Question 1: Palette 1/251

JEE Mains 2026 MCQ

Let [.] denote the greatest integer function. Then

$ \int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \frac{12(3+[x])}{3+\left[\sin x\right]+\left[\cos x\right]} \right) dx $
is equal to :

A

$12\pi+5$

B

$11\pi+2$

C

$15\pi+4$

D

$13\pi+1$

Question 2: Palette 2/251

JEE Mains 2026 MCQ

Let $f$ be a polynomial function such that $f\left(x^2+1\right)=x^4+5 x^2+2$, for all $x \in \mathbb{R}$.

Then $\int\limits_0^3 f(x) d x$ is equal to

A

$\frac{33}{2}$

B

$\frac{5}{3}$

C

$\frac{27}{2}$

D

$\frac{41}{3}$

Question 3: Palette 3/251

JEE Mains 2026 MCQ

The value of the integral $\int_{\frac{\pi}{24}}^{\frac{5 \pi}{24}} \frac{\mathrm{~d} x}{1+\sqrt[3]{\tan 2 x}}$ is :

A

$\frac{\pi}{3}$

B

$\frac{\pi}{18}$

C

$\frac{\pi}{6}$

D

$\frac{\pi}{12}$

Question 4: Palette 4/251

JEE Mains 2026 MCQ

The value of $\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{1}{[x]+4}\right) d x$, where $[\cdot]$ denotes the greatest integer function, is

A

$\frac{1}{60}(21 \pi-1)$

B

$\frac{1}{60}(\pi-7)$

C

$\frac{7}{60}(\pi-3)$

D

$\frac{7}{60}(3 \pi-1)$

Question 5: Palette 5/251

JEE Mains 2026 MCQ

Let $f:[1, \infty) \rightarrow \mathbb{R}$ be a differentiable function. If $6 \int\limits_1^x f(t) d t=3 x f(x)+x^3-4$ for all $x \geq 1$, then the value of $f(2)-f(3)$ is :

A

4

B

3

C

-4

D

-3

Question 6: Palette 6/251

JEE Mains 2026 MCQ

The value of $\int\limits_{-\pi / 6}^{\pi / 6}\left(\frac{\pi+4 x^{11}}{1-\sin (|x|+\pi / 6)}\right) d x$ is equal to:

A

$8 \pi$

B

$4 \pi$

C

$2 \pi$

D

$6 \pi$

Question 7: Palette 7/251

JEE Mains 2026 MCQ

The value of the integral $\int\limits_0^2 \frac{\sqrt{x\left(x^2+x+1\right)}}{(\sqrt{x+1})\left(\sqrt{x^4+x^2+1}\right)} \mathrm{d} x$ is equal to:

A

$\frac{1}{3} \log _e(3-2 \sqrt{2})$

B

$\frac{2}{3} \log _e(4+\sqrt{2})$

C

$\frac{2}{3} \log _e(3+2 \sqrt{2})$

D

$\frac{1}{3} \log _e(1+6 \sqrt{2})$

Question 8: Palette 8/251

JEE Mains 2026 MCQ

Let $f(x)=\left\{\begin{array}{cc}\frac{1}{3}, & x \leq \pi / 2 \\ \frac{\mathrm{~b}(1-\sin x)}{(\pi-2 x)^2}, & x>\pi / 2\end{array}\right.$. If $f$ is continuous at $x=\pi / 2$, then the value of $\int\limits_0^{3 \mathrm{~b}-6}\left|x^2+2 x-3\right| \mathrm{d} x$ is :

A

5

B

2

C

3

D

4

Question 9: Palette 9/251

JEE Mains 2026 MCQ

Let $\mathrm{A}=\left[\begin{array}{ccc}1 & 3 & -1 \\ 2 & 1 & \alpha \\ 0 & 1 & -1\end{array}\right]$ be a singular matrix. Let $f(x)=\int_0^x\left(\mathrm{t}^2+2 \mathrm{t}+3\right) \mathrm{dt}, x \in[1, \alpha]$. If M and m are respectively the maximum and the minimum values of $f$ in $[1, \alpha]$, then $3(\mathrm{M}-\mathrm{m})$ is equal to :

A

64

B

68

C

72

D

76

Question 10: Palette 10/251

JEE Mains 2026 MCQ

The value of the integral $\int\limits_{-1}^1\left(\frac{x^3+|x|+1}{x^2+2|x|+1}\right) \mathrm{d} x$ is equal to:

A

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

B

$2 \log _{\mathrm{e}} 2$

C

$5 \log _{\mathrm{e}} 3$

D

$ 3 \log _{\mathrm{e}} 3$

Question 11: Palette 11/251

JEE Mains 2026 MCQ

The value of the integral $\int\limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\left(\frac{32 \cos ^4 x}{1+e^{\sin x}}\right) d x$ is :

A

$4 \pi+2$

B

$3 \pi+8$

C

$3 \pi+4$

D

$4 \pi+3$

Question 12: Palette 12/251

JEE Mains 2026 MCQ

Let $\left(2^{1-\mathrm{a}}+2^{1+\mathrm{a}}\right), f(\mathrm{a}),\left(3^{\mathrm{a}}+3^{-\mathrm{a}}\right)$ be in A.P. and $\alpha$ be the minimum value of $f(\mathrm{a})$. Then the value of the integral $\int\limits_{\log _e(\alpha-1)}^{\log _e(\alpha)} \frac{d x}{\left(e^{2 x}-e^{-2 x}\right)}$ is :

A

$ \frac{1}{2} \log _e\left(\frac{4}{3}\right) $

B

$ \frac{1}{4} \log _e\left(\frac{4}{3}\right) $

C

$ \frac{1}{2} \log _e\left(\frac{8}{5}\right) $

D

$ \frac{1}{4} \log _e\left(\frac{8}{5}\right) $

Question 13: Palette 13/251

JEE Mains 2026 MCQ

The value of the integral $\int\limits_0^{\infty} \frac{\log _e(x)}{x^2+4} d x$ is:

A

$\frac{\pi \log _e(2)}{2}$

B

$\frac{\pi \log _e(2)}{4}$

C

$ 1+\pi \log _e(2) $

D

$ 2+\pi \log _e(2) $

Question 14: Palette 14/251

JEE Mains 2026 MCQ

The value of the integral $\int\limits_{\frac{\pi}{6}}^{\frac{\pi}{3}}\left(\frac{4-\operatorname{cosec}^2 x}{\cos ^4 x}\right) d x$ is :

A

$\frac{11}{\sqrt{3}}$

B

$\frac{16}{\sqrt{3}}$

C

$\frac{32}{3 \sqrt{3}}$

D

$\frac{64}{3 \sqrt{3}}$

Question 15: Palette 15/251

JEE Mains 2026 MCQ

If $\alpha=1$ and $\beta=1+i \sqrt{2}$, where $i=\sqrt{-1}$ are two roots of the equation

$x^3+a x^2+b x+c=0, a, b, \in \mathbb{R}$, then $\int_{-1}^1\left(x^3+a x^2+b x+c\right) d x$ is equal to:

