CUET Mathematics - Calculus - Integrals | Free Questions - Page 9

Q1:

2025: 13 May Shift 1

Calculus

>

Integrals

Medium

Applied

The value of the integral $I = \int_0^1 \frac{1}{\sqrt{1+3\sqrt{x}}} dx$ is

16/27

8/3

2/9

8/9

Correct Answer

Option 1

Correct Answer

Explanation

Q2:

2024: 16 May Shift 1

Calculus

>

Integrals

Medium

Common

$\int \dfrac{\pi}{x^{n+1}-x} d x=$

$\frac{\pi}{n} \log _{e}\left|\frac{x^{n}-1}{x^{n}}\right|+C$

$\quad \log _{e}\left|\frac{x^{n}+1}{x^{n}-1}\right|+C$

$\frac{\pi}{n} \log _{e}\left|\frac{x^{n}+1}{x^{n}}\right|+C$

$\pi \log _{e}\left|\frac{x^{n}}{x^{n}-1}\right|+C$

Correct Answer

Option 1

Correct Answer

Explanation

Q3:

2024: 16 May Shift 1

Calculus

>

Integrals

Medium

Common

The value of $\int_{0}^{1} \frac{a-b x^{2}}{\left(a+b x^{2}\right)^{2}} d x$ is :

$\frac{a-b}{a+b}$

$\frac{1}{a-b}$

$\frac{a+b}{2}$

$\frac{1}{a+b}$

Correct Answer

Option 4

Correct Answer

Explanation

Q4:

2024: 16 May Shift 1

Calculus

>

Integrals

Medium

Core

The value of the integral $\int_{\log _{e} 2}^{\log _{e} 3} \frac{e^{2 x}-1}{e^{2 x}+1} d x$ is :

$\log _{e} 3$

$\log _{e} 4-\log _{e} 3$

$\log _{e} 9-\log _{e} 4$

$\log _{e} 3-\log _{e} 2$

Correct Answer

Option 2

Correct Answer

Explanation

Q5:

2024: 16 May Shift 1

Calculus

>

Integrals

Medium

Core

$\int_{0}^{\frac{\pi}{2}} \frac{1-\cot x}{\operatorname{cosec} x+\cos x} d x=$

$0$

$\frac{\pi}{4}$

$\infty$

$\frac{\pi}{12}$

Correct Answer

Option 1

Correct Answer

Explanation

Q6:

2024: 16 May Shift 1

Calculus

>

Integrals

Medium

Core

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

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

$-\mathrm{e}^{\mathrm{x}} \sqrt{\mathrm{x}}+C$

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

$e^{x} \sqrt{x}+C$

Correct Answer

Option 4

Correct Answer

Explanation

Q7:

2023: 23 May Shift 3

Calculus

>

Integrals

Medium

Common

The integral $\int e^x \left(\frac{x-1}{2x^2}\right) dx$ is equal to:

$\frac{e^x}{x} + C$ , where C is constant of integration

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

$e^x x + C$ , where C is constant of integration

$x^2 e^x + C$ , where C is constant of integration

Correct Answer

Option 2

Correct Answer

Explanation

Q8:

2023: 23 May Shift 3

Calculus

>

Integrals

Easy

Common

The value of $\int \left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^2 dx$ is:

$\frac{x^2}{2} + \log|x| + C$ , (where C is constant of integration)

$\frac{x^2}{2} + \log|x| + 2x + C$ , (where C is constant of integration)

$\frac{3}{2}\left(\sqrt{x} + \frac{1}{\sqrt{x}}\right) + C$ , (where C is constant of integration)

$\frac{2}{3} x^{\frac{3}{2}} + 2\sqrt{x} + C$ , (where C is constant of integration)

Correct Answer

Option 2

Correct Answer

Explanation

Q9:

2023: 23 May Shift 3

Calculus

>

Integrals

Hard

Core

$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{dx}{1 + \sqrt{\tan x}}$ is equal to :

$\frac{\pi}{6}$

$\frac{\pi}{12}$

$\frac{\pi}{4}$

$\frac{\pi}{3}$

Correct Answer

Option 2

Correct Answer

Explanation

Q10:

