

Core

$\int e^x \left(\frac{1}{x} - \frac{2}{x^3}\right) dx =$

$e^x \left(\frac{x+1}{x^2}\right) + C$

$e^x \left(\frac{x-1}{x^2}\right) + C$

$e^x \left(\frac{x-2}{x^2}\right) + C$

$e^x \left(\frac{x^2-2}{x^3}\right) + C$

Correct Answer

Option 1

Correct Answer

Explanation

Q6:

2022: 4 Aug Shift 1

Calculus

>

Integrals

Easy

Core

Match List I with List - II

List - I	List - II
A. An even function	I. $x^2 + \cos x$
B. For an even function, $\int_{-a}^{a} f(x)dx =$	II. 0
C. If $f(2a-x) = -f(x)$ , then $\int_{0}^{2a} f(x)dx =$	III. $2\int_{0}^{a} f(x)dx$
D. An odd function	IV. $x^3 + \sin x$

Choose the correct answer from the options given below:

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

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

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

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

Correct Answer

Option 4

Correct Answer

Explanation

Q7:

2022: 4 Aug Shift 1

Calculus

>

Integrals

Medium

Core

$\int_{0}^{\pi} \frac{e^{\cos x}}{e^{\cos x} + e^{-\cos x}} dx =$

$\pi/4$

$\pi/3$

$\pi/6$

$\pi/2$

Correct Answer

Option 4

Correct Answer

Explanation

Q8:

2022: 10 Aug Shift 1

Calculus

>

Integrals

Medium

Core

If $\int (x + \sqrt{x^2 - 1})^2 \, dx = \alpha \cdot x + \beta x^3 + \gamma (x^2 - 1)^{\frac{3}{2}} + C$ , where $C$ is arbitrary constant, then the value of $3(\alpha + \beta + \gamma)$ is

44

23

11

1

Correct Answer

Option 4

Correct Answer

Explanation

Q9:

2022: 10 Aug Shift 1

Calculus

>

Integrals

Medium

Core

$\int_0^1 \frac{dx}{x^2 + x + 1}$

$\frac{\pi}{3\sqrt{3}}$

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

$\frac{\pi}{6\sqrt{3}}$

$\frac{\pi}{\sqrt{3}}$

Correct Answer

Option 1

Correct Answer

Explanation

Q10:

2022: 10 Aug Shift 1

Calculus

>

Integrals

Medium

Core

Match List I with List II

List - I	List - II
A. $\int_{-\pi/2}^{\pi/2} \sin^7 x \, dx$	I. $\frac{\pi}{2}$
B. $\int_{-\pi/2}^{\pi/2} \sin^2 x \, dx$	II. $\frac{\pi}{4}$
C. $\int_0^{\pi/2} \frac{1}{1 + \tan x} \, dx$	III. 0
D. $\int_0^{\pi} \lvert \cos x \rvert \, dx$	IV. 2

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

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

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

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

Correct Answer

Option 2

Correct Answer

Explanation

Q11:

2022: 10 Aug Shift 1

Calculus

>

Integrals

Hard

Core

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

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

$\frac{-e^x}{1+x^2} + C$

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

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

Correct Answer

Option 1

Correct Answer

Explanation

Q12:

2022: 10 Aug Shift 1

Calculus

>

Integrals

Easy

Core

$\int \frac{\sin(\tan^{-1} x)}{1 + x^2} dx =$

$-\cos(\tan^{-1} x) + C$

$\cos(\tan^{-1} x) + C$

$\tan(\cos^{-1} x) + C$

$-\tan(\cos^{-1} x) + C$

Correct Answer

Option 1

Correct Answer

Explanation

Q13:

2022: 10 Aug Shift 1

Calculus

>

Integrals

Hard

Core

$\int \frac{x^2 + 1}{x^4 + 1} dx =$

$\log(x^4 + 1) + C$

$\log\left(\frac{x^2 - 1}{x^2 + 1}\right) + C$

$\frac{1}{\sqrt{2}} \tan^{-1}\left(\frac{x^2 - 1}{\sqrt{2} \, x}\right) + C$

$\frac{1}{\sqrt{2}} \tan^{-1}\left(\frac{x^2 + 1}{\sqrt{2} \, x}\right) + C$

Correct Answer

Option 3

Correct Answer

Explanation

Q14:

2022: 10 Aug Shift 1

Calculus

>

Integrals

Hard

Core

$\int_0^{4\pi} \frac{x}{1 + |\cos x|} dx =$

$4\pi$

$16\pi$

$32\pi$

$64\pi$

Correct Answer

Option 2

Correct Answer

Explanation

Q15:

2022: 10 Aug Shift 1

Calculus

>

Integrals

Hard

Core

If $\int_0^{\pi/2} \sqrt{\tan x} \, dx = \frac{\lambda}{\sqrt{2}}$ , then the value of $\lambda$ is

$\pi$

$-1$

$\frac{\pi}{2}$

$\frac{3}{\sqrt{2}}$

Correct Answer

Option 1

Correct Answer

Explanation

Q16:

2022: 30 Aug Shift 1

Calculus

>

Integrals

Medium

Core

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

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

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

$\frac{x}{x+1} + c$

$\frac{x}{x-1} + c$

Correct Answer

Option 1

Correct Answer

Explanation

Q17:

2022: 30 Aug Shift 1

Calculus

>

Integrals

Hard

Core

$\int_{\frac{1}{3}}^{1} \frac{(x - x^3)^{\frac{1}{3}}}{x^4} dx =$

3

4

6

0

Correct Answer

Option 3

Correct Answer

Explanation

Q18:

2022: 30 Aug Shift 1

Calculus

>

Integrals

Medium

Core

$\int \sqrt{1 - 49x^2} \, dx$ is equal to

$\frac{x}{2}\left(\sqrt{1 - 49x^2}\right) + \frac{1}{98}\sin^{-1}7x + C$

$\frac{7x}{2}\sqrt{1 + 49x^2} + \frac{1}{49}\sin^{-1}x + C$

$\frac{x}{2}\sqrt{1 + \frac{1}{7x^2}} - \frac{1}{49}\sin^{-1}7x + C$

$\frac{x}{2}\sqrt{1 - 49x^2} + \frac{1}{14}\sin^{-1}7x + C$

Correct Answer

Option 4

Correct Answer

Explanation

Q19:

2022: 30 Aug Shift 1

Calculus

>

Integrals

Medium

Core

$\int x\sqrt{x + 2} \, dx$ is equal to :

$\frac{2}{5}(x+2)^{\frac{5}{2}} - \frac{4}{3}(x+2)^{\frac{3}{2}} + C$

$\frac{2}{5}(x+2)^{\frac{5}{2}} + \frac{4}{3}(x+2)^{\frac{3}{2}} + C$

$\frac{1}{5}(x+2)^{\frac{5}{2}} - \frac{2}{3}(x+2)^{\frac{3}{2}} + C$

$\frac{2}{5}(x+2)^{\frac{5}{2}} + \frac{4}{3}(x+2)^{\frac{3}{2}} + C$

Correct Answer

Option 1

Correct Answer

Explanation

Q20:

2022: 16 July Shift 2

Calculus

>

Integrals

Medium

Core

The value of $\int_{-3}^{3} \log_e\left(\frac{4+x}{4-x}\right) dx$ is

2

0

-1

3

Correct Answer

Option 2

Correct Answer

Explanation

Q21:

2022: 16 July Shift 2

Calculus

>

Integrals

Hard

Core

$\int \frac{x^2+1}{(x+1)^2} e^x \, dx$ is equal to

$\left(\frac{x-1}{x+1}\right) e^x + c$ , where c is a constant of integration

$\left(\frac{1-x}{x+1}\right) e^x + c$ , where c is a constant of integration

$\left(\frac{x}{x+1}\right) e^x + c$ , where c is a constant of integration

$\frac{2e^x}{x+1} + c$ , where c is a constant of integration

Correct Answer

Option 1

Correct Answer

Explanation

Q22:

2022: 16 July Shift 2

Calculus

>

Integrals

Medium

Core

$\int \frac{x}{(x-1)(x-2)} dx =$

(where c is an arbitrary constant)