A

-2

B

-4

C

-8

D

-10

Question 16: Palette 16/251

JEE Mains 2026 MCQ

The integral $\int\limits_0^1 \cot ^{-1}\left(1+x+x^2\right) d x$ is equal to :

A

$ 2 \tan ^{-1} 2+\frac{1}{2} \log _e\left(\frac{5}{4}\right)+\frac{\pi}{2} $

B

$ 2 \tan ^{-1} 2+\frac{1}{2} \log _e\left(\frac{5}{4}\right)-\frac{\pi}{2} $

C

$ 2 \tan ^{-1} 2-\frac{1}{2} \log _e\left(\frac{5}{4}\right)+\frac{\pi}{2} $

D

$ 2 \tan ^{-1} 2-\frac{1}{2} \log _e\left(\frac{5}{4}\right)-\frac{\pi}{2} $

Question 17: Palette 17/251

JEE Mains 2026 MCQ

Let $f$ be a real polynomial of degree $n$ such that $f(x)=f^{\prime}(x) f^{\prime \prime}(x)$, for all $x \in \mathbb{R}$. If $f(0)=0$, then $36\left(f^{\prime}(2)+f^{\prime \prime}(2)+\int_0^2 f(x) d x\right)$ is equal to:

A

42

B

46

C

56

D

66

Question 18: Palette 18/251

JEE Mains 2026 MCQ

Let $\int\limits_{-2}^2(|\sin x|+[x \sin x]) d x=2(3-\cos 2)+\beta$, where [ ⋅ ] is the greatest integer function. Then $\beta \sin \left(\frac{\beta}{2}\right)$ equals:

A

1

B

2

C

4

D

8

Question 19: Palette 19/251

JEE Mains 2026 MCQ

The value of $\int\limits_{0}^{20\pi} (\sin^4 x + \cos^4 x) dx$ is equal to:

A

$\frac{15\pi}{2}$

B

$25\pi$

C

$15\pi$

D

$\frac{25\pi}{2}$

Question 20: Palette 20/251

JEE Mains 2026 MCQ

Let [.] denote the greatest integer function. Then the value of $\int\limits_{0}^{3} \left( \frac{e^{x} + e^{-x}}{[x]!} \right) dx$ is :

A

$e^2 + e^3 - \frac{1}{e^2} - \frac{1}{e^3}$

B

$\frac{1}{2} \left( e^2 + e^3 - \frac{1}{e^2} - \frac{1}{e^3} \right)$

C

$e^2 + e^3 - \frac{1}{2e^2} - \frac{1}{2e^3}$

D

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

Question 21: Palette 21/251

JEE Mains 2025 MCQ

The integral $\int\limits_{-1}^{\frac{3}{2}} \left(| \pi^2 x \sin(\pi x) \right|) dx$ is equal to:

A

$2 + 3\pi$

B

$4 + \pi$

C

$1 + 3\pi$

D

$3 + 2\pi$

Question 22: Palette 22/251

JEE Mains 2025 MCQ

Let f(x) be a positive function and $I_{1} = \int\limits_{-\frac{1}{2}}^{1} 2x \, f(2x(1-2x)) \, dx$ and $I_{2} = \int\limits_{-1}^{2} f(x(1-x)) \, dx$. Then the value of $\frac{I_{2}}{I_{1}}$ is equal to ________

A

12

B

9

C

6

D

4

Question 23: Palette 23/251

JEE Mains 2025 MCQ

The integral $\int_0^\pi \frac{(x+3) \sin x}{1+3 \cos ^2 x} d x$ is equal to

A

$\frac{\pi}{\sqrt{3}}(\pi+1)$

B

$\frac{\pi}{3 \sqrt{3}}(\pi+6)$

C

$\frac{\pi}{\sqrt{3}}(\pi+2)$

D

$\frac{\pi}{2 \sqrt{3}}(\pi+4)$

Question 24: Palette 24/251

JEE Mains 2025 MCQ

Let $f(x)+2 f\left(\frac{1}{x}\right)=x^2+5$ and $2 g(x)-3 g\left(\frac{1}{x}\right)=x, x>0$. If $\alpha=\int_1^2 f(x) \mathrm{d} x$, and $\beta=\int_1^2 g(x) \mathrm{d} x$, then the value of $9 \alpha+\beta$ is :

A

0

B

10

C

1

D

11

Question 25: Palette 25/251

JEE Mains 2025 MCQ

The value of $\int_\limits{-1}^1 \frac{(1+\sqrt{|x|-x}) e^x+(\sqrt{|x|-x}) e^{-x}}{e^x+e^{-x}} d x$ is equal to

A

$1+\frac{2 \sqrt{2}}{3}$

B

$1-\frac{2 \sqrt{2}}{3}$

C

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

D

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

Question 26: Palette 26/251

JEE Mains 2025 MCQ

The integral $\int_0^\pi \frac{8 x d x}{4 \cos ^2 x+\sin ^2 x}$ is equal to

A

$2 \pi^2$

B

$4 \pi^2$

C

$\pi^2$

D

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

Question 27: Palette 27/251

JEE Mains 2025 MCQ

Let the domain of the function $f(x)=\log _2 \log _4 \log _6\left(3+4 x-x^2\right)$ be $(a, b)$. If $\int_0^{b-a}\left[x^2\right] d x=p-\sqrt{q}-\sqrt{r}, p, q, r \in \mathbb{N}, \operatorname{gcd}(p, q, r)=1$, where $[\cdot]$ is the greatest integer function, then $p+q+r$ is equal to

A

10

B

11

C

9

D

8

Question 28: Palette 28/251

JEE Mains 2025 MCQ

Let $f:[1, \infty) \rightarrow[2, \infty)$ be a differentiable function. If $10 \int_1^x f(\mathrm{t}) \mathrm{dt}=5 x f(x)-x^5-9$ for all $x \geqslant 1$, then the value of $f(3)$ is :

A

22

B

26

C

32

D

18

Question 29: Palette 29/251

JEE Mains 2025 MCQ

Let $(a, b)$ be the point of intersection of the curve $x^2=2 y$ and the straight line $y-2 x-6=0$ in the second quadrant. Then the integral $\mathrm{I}=\int_{\mathrm{a}}^{\mathrm{b}} \frac{9 x^2}{1+5^x} \mathrm{~d} x$ is equal to :

A

27

B

18

C

24

D

21

Question 30: Palette 30/251

JEE Mains 2025 MCQ

$4 \int_0^1\left(\frac{1}{\sqrt{3+x^2}+\sqrt{1+x^2}}\right) d x-3 \log _e(\sqrt{3})$ is equal to :

A

$2-\sqrt{2}-\log _{\mathrm{e}}(1+\sqrt{2})$

B

$2+\sqrt{2}+\log _{\mathrm{e}}(1+\sqrt{2})$

C

$2+\sqrt{2}-\log _{\mathrm{e}}(1+\sqrt{2})$

D

$2-\sqrt{2}+\log _e(1+\sqrt{2})$

Question 31: Palette 31/251

JEE Mains 2025 MCQ

Let $f(x)=\int\limits_0^x \mathrm{t}\left(\mathrm{t}^2-9 \mathrm{t}+20\right) \mathrm{dt}, 1 \leq x \leq 5$. If the range of $f$ is $[\alpha, \beta]$, then $4(\alpha+\beta)$ equals :