2023: 23 May Shift 3

Calculus

>

Integrals

Medium

Core

The value of integral $\int \sqrt{4x^2 + 9}\, dx$ is

$\frac{x}{2}\sqrt{4x^2 + 9} + \frac{9}{2}\log|2x + \sqrt{4x^2 + 9}| + C$

$\frac{x}{2}\sqrt{4x^2 + 9} + \frac{3}{2}\log|2x + \sqrt{4x^2 + 9}| + C$

$2x\sqrt{4x^2 + 9} + \frac{9}{2}\log|2x + \sqrt{4x^2 + 9}| + C$

$x\sqrt{4x^2 + 9} + \frac{9}{4}\log|2x + \sqrt{4x^2 + 9}| + C$

Correct Answer

Option 1

Correct Answer

Explanation

Q11:

2023: 22 May Shift 3

Calculus

>

Integrals

Medium

Common

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

$\tan^{-1} \frac{32}{27}$

$\tan^{-1} \frac{27}{32}$

$\log \frac{32}{27}$

$\log \frac{27}{32}$

Correct Answer

Option 3

Correct Answer

Explanation

Q12:

2023: 22 May Shift 3

Calculus

>

Integrals

Medium

Common

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

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

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

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

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

Correct Answer

Option 1

Correct Answer

Explanation

Q13:

2023: 22 May Shift 3

Calculus

>

Integrals

Medium

Core

$\int_0^{\pi/2} \sqrt{1 - \sin 2x} \, dx$ is equal to :

$2\sqrt{2}$

$2(\sqrt{2} + 1)$

$2$

$2(\sqrt{2} - 1)$

Correct Answer

Option 4

Correct Answer

Explanation

Q14:

2023: 22 May Shift 3

Calculus

>

Integrals

Medium

Core

$\int e^x (\tan x + \log_e \sec x) \, dx =$

$e^x \log_e \sec x + C$

$\log_e \sec x + C$

$e^x \tan x + C$

$e^x \sec x + C$

Correct Answer

Option 1

Correct Answer

Explanation

Q15:

2023: 30 May Shift 3

Calculus

>

Integrals

Easy

Common

$\int \left(x + \frac{1}{x}\right)^2 dx$ equals :

$\frac{x^3}{3} + \frac{1}{x} - 2x + c$

$\frac{x^3}{3} - \frac{1}{x} + 2x + c$

$\frac{x^3}{3} - \frac{1}{x} - 2x + c$

$\frac{x^3}{3} + \frac{1}{x} + 2x + c$

Correct Answer

Option 2

Correct Answer

Explanation

Q16:

2023: 30 May Shift 3

Calculus

>

Integrals

Medium

Core

$\int \frac{\sqrt{\tan x}}{\sin x \cos x} dx$ equals :

$2\sqrt{\tan x} + c$

$2\sqrt{\cot x} + c$

$\sqrt{\tan x} + c$

$\frac{2}{\sqrt{\tan x}} + c$

Correct Answer

Option 1

Correct Answer

Explanation

Q17:

2023: 30 May Shift 3

Calculus

>

Integrals

Medium

Core

$\int e^x \sec x (1 + \tan x) dx$ equals :

$e^x \sec x + c$

$e^x \tan x + c$

$e^x \sin x + c$

$e^x \cos x + c$

Correct Answer

Option 1

Correct Answer

Explanation

Q18:

2023: 15 June Shift 2

Calculus

>

Integrals

Easy

Common

The value of the integral $\int_{2}^{4} \frac{x}{x^2+1} dx$ is :

$\frac{1}{2}\log\left(\frac{5}{17}\right)$

$\frac{1}{2}\log\left(\frac{17}{5}\right)$

$2\log\left(\frac{5}{17}\right)$

$2\log\left(\frac{17}{5}\right)$

Correct Answer

Option 2

Correct Answer

Explanation

Q19:

2023: 15 June Shift 2

Calculus

>

Integrals

Easy

Core

The value of $\int_{0}^{3} |2x - 6| dx$ is :

$-\frac{9}{4}$

$\frac{19}{2}$

9

$-3$

Correct Answer

Option 3

Correct Answer

Explanation

Q20:

2023: 15 June Shift 2

Calculus

>

Integrals

Hard

Core

The integral $\int \frac{dx}{x^2(x^4+1)^{\frac{3}{4}}}$ equals __________.