$\log\left|\frac{x^2-4x+4}{x-1}\right| + c$

$\log\left|\frac{x-1}{x-2}\right| + c$

$\log\left|\frac{x^2-2x+1}{x-2}\right| + c$

$\log\left|\frac{x-2}{x+1}\right| + c$

Correct Answer

Option 1

Correct Answer

Explanation

Q23:

2022: 16 July Shift 2

Calculus

>

Integrals

Medium

Core

The value of $\int_{\pi/2}^{\pi} \frac{1}{1 + \cot x} dx$ is equal to:

$\frac{\pi}{2}$

$\frac{\pi}{4}$

$\frac{\pi}{6}$

$\pi$

Correct Answer

Option 2

Correct Answer

Explanation

Q24:

2022: 16 July Shift 2

Calculus

>

Integrals

Easy

Core

Match List I with List II

List I	List II
A. $\int \frac{dx}{x^2 - a^2}$ is equal to	I. $\frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2}\log\left\lvert x + \sqrt{x^2 + a^2}\right\rvert + C$
B. $\int \frac{dx}{\sqrt{x^2 + a^2}}$ is equal to	II. $\frac{1}{2a}\log\left\lvert \frac{x-a}{x+a}\right\rvert + C$
C. $\int \sqrt{a^2 - x^2} \, dx$ is equal to	III. $\frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C$
D. $\int \sqrt{a^2 + x^2} \, dx$ is equal to	IV. $\log\left\lvert x + \sqrt{x^2 + a^2}\right\rvert + C$

Choose the correct answer from the options given below:

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

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

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

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

Correct Answer

Option 3

Correct Answer

Explanation

Q25:

2022: 23 Aug Shift 1

Calculus

>

Integrals

Medium

Core

$\int \frac{dx}{(e^x - 1)} =$

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

$\log\left(\frac{1 - e^{2x}}{e^x}\right) + C$

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

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

Correct Answer

Option 3

Correct Answer

Explanation

Q26:

2022: 23 Aug Shift 1

Calculus

>

Integrals

Medium

Core

If $\int (1 + e^{-x} + e^{-2x} + ...)dx = \log\phi(x) + C$ , then $\phi(x)$ is equal to :

$x + 1$

$1 - x$

$e^x + 1$

$e^x - 1$

Correct Answer

Option 4

Correct Answer

Explanation

Q27:

2022: 23 Aug Shift 1

Calculus

>

Integrals

Hard

Core

Let $\int \frac{dx}{\left(\sqrt{x} - \sqrt{x-1}\right)^2} = \alpha u(x) + \beta v(x) + C$ , where $u(x) = x^2 - x + \left(x - \frac{1}{2}\right)\sqrt{x^2 - x}$ and $v(x) = \log\left|x - \frac{1}{2} + \sqrt{x^2 - x}\right|$ . The value of $\alpha + \beta$ is :

$\frac{3}{4}$

$\frac{1}{4}$

$-\frac{1}{4}$

$\frac{9}{4}$

Correct Answer

Option 1

Correct Answer

Explanation

Q28:

2022: 23 Aug Shift 1

Calculus

>

Integrals

Medium

Core

$\int \frac{x^2 - 4}{(x+2)(x-1)(x-3)} dx =$

$2 \log |C(x-1)(x-3)|$

$\log_e |C(x-1)(x-3)|$

$\frac{1}{2}\log_e |C(x-1)(x-3)|$

$\frac{1}{2}\log_e |C(x+2)(x-3)|$

Correct Answer

Option 3

Correct Answer

Explanation

Q29:

2022: 23 Aug Shift 1

Calculus

>

Integrals

Easy

Core

$\int_{-\pi/2}^{\pi/2} \sin^7 x \, dx =$

$-1$

$1$

$0$

$\pi$

Correct Answer

Option 3

Correct Answer

Explanation

Q30:

2022: 23 Aug Shift 1

Calculus

>

Integrals

Easy

Core

$\int e^x \left(\frac{1}{x} - \frac{1}{x^2}\right) dx =$

$e^x \cdot \frac{1}{x} + C$

$-e^x \cdot \frac{1}{x} + C$

$e^x + C$

$e^x \cdot \frac{1}{x^2} + C$

Correct Answer

Option 1

Correct Answer

Explanation

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