A

253

B

157

C

154

D

125

Question 32: Palette 32/251

JEE Mains 2025 MCQ

The integral $80 \int\limits_0^{\frac{\pi}{4}}\left(\frac{\sin \theta+\cos \theta}{9+16 \sin 2 \theta}\right) d \theta$ is equal to :

A

3 $ \log 4 $

B

4 $ \log 3 $

C

6 $ \log \frac{4}{3} $

D

2 $ \log 3 $

Question 33: Palette 33/251

JEE Mains 2025 MCQ

Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be a twice differentiable function such that $f(2)=1$. If $\mathrm{F}(\mathrm{x})=\mathrm{x} f(\mathrm{x})$ for all $\mathrm{x} \in \mathrm{R}$, $\int\limits_0^2 x F^{\prime}(x) d x=6$ and $\int\limits_0^2 x^2 F^{\prime \prime}(x) d x=40$, then $F^{\prime}(2)+\int\limits_0^2 F(x) d x$ is equal to :

A

13

B

11

C

9

D

15

Question 34: Palette 34/251

JEE Mains 2025 MCQ

Let $f$ be a real valued continuous function defined on the positive real axis such that $g(x)=\int\limits_0^x t f(t) d t$. If $g\left(x^3\right)=x^6+x^7$, then value of $\sum\limits_{r=1}^{15} f\left(r^3\right)$ is :

A

270

B

340

C

310

D

320

Question 35: Palette 35/251

JEE Mains 2025 MCQ

If $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{96 x^2 \cos ^2 x}{\left(1+e^x\right)} \mathrm{d} x=\pi\left(\alpha \pi^2+\beta\right), \alpha, \beta \in \mathbb{Z}$, then $(\alpha+\beta)^2$ equals

A

196

B

100

C

64

D

144

Question 36: Palette 36/251

JEE Mains 2025 MCQ

If $I(m, n)=\int_0^1 x^{m-1}(1-x)^{n-1} d x, m, n>0$, then $I(9,14)+I(10,13)$ is

A

$I(9,1)$

B

$I(1,13)$

C

$\mathrm{I}(19,27)$

D

$\mathrm{I}(9,13)$

Question 37: Palette 37/251

JEE Mains 2025 MCQ

If $\mathrm{I}=\int_0^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x}{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x} \mathrm{~d} x$, then $\int_0^{2I} \frac{x \sin x \cos x}{\sin ^4 x+\cos ^4 x} \mathrm{~d} x$ equals :

A

$\frac{\pi^2}{12}$

B

$\frac{\pi^2}{4}$

C

$\frac{\pi^2}{16}$

D

$\frac{\pi^2}{8}$

Question 38: Palette 38/251

JEE Mains 2025 MCQ

The value of $\int_{e^2}^{e^4} \frac{1}{x}\left(\frac{e^{\left(\left(\log _e x\right)^2+1\right)^{-1}}}{e^{\left(\left(\log _e x\right)^2+1\right)^{-1}}+e^{\left(\left(6-\log _e x\right)^2+1\right)^{-1}}}\right) d x$ is

A

1

B

$\log_e2$

C

$e^2$

D

2

Question 39: Palette 39/251

JEE Mains 2025 MCQ

Let for $f(x)=7 \tan ^8 x+7 \tan ^6 x-3 \tan ^4 x-3 \tan ^2 x, \quad \mathrm{I}_1=\int_0^{\pi / 4} f(x) \mathrm{d} x$ and $\mathrm{I}_2=\int_0^{\pi / 4} x f(x) \mathrm{d} x$. Then $7 \mathrm{I}_1+12 \mathrm{I}_2$ is equal to :

A

2$\pi$

B

1

C

$\pi$

D

2

Question 40: Palette 40/251

JEE Mains 2024 MCQ

The integral $\int_\limits{1 / 4}^{3 / 4} \cos \left(2 \cot ^{-1} \sqrt{\frac{1-x}{1+x}}\right) d x$ is equal to

A

$-1/2$

B

$-1/4$

C

1/4

D

1/2

Question 41: Palette 41/251

JEE Mains 2024 MCQ

$\lim _\limits{x \rightarrow \frac{\pi}{2}}\left(\frac{\int_{x^3}^{(\pi / 2)^3}\left(\sin \left(2 t^{1 / 3}\right)+\cos \left(t^{1 / 3}\right)\right) d t}{\left(x-\frac{\pi}{2}\right)^2}\right)$ is equal to

A

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

B

$\frac{9 \pi^2}{8}$

C

$\frac{5 \pi^2}{9}$

D

$\frac{11 \pi^2}{10}$

Question 42: Palette 42/251

JEE Mains 2024 MCQ

The value of the integral $\int_\limits{-1}^2 \log _e\left(x+\sqrt{x^2+1}\right) d x$ is

A

$\sqrt{5}-\sqrt{2}+\log _e\left(\frac{7+4 \sqrt{5}}{1+\sqrt{2}}\right)$

B

$\sqrt{2}-\sqrt{5}+\log _e\left(\frac{7+4 \sqrt{5}}{1+\sqrt{2}}\right)$

C

$\sqrt{5}-\sqrt{2}+\log _e\left(\frac{9+4 \sqrt{5}}{1+\sqrt{2}}\right)$

D

$\sqrt{2}-\sqrt{5}+\log _e\left(\frac{9+4 \sqrt{5}}{1+\sqrt{2}}\right)$

Question 43: Palette 43/251

JEE Mains 2024 MCQ

Let $\int_\limits\alpha^{\log _e 4} \frac{\mathrm{d} x}{\sqrt{\mathrm{e}^x-1}}=\frac{\pi}{6}$. Then $\mathrm{e}^\alpha$ and $\mathrm{e}^{-\alpha}$ are the roots of the equation :

A

$2 x^2-5 x+2=0$

B

$x^2-2 x-8=0$

C

$2 x^2-5 x-2=0$

D

$x^2+2 x-8=0$

Question 44: Palette 44/251

JEE Mains 2024 MCQ

The value of $k \in \mathbb{N}$ for which the integral $I_n=\int_0^1\left(1-x^k\right)^n d x, n \in \mathbb{N}$, satisfies $147 I_{20}=148 I_{21}$ is

A

8

B

14

C

7

D

10

Question 45: Palette 45/251

JEE Mains 2024 MCQ

$\int_\limits0^{\pi / 4} \frac{\cos ^2 x \sin ^2 x}{\left(\cos ^3 x+\sin ^3 x\right)^2} d x \text { is equal to }$

A

1/9

B

1/6

C

1/3

D

1/12

Question 46: Palette 46/251

JEE Mains 2024 MCQ

Let $\beta(\mathrm{m}, \mathrm{n})=\int_\limits0^1 x^{\mathrm{m}-1}(1-x)^{\mathrm{n}-1} \mathrm{~d} x, \mathrm{~m}, \mathrm{n}>0$. If $\int_\limits0^1\left(1-x^{10}\right)^{20} \mathrm{~d} x=\mathrm{a} \times \beta(\mathrm{b}, \mathrm{c})$, then $100(\mathrm{a}+\mathrm{b}+\mathrm{c})$ equals _________.