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

$(x^4+1)^{\frac{1}{4}} + C$

$-(x^4+1)^{\frac{1}{4}} + C$

$-\left(\frac{x^4+1}{x^4}\right)^{\frac{1}{4}} + C$

Correct Answer

Option 4

Correct Answer

Explanation

Q21:

2023: 15 June Shift 2

Calculus

>

Integrals

Medium

Core

The integral $\int_{0}^{1} x(1-x)^n dx$ is equal to :

$\frac{1}{(n+2)(n+3)}$

$\frac{1}{(n+1)(n+2)}$

$\frac{1}{n(n+1)}$

$\frac{1}{(n-1)(n-2)}$

Correct Answer

Option 2

Correct Answer

Explanation

Q22:

2022: 7 Aug Shift 2

Calculus

>

Integrals

Easy

Core

$\int_{0}^{1} x^2 e^{x^3} \, dx$ is equal to :

$\frac{1}{3}(e - 1)$

$\frac{2}{3}(e - 1)$

$\frac{3}{4}(e - 1)$

$\frac{3}{7}(e - 1)$

Correct Answer

Option 1

Correct Answer

Explanation

Q23:

2022: 7 Aug Shift 2

Calculus

>

Integrals

Medium

Core

If $\frac{d}{dx}(f(x)) = 5x^4 - \frac{4}{x^5}$ such that $f(1) = 0$ . Then $f(2) - 2f\left(\frac{1}{2}\right)$ is equal to :

$-2$

2

0

1

Correct Answer

Option 2

Correct Answer

Explanation

Q24:

2022: 7 Aug Shift 2

Calculus

>

Integrals

Easy

Core

If $f(x) = \begin{cases} 1 - 2x, & x \leq 0 \\ 1 + 2x, & x > 0 \end{cases}$ , then $\int_{-1}^{1} f(x) \, dx =$

0

2

3

4

Correct Answer

Option 4

Correct Answer

Explanation

Q25:

2022: 7 Aug Shift 2

Calculus

>

Integrals

Easy

Core

$\int_{1}^{5} |x - 2| \, dx =$

5

4

9

1

Correct Answer

Option 1

Correct Answer

Explanation

Q26:

2022: 7 Aug Shift 2

Calculus

>

Integrals

Easy

Core

$\int \frac{1}{\cos^2 x (1 + \tan x)^3} \, dx =$

$\frac{-2}{(1 + \tan x)^2} + C$ , where C is constant of Integration

$\frac{-4}{(1 + \tan x)^4} + C$ , where C is constant of Integration

$\frac{-1}{2(1 + \tan x)^2} + C$ , where C is constant of Integration

$\log(1 + \tan x)^3 + C$ , where C is constant of Integration

Correct Answer

Option 3

Correct Answer

Explanation

Q27:

2022: 7 Aug Shift 2

Calculus

>

Integrals

Medium

Core

$\int \frac{x^2 + 4}{(x^2 + 3)(x^2 + 5)} \, dx = u \tan^{-1}\left(\frac{x}{\sqrt{3}}\right) + v \tan^{-1}\left(\frac{x}{\sqrt{5}}\right) + c$ , where c is arbitrary constant, then value of $\frac{1}{u^2} + \frac{1}{v^2}$ is equal to :

32

16

08

04

Correct Answer

Option 1

Correct Answer

Explanation

Q28:

2022: 7 Aug Shift 2

Calculus

>

Integrals

Medium

Core

For $a, b \in R$ and $a < b$ , $\int_{a}^{b} \frac{f(x)}{f(x) + f(a + b - x)} \, dx =$

$a - b$

$\frac{a + b}{2}$

$2b$

$\frac{b - a}{2}$

Correct Answer

Option 4

Correct Answer

Explanation

Q29:

2022: 17 Aug Shift 2

Calculus

>

Integrals

Hard

Core

If $\int(\sqrt{x+1} + \sqrt{x-1})^2 dx = \alpha x^2 + \beta x\sqrt{x^2-1} + \gamma \log|x+\sqrt{x^2-1}| + C$ , then value of $\alpha + \beta - 2\gamma$ is :

$-1$

$0$

$1$

$6$

Correct Answer

Option 2

Correct Answer

Explanation

Q30:

2022: 17 Aug Shift 2

Calculus

>

Integrals

Hard

Core

If $g(x) = \int \frac{dx}{x^{1/2} + x^{1/6}}$ , then $g(1) - g(0)$ is :

$4 - \frac{3\pi}{2}$

$5 - 6\log_e 2$

$5 + 6\log_e 2$

$-4 + \frac{3\pi}{2}$

Correct Answer

Option 4

Correct Answer

Explanation

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