A

2012

B

1021

C

1120

D

2120

Question 47: Palette 47/251

JEE Mains 2024 MCQ

The integral $\int_\limits0^{\pi / 4} \frac{136 \sin x}{3 \sin x+5 \cos x} \mathrm{~d} x$ is equal to :

A

$3 \pi-50 \log _e 2+20 \log _e 5$

B

$3 \pi-25 \log _e 2+10 \log _e 5$

C

$3 \pi-10 \log _{\mathrm{e}}(2 \sqrt{2})+10 \log _{\mathrm{e}} 5$

D

$3 \pi-30 \log _e 2+20 \log _e 5$

Question 48: Palette 48/251

JEE Mains 2024 MCQ

The value of $\int_\limits{-\pi}^\pi \frac{2 y(1+\sin y)}{1+\cos ^2 y} d y$ is :

A

$\frac{\pi}{2}$

B

$\pi^2$

C

$\frac{\pi^2}{2}$

D

$2 \pi^2$

Question 49: Palette 49/251

JEE Mains 2024 MCQ

Let $f(x)=\int_0^x\left(t+\sin \left(1-e^t\right)\right) d t, x \in \mathbb{R}$. Then, $\lim _\limits{x \rightarrow 0} \frac{f(x)}{x^3}$ is equal to

A

$\frac{1}{6}$

B

$-\frac{1}{6}$

C

$\frac{2}{3}$

D

$-\frac{2}{3}$

Question 50: Palette 50/251

JEE Mains 2024 MCQ

If the value of the integral $\int\limits_{-1}^1 \frac{\cos \alpha x}{1+3^x} d x$ is $\frac{2}{\pi}$.Then, a value of $\alpha$ is

A

$\frac{\pi}{2}$

B

$\frac{\pi}{4}$

C

$\frac{\pi}{3}$

D

$\frac{\pi}{6}$

Question 51: Palette 51/251

JEE Mains 2024 MCQ

$\text { Let } f(x)=\left\{\begin{array}{lr} -2, & -2 \leq x \leq 0 \\ x-2, & 0< x \leq 2 \end{array} \text { and } \mathrm{h}(x)=f(|x|)+|f(x)| \text {. Then } \int_\limits{-2}^2 \mathrm{~h}(x) \mathrm{d} x\right. \text { is equal to: }$

A

2

B

6

C

4

D

1

Question 52: Palette 52/251

JEE Mains 2024 MCQ

If $\int\limits_0^{\frac{\pi}{3}} \cos ^4 x \mathrm{~d} x=\mathrm{a} \pi+\mathrm{b} \sqrt{3}$, where $\mathrm{a}$ and $\mathrm{b}$ are rational numbers, then $9 \mathrm{a}+8 \mathrm{b}$ is equal to :

A

2

B

1

C

3

D

$\frac{3}{2}$

Question 53: Palette 53/251

JEE Mains 2024 MCQ

The value of $\int\limits_0^1\left(2 x^3-3 x^2-x+1\right)^{\frac{1}{3}} \mathrm{~d} x$ is equal to :

A

-1

B

2

C

0

D

1

Question 54: Palette 54/251

JEE Mains 2024 MCQ

The value of the integral $\int\limits_0^{\pi / 4} \frac{x \mathrm{~d} x}{\sin ^4(2 x)+\cos ^4(2 x)}$ equals :

A

$\frac{\sqrt{2} \pi^2}{8}$

B

$\frac{\sqrt{2} \pi^2}{16}$

C

$\frac{\sqrt{2} \pi^2}{32}$

D

$\frac{\sqrt{2} \pi^2}{64}$

Question 55: Palette 55/251

JEE Mains 2024 MCQ

Let $f, g:(0, \infty) \rightarrow \mathbb{R}$ be two functions defined by $f(x)=\int\limits_{-x}^x\left(|t|-t^2\right) e^{-t^2} d t$ and $g(x)=\int\limits_0^{x^2} t^{1 / 2} e^{-t} d t$. Then, the value of $9\left(f\left(\sqrt{\log _e 9}\right)+g\left(\sqrt{\log _e 9}\right)\right)$ is equal to :

A

10

B

9

C

8

D

6

Question 56: Palette 56/251

JEE Mains 2024 MCQ

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $f(x)=\frac{x}{\left(1+x^4\right)^{1 / 4}}$, and $g(x)=f(f(f(f(x))))$. Then, $18 \int_0^{\sqrt{2 \sqrt{5}}} x^2 g(x) d x$ is equal to

A

36

B

33

C

39

D

42

Question 57: Palette 57/251

JEE Mains 2024 MCQ

Let $y=f(x)$ be a thrice differentiable function in $(-5,5)$. Let the tangents to the curve $y=f(x)$ at $(1, f(1))$ and $(3, f(3))$ make angles $\pi / 6$ and $\pi / 4$, respectively with positive $x$-axis. If $27 \int_\limits1^3\left(\left(f^{\prime}(t)\right)^2+1\right) f^{\prime \prime}(t) d t=\alpha+\beta \sqrt{3}$ where $\alpha, \beta$ are integers, then the value of $\alpha+\beta$ equals

A

26

B

$-$16

C

36

D

$-$14

Question 58: Palette 58/251

JEE Mains 2024 MCQ

Let $a$ and $b$ be real constants such that the function $f$ defined by $f(x)=\left\{\begin{array}{ll}x^2+3 x+a & , x \leq 1 \\ b x+2 & , x>1\end{array}\right.$ be differentiable on $\mathbb{R}$. Then, the value of $\int_\limits{-2}^2 f(x) d x$ equals

A

21

B

19/6

C

17

D

15/6

Question 59: Palette 59/251

JEE Mains 2024 MCQ

Let $\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$ be defined as $f(x)=a e^{2 x}+b e^x+c x$. If $f(0)=-1, f^{\prime}\left(\log _e 2\right)=21$ and $\int_0^{\log _e 4}(f(x)-c x) d x=\frac{39}{2}$, then the value of $|a+b+c|$ equals

A

16

B

12

C

8

D

10

Question 60: Palette 60/251

JEE Mains 2024 MCQ

The value of $\lim _\limits{n \rightarrow \infty} \sum_\limits{k=1}^n \frac{n^3}{\left(n^2+k^2\right)\left(n^2+3 k^2\right)}$ is :

A

$\frac{\pi}{8(2 \sqrt{3}+3)}$

B

$\frac{(2 \sqrt{3}+3) \pi}{24}$

C

$\frac{13 \pi}{8(4 \sqrt{3}+3)}$

D

$\frac{13(2 \sqrt{3}-3) \pi}{8}$

Question 61: Palette 61/251

JEE Mains 2024 MCQ

Let $f:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow \mathbf{R}$ be a differentiable function such that $f(0)=\frac{1}{2}$. If the $\lim _\limits{x \rightarrow 0} \frac{x \int_0^x f(\mathrm{t}) \mathrm{dt}}{\mathrm{e}^{x^2}-1}=\alpha$, then $8 \alpha^2$ is equal to :

A

4

B

2

C

1

D

16

Question 62: Palette 62/251

JEE Mains 2024 MCQ

$\mathop {\lim }\limits_{x \to {\pi \over 2}} \left( {{1 \over {{{\left( {x - {\pi \over 2}} \right)}^2}}}\int\limits_{{x^3}}^{{{\left( {{\pi \over 2}} \right)}^3}} {\cos \left( {{t^{{1 \over 3}}}} \right)dt} } \right)$ is equal to

A

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

B

$\frac{3 \pi^2}{8}$

C

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

D

$\frac{3 \pi}{8}$

Question 63: Palette 63/251

JEE Mains 2024 MCQ

If the value of the integral $\int_\limits{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{x^2 \cos x}{1+\pi^x}+\frac{1+\sin ^2 x}{1+e^{\sin x^{2123}}}\right) d x=\frac{\pi}{4}(\pi+a)-2$, then the value of $a$ is

A

$-\frac{3}{2}$

B

3

C

$\frac{3}{2}$

D

2

Question 64: Palette 64/251

JEE Mains 2024 MCQ

For $0 < \mathrm{a} < 1$, the value of the integral $\int_\limits0^\pi \frac{\mathrm{d} x}{1-2 \mathrm{a} \cos x+\mathrm{a}^2}$ is :

A

$\frac{\pi^2}{\pi+a^2}$

B

$\frac{\pi^2}{\pi-a^2}$

C

$\frac{\pi}{1-\mathrm{a}^2}$

D

$\frac{\pi}{1+\mathrm{a}^2}$

Question 65: Palette 65/251

JEE Mains 2024 MCQ

If $\int\limits_0^1 \frac{1}{\sqrt{3+x}+\sqrt{1+x}} \mathrm{~d} x=\mathrm{a}+\mathrm{b} \sqrt{2}+\mathrm{c} \sqrt{3}$, where $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are rational numbers, then $2 \mathrm{a}+3 \mathrm{~b}-4 \mathrm{c}$ is equal to :

A

10

B

7

C

4

D

8

Question 66: Palette 66/251

JEE Mains 2024 MCQ

If $(a, b)$ be the orthocentre of the triangle whose vertices are $(1,2),(2,3)$ and $(3,1)$, and $\mathrm{I}_1=\int\limits_{\mathrm{a}}^{\mathrm{b}} x \sin \left(4 x-x^2\right) \mathrm{d} x, \mathrm{I}_2=\int\limits_{\mathrm{a}}^{\mathrm{b}} \sin \left(4 x-x^2\right) \mathrm{d} x$, then $36 \frac{\mathrm{I}_1}{\mathrm{I}_2}$ is equal to :

A

80

B

72

C

66

D

88

Question 67: Palette 67/251

JEE Mains 2023 MCQ

If $\int\limits_{0}^{1} \frac{1}{\left(5+2 x-2 x^{2}\right)\left(1+e^{(2-4 x)}\right)} d x=\frac{1}{\alpha} \log _{e}\left(\frac{\alpha+1}{\beta}\right), \alpha, \beta>0$, then $\alpha^{4}-\beta^{4}$ is equal to :

A

-21

B

21

C

19

D

0

Question 68: Palette 68/251

JEE Mains 2023 MCQ

The value of ${{{e^{ - {\pi \over 4}}} + \int\limits_0^{{\pi \over 4}} {{e^{ - x}}{{\tan }^{50}}xdx} } \over {\int\limits_0^{{\pi \over 4}} {{e^{ - x}}({{\tan }^{49}}x + {{\tan }^{51}}x)dx} }}$ is

A

51

B

50

C

25

D

49

Question 69: Palette 69/251

JEE Mains 2023 MCQ

Among

(S1): $\lim_\limits{n \rightarrow \infty} \frac{1}{n^{2}}(2+4+6+\ldots \ldots+2 n)=1$

(S2) : $\lim_\limits{n \rightarrow \infty} \frac{1}{n^{16}}\left(1^{15}+2^{15}+3^{15}+\ldots \ldots+n^{15}\right)=\frac{1}{16}$

A

Only (S1) is true

B

Both (S1) and (S2) are true

C

Both (S1) and (S2) are false

D

Only (S2) is true

Question 70: Palette 70/251

JEE Mains 2023 MCQ

$\int_\limits{0}^{\infty} \frac{6}{e^{3 x}+6 e^{2 x}+11 e^{x}+6} d x=$

A

$\log _{e}\left(\frac{256}{81}\right)$

B

$\log _{e}\left(\frac{64}{27}\right)$

C

$\log _{e}\left(\frac{32}{27}\right)$

D

$\log _{e}\left(\frac{512}{81}\right)$

Question 71: Palette 71/251

JEE Mains 2023 MCQ

If $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function satisfying $\int_\limits{0}^{\frac{\pi}{2}} f(\sin 2 x) \sin x d x+\alpha \int_\limits{0}^{\frac{\pi}{4}} f(\cos 2 x) \cos x d x=0$, then the value of $\alpha$ is :

A

$-\sqrt{3}$

B

$\sqrt{2}$

C

$-\sqrt{2}$

D

$\sqrt{3}$

Question 72: Palette 72/251

JEE Mains 2023 MCQ

Let the function $f:[0,2] \rightarrow \mathbb{R}$ be defined as

$f(x)= \begin{cases}e^{\min \left\{x^{2}, x-[x]\right\},} & x \in[0,1) \\ e^{\left[x-\log _{e} x\right]}, & x \in[1,2]\end{cases}$

where $[t]$ denotes the greatest integer less than or equal to $t$. Then the value of the integral $\int_\limits{0}^{2} x f(x) d x$ is :

A

$2 e-1$

B

$2 e-\frac{1}{2}$

C

$1+\frac{3 e}{2}$

D

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

Question 73: Palette 73/251

JEE Mains 2023 MCQ

The value of the integral $\int_\limits{-\log _{e} 2}^{\log _{e} 2} e^{x}\left(\log _{e}\left(e^{x}+\sqrt{1+e^{2 x}}\right)\right) d x$ is equal to :

A

$\log _{e}\left(\frac{(2+\sqrt{5})^{2}}{\sqrt{1+\sqrt{5}}}\right)+\frac{\sqrt{5}}{2}$

B

$\log _{e}\left(\frac{\sqrt{2}(2+\sqrt{5})^{2}}{\sqrt{1+\sqrt{5}}}\right)-\frac{\sqrt{5}}{2}$

C

$\log _{e}\left(\frac{2(2+\sqrt{5})}{\sqrt{1+\sqrt{5}}}\right)-\frac{\sqrt{5}}{2}$

D

$\log _{e}\left(\frac{\sqrt{2}(3-\sqrt{5})^{2}}{\sqrt{1+\sqrt{5}}}\right)+\frac{\sqrt{5}}{2}$

Question 74: Palette 74/251

JEE Mains 2023 MCQ

Let $f$ be a continuous function satisfying $\int_\limits{0}^{t^{2}}\left(f(x)+x^{2}\right) d x=\frac{4}{3} t^{3}, \forall t > 0$. Then $f\left(\frac{\pi^{2}}{4}\right)$ is equal to :

A

$-\pi\left(1+\frac{\pi^{3}}{16}\right)$

B

$\pi\left(1-\frac{\pi^{3}}{16}\right)$

C

$-\pi^{2}\left(1+\frac{\pi^{2}}{16}\right)$

D

$\pi^{2}\left(1-\frac{\pi^{2}}{16}\right)$

Question 75: Palette 75/251

JEE Mains 2023 MCQ

Let $f(x)$ be a function satisfying $f(x)+f(\pi-x)=\pi^{2}, \forall x \in \mathbb{R}$. Then $\int_\limits{0}^{\pi} f(x) \sin x d x$ is equal to :

A

$\pi^{2}$

B

$\frac{\pi^{2}}{2}$

C

$2 \pi^{2}$

D

$\frac{\pi^{2}}{4}$

Question 76: Palette 76/251

JEE Mains 2023 MCQ

$\lim _\limits{n \rightarrow \infty}\left\{\left(2^{\frac{1}{2}}-2^{\frac{1}{3}}\right)\left(2^{\frac{1}{2}}-2^{\frac{1}{5}}\right) \ldots . .\left(2^{\frac{1}{2}}-2^{\frac{1}{2 n+1}}\right)\right\}$ is equal to :

A

$\sqrt{2}$

B

1

C

$\frac{1}{\sqrt{2}}$

D

0

Question 77: Palette 77/251

JEE Mains 2023 MCQ

Let $5 f(x)+4 f\left(\frac{1}{x}\right)=\frac{1}{x}+3, x > 0$. Then $18 \int_\limits{1}^{2} f(x) d x$ is equal to :

A

$10 \log _{\mathrm{e}} 2+6$

B

$5 \log _{e} 2-3$

C

$10 \log _{\mathrm{e}} 2-6$

D

$5 \log _{\mathrm{e}} 2+3$

Question 78: Palette 78/251

JEE Mains 2023 MCQ

The value of the integral

$\int\limits_{ - {\pi \over 4}}^{{\pi \over 4}} {{{x + {\pi \over 4}} \over {2 - \cos 2x}}dx} $ is :

A

${{{\pi ^2}} \over {6\sqrt 3 }}$

B

${{{\pi ^2}} \over 6}$

C

${{{\pi ^2}} \over {3\sqrt 3 }}$

D

${{{\pi ^2}} \over {12\sqrt 3 }}$

Question 79: Palette 79/251

JEE Mains 2023 MCQ

$\mathop {\lim }\limits_{n \to \infty } \left[ {{1 \over {1 + n}} + {1 \over {2 + n}} + {1 \over {3 + n}}\, + \,...\, + \,{1 \over {2n}}} \right]$ is equal to

A

0

B

${\log _e}2$

C

${\log _e}\left( {{2 \over 3}} \right)$

D

${\log _e}\left( {{3 \over 2}} \right)$

Question 80: Palette 80/251

JEE Mains 2023 MCQ

Let $\alpha>0$. If $\int\limits_0^\alpha \frac{x}{\sqrt{x+\alpha}-\sqrt{x}} \mathrm{~d} x=\frac{16+20 \sqrt{2}}{15}$, then $\alpha$ is equal to :

A

4

B

2

C

$2 \sqrt{2}$

D

$\sqrt{2}$

Question 81: Palette 81/251

JEE Mains 2023 MCQ

If $\phi(x)=\frac{1}{\sqrt{x}} \int\limits_{\frac{\pi}{4}}^x\left(4 \sqrt{2} \sin t-3 \phi^{\prime}(t)\right) d t, x>0$,

then $\emptyset^{\prime}\left(\frac{\pi}{4}\right)$ is equal to :

A

$\frac{4}{6+\sqrt{\pi}}$

B

$\frac{4}{6-\sqrt{\pi}}$

C

$\frac{8}{\sqrt{\pi}}$

D

$\frac{8}{6+\sqrt{\pi}}$

Question 82: Palette 82/251

JEE Mains 2023 MCQ

Let $\alpha \in (0,1)$ and $\beta = {\log _e}(1 - \alpha )$. Let ${P_n}(x) = x + {{{x^2}} \over 2} + {{{x^3}} \over 3}\, + \,...\, + \,{{{x^n}} \over n},x \in (0,1)$. Then the integral $\int\limits_0^\alpha {{{{t^{50}}} \over {1 - t}}dt} $ is equal to

A

$ - \left( {\beta + {P_{50}}\left( \alpha \right)} \right)$

B

$\beta - {P_{50}}(\alpha )$

C

${P_{50}}(\alpha ) - \beta $

D

$\beta + {P_{50}} - (\alpha )$

Question 83: Palette 83/251

JEE Mains 2023 MCQ

The value of $\int_\limits{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{(2+3 \sin x)}{\sin x(1+\cos x)} d x$ is equal to :

A

$\frac{10}{3}-\sqrt{3}+\log _{e} \sqrt{3}$

B

$\frac{7}{2}-\sqrt{3}-\log _{e} \sqrt{3}$

C

$\frac{10}{3}-\sqrt{3}-\log _{e} \sqrt{3}$

D

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

Question 84: Palette 84/251

JEE Mains 2023 MCQ

$\lim\limits_{n \rightarrow \infty} \frac{3}{n}\left\{4+\left(2+\frac{1}{n}\right)^2+\left(2+\frac{2}{n}\right)^2+\ldots+\left(3-\frac{1}{n}\right)^2\right\}$ is equal to :

A

0

B

$\frac{19}{3}$

C

19

D

12

Question 85: Palette 85/251

JEE Mains 2023 MCQ

If [t] denotes the greatest integer $\le \mathrm{t}$, then the value of ${{3(e - 1)} \over e}\int\limits_1^2 {{x^2}{e^{[x] + [{x^3}]}}dx} $ is :

A

$\mathrm{e^8-e}$

B

$\mathrm{e^7-1}$

C

$\mathrm{e^9-e}$

D

$\mathrm{e^8-1}$

Question 86: Palette 86/251

JEE Mains 2023 MCQ

The value of the integral $\int_1^2 {\left( {{{{t^4} + 1} \over {{t^6} + 1}}} \right)dt} $ is

A

${\tan ^{ - 1}}{1 \over 2} - {1 \over 3}{\tan ^{ - 1}}8 + {\pi \over 3}$

B

${\tan ^{ - 1}}2 - {1 \over 3}{\tan ^{ - 1}}8 + {\pi \over 3}$

C

${\tan ^{ - 1}}2 + {1 \over 3}{\tan ^{ - 1}}8 - {\pi \over 3}$

D

${\tan ^{ - 1}}{1 \over 2} + {1 \over 3}{\tan ^{ - 1}}8 - {\pi \over 3}$

Question 87: Palette 87/251

JEE Mains 2023 MCQ

The value of the integral $\int\limits_{1/2}^2 {{{{{\tan }^{ - 1}}x} \over x}dx} $ is equal to :

A

${\pi \over 2}{\log _e}2$

B

${\pi \over 4}{\log _e}2$

C

${1 \over 2}{\log _e}2$

D

$\pi {\log _e}2$

Question 88: Palette 88/251

JEE Mains 2023 MCQ

Let $f(x) = x + {a \over {{\pi ^2} - 4}}\sin x + {b \over {{\pi ^2} - 4}}\cos x,x \in R$ be a function which

satisfies $f(x) = x + \int\limits_0^{\pi /2} {\sin (x + y)f(y)dy} $. then $(a+b)$ is equal to

A

$ - 2\pi (\pi + 2)$

B

$ - \pi (\pi - 2)$

C

$ - \pi (\pi + 2)$

D

$ - 2\pi (\pi - 2)$

Question 89: Palette 89/251

JEE Mains 2023 MCQ

The integral $16\int\limits_1^2 {{{dx} \over {{x^3}{{\left( {{x^2} + 2} \right)}^2}}}} $ is equal to

A

${{11} \over {12}} + {\log _e}4$

B

${{11} \over 6} + {\log _e}4$

C

${{11} \over {12}} - {\log _e}4$

D

${{11} \over 6} - {\log _e}4$

Question 90: Palette 90/251

JEE Mains 2023 MCQ

The minimum value of the function $f(x) = \int\limits_0^2 {{e^{|x - t|}}dt} $ is :

A

2

B

$2(e-1)$

C

$e(e-1)$

D

$2e-1$

Question 91: Palette 91/251

JEE Mains 2023 MCQ

$\int\limits_{{{3\sqrt 2 } \over 4}}^{{{3\sqrt 3 } \over 4}} {{{48} \over {\sqrt {9 - 4{x^2}} }}dx} $ is equal to :

A

${\pi \over 2}$

B

${\pi \over 3}$

C

${\pi \over 6}$

D

$2\pi $

Question 92: Palette 92/251

JEE Mains 2022 MCQ

If $[t]$ denotes the greatest integer $\leq t$, then the value of $\int_{0}^{1}\left[2 x-\left|3 x^{2}-5 x+2\right|+1\right] \mathrm{d} x$ is :

A

$\frac{\sqrt{37}+\sqrt{13}-4}{6}$

B

$\frac{\sqrt{37}-\sqrt{13}-4}{6}$

C

$\frac{-\sqrt{37}-\sqrt{13}+4}{6}$

D

$\frac{-\sqrt{37}+\sqrt{13}+4}{6}$

Question 93: Palette 93/251

JEE Mains 2022 MCQ

The integral $\int\limits_{0}^{\frac{\pi}{2}} \frac{1}{3+2 \sin x+\cos x} \mathrm{~d} x$ is equal to :

A

$\tan ^{-1}(2)$

B

$\tan ^{-1}(2)-\frac{\pi}{4}$

C

$\frac{1}{2} \tan ^{-1}(2)-\frac{\pi}{8}$

D

$\frac{1}{2}$

Question 94: Palette 94/251

JEE Mains 2022 MCQ

If $f(\alpha)=\int\limits_{1}^{\alpha} \frac{\log _{10} \mathrm{t}}{1+\mathrm{t}} \mathrm{dt}, \alpha>0$, then $f\left(\mathrm{e}^{3}\right)+f\left(\mathrm{e}^{-3}\right)$ is equal to :

A

9

B

$\frac{9}{2}$

C

$\frac{9}{\log _{e}(10)}$

D

$\frac{9}{2 \log _{e}(10)}$

Question 95: Palette 95/251

JEE Mains 2022 MCQ

Let $I_{n}(x)=\int_{0}^{x} \frac{1}{\left(t^{2}+5\right)^{n}} d t, n=1,2,3, \ldots .$ Then :

A

$50 I_{6}-9 I_{5}=x I_{5}^{\prime}$

B

$50 I_{6}-11 I_{5}=x I_{5}^{\prime}$

C

$50 I_{6}-9 I_{5}=I_{5}^{\prime}$

D

$50 I_{6}-11 I_{5}=I_{5}^{\prime}$

Question 96: Palette 96/251

JEE Mains 2022 MCQ

The minimum value of the twice differentiable function $f(x)=\int\limits_{0}^{x} \mathrm{e}^{x-\mathrm{t}} f^{\prime}(\mathrm{t}) \mathrm{dt}-\left(x^{2}-x+1\right) \mathrm{e}^{x}$, $x \in \mathbf{R}$, is :

A

$-\frac{2}{\sqrt{\mathrm{e}}}$

B

$-2 \sqrt{\mathrm{e}}$

C

$-\sqrt{\mathrm{e}}$

D

$\frac{2}{\sqrt{\mathrm{e}}}$

Question 97: Palette 97/251

JEE Mains 2022 MCQ

Let $f(x)=2+|x|-|x-1|+|x+1|, x \in \mathbf{R}$.

Consider

$(\mathrm{S} 1): f^{\prime}\left(-\frac{3}{2}\right)+f^{\prime}\left(-\frac{1}{2}\right)+f^{\prime}\left(\frac{1}{2}\right)+f^{\prime}\left(\frac{3}{2}\right)=2$

$(\mathrm{S} 2): \int\limits_{-2}^{2} f(x) \mathrm{d} x=12$

Then,

A

both (S1) and (S2) are correct

B

both (S1) and (S2) are wrong

C

only (S1) is correct

D

only (S2) is correct

Question 98: Palette 98/251

JEE Mains 2022 MCQ

$\int\limits_{0}^{2}\left(\left|2 x^{2}-3 x\right|+\left[x-\frac{1}{2}\right]\right) \mathrm{d} x$, where [t] is the greatest integer function, is equal to :

A

$\frac{7}{6}$

B

$\frac{19}{12}$

C

$\frac{31}{12}$

D

$\frac{3}{2}$

Question 99: Palette 99/251

JEE Mains 2022 MCQ

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined as

$f(x)=a \sin \left(\frac{\pi[x]}{2}\right)+[2-x], a \in \mathbb{R}$ where $[t]$ is the greatest integer less than or equal to $t$. If $\mathop {\lim }\limits_{x \to -1 } f(x)$ exists, then the value of $\int\limits_{0}^{4} f(x) d x$ is equal to

A

$-$1

B

$-$2

C

1

D

2

Question 100: Palette 100/251

JEE Mains 2022 MCQ

Let $ I=\int_{\pi / 4}^{\pi / 3}\left(\frac{8 \sin x-\sin 2 x}{x}\right) d x $. Then

A

${\pi \over 2} < I < {{3\pi } \over 4}$

B

${\pi \over 5} < I < {{5\pi } \over {12}}$

C

${{5\pi } \over {12}} < I < {{\sqrt 2 } \over 3}\pi $

D

${{3\pi } \over 4} < I < \pi $

Question 101: Palette 101/251

JEE Mains 2022 MCQ

Let a function $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as :

$f(x)= \begin{cases}\int\limits_{0}^{x}(5-|t-3|) d t, & x>4 \\ x^{2}+b x & , x \leq 4\end{cases}$

where $\mathrm{b} \in \mathbb{R}$. If $f$ is continuous at $x=4$, then which of the following statements is NOT true?

A

$f$ is not differentiable at $x=4$

B

$f^{\prime}(3)+f^{\prime}(5)=\frac{35}{4}$

C

$f$ is increasing in $\left(-\infty, \frac{1}{8}\right) \cup(8, \infty)$

D

$f$ has a local minima at $x=\frac{1}{8}$

Question 102: Palette 102/251

JEE Mains 2022 MCQ

$ \int\limits_{0}^{20 \pi}(|\sin x|+|\cos x|)^{2} d x \text { is equal to } $

A

$10(\pi+4)$

B

$10(\pi+2)$

C

$20(\pi-2)$

D

$20(\pi+2)$

Question 103: Palette 103/251

JEE Mains 2022 MCQ

If $a = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {{{2n} \over {{n^2} + {k^2}}}} $ and $f(x) = \sqrt {{{1 - \cos x} \over {1 + \cos x}}} $, $x \in (0,1)$, then :

A

$2\sqrt 2 f\left( {{a \over 2}} \right) = f'\left( {{a \over 2}} \right)$

B

$f\left( {{a \over 2}} \right)f'\left( {{a \over 2}} \right) = \sqrt 2 $

C

$\sqrt 2 f\left( {{a \over 2}} \right) = f'\left( {{a \over 2}} \right)$

D

$f\left( {{a \over 2}} \right) = \sqrt 2 f'\left( {{a \over 2}} \right)$

Question 104: Palette 104/251

JEE Mains 2022 MCQ

$\mathop {\lim }\limits_{n \to \infty } {1 \over {{2^n}}}\left( {{1 \over {\sqrt {1 - {1 \over {{2^n}}}} }} + {1 \over {\sqrt {1 - {2 \over {{2^n}}}} }} + {1 \over {\sqrt {1 - {3 \over {{2^n}}}} }} + \,\,...\,\, + \,\,{1 \over {\sqrt {1 - {{{2^n} - 1} \over {{2^n}}}} }}} \right)$ is equal to

A

$\frac{1}{2}$

B

1

C

2

D

$-$2

Question 105: Palette 105/251

JEE Mains 2022 MCQ

Let $[t]$ denote the greatest integer less than or equal to $t$. Then the value of the integral $\int_{-3}^{101}\left([\sin (\pi x)]+e^{[\cos (2 \pi x)]}\right) d x$ is equal to

A

$\frac{52(1-e)}{e}$

B

$\frac{52}{e}$

C

$\frac{52(2+e)}{e}$

D

$\frac{104}{e}$

Question 106: Palette 106/251

JEE Mains 2022 MCQ

For any real number $x$, let $[x]$ denote the largest integer less than equal to $x$. Let $f$ be a real valued function defined on the interval $[-10,10]$ by $f(x)=\left\{\begin{array}{l}x-[x], \text { if }[x] \text { is odd } \\ 1+[x]-x, \text { if }[x] \text { is even } .\end{array}\right.$ Then the value of $\frac{\pi^{2}}{10} \int_{-10}^{10} f(x) \cos \pi x \,d x$ is :

A

4

B

2

C

1

D

0

Question 107: Palette 107/251

JEE Mains 2022 MCQ

$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {{r \over {2{r^2} - 7rn + 6{n^2}}}} $ is equal to :

A

${\log _e}\left( {{{\sqrt 3 } \over 2}} \right)$

B

${\log _e}\left( {{{3\sqrt 3 } \over 4}} \right)$

C

${\log _e}\left( {{{27} \over 4}} \right)$

D

${\log _e}\left( {{4 \over 3}} \right)$

Question 108: Palette 108/251

JEE Mains 2022 MCQ

Let f be a real valued continuous function on [0, 1] and $f(x) = x + \int\limits_0^1 {(x - t)f(t)dt} $.

Then, which of the following points (x, y) lies on the curve y = f(x) ?

A

(2, 4)

B

(1, 2)

C

(4, 17)

D

(6, 8)

Question 109: Palette 109/251

JEE Mains 2022 MCQ

If $\int\limits_0^2 {\left( {\sqrt {2x} - \sqrt {2x - {x^2}} } \right)dx = \int\limits_0^1 {\left( {1 - \sqrt {1 - {y^2}} - {{{y^2}} \over 2}} \right)dy + \int\limits_1^2 {\left( {2 - {{{y^2}} \over 2}} \right)dy + I} } } $, then I equals

A

$\int\limits_0^1 {\left( {1 + \sqrt {1 - {y^2}} } \right)dy} $

B

$\int\limits_0^1 {\left( {{{{y^2}} \over 2} - \sqrt {1 - {y^2}} + 1} \right)dy} $

C

$\int\limits_0^1 {\left( {1 - \sqrt {1 - {y^2}} } \right)dy} $

D

$\int\limits_0^1 {\left( {{{{y^2}} \over 2} + \sqrt {1 - {y^2}} + 1} \right)dy} $

Question 110: Palette 110/251

JEE Mains 2022 MCQ

Let $f:R \to R$ be a function defined by :

$f(x) = \left\{ {\matrix{ {\max \,\{ {t^3} - 3t\} \,t \le x} & ; & {x \le 2} \cr {{x^2} + 2x - 6} & ; & {2 < x < 3} \cr {[x - 3] + 9} & ; & {3 \le x \le 5} \cr {2x + 1} & ; & {x > 5} \cr } } \right.$

where [t] is the greatest integer less than or equal to t. Let m be the number of points where f is not differentiable and $I = \int\limits_{ - 2}^2 {f(x)\,dx} $. Then the ordered pair (m, I) is equal to :

A

$\left( {3,\,{{27} \over 4}} \right)$

B

$\left( {3,\,{{23} \over 4}} \right)$

C

$\left( {4,\,{{27} \over 4}} \right)$

D

$\left( {4,\,{{23} \over 4}} \right)$

Question 111: Palette 111/251

JEE Mains 2022 MCQ

$\int_0^5 {\cos \left( {\pi \left( {x - \left[ {{x \over 2}} \right]} \right)} \right)dx} $,

where [t] denotes greatest integer less than or equal to t, is equal to:

A

$-$3

B

$-$2

C

2

D

0

Question 112: Palette 112/251

JEE Mains 2022 MCQ

Let f : R $\to$ R be a differentiable function such that $f\left( {{\pi \over 4}} \right) = \sqrt 2 ,\,f\left( {{\pi \over 2}} \right) = 0$ and $f'\left( {{\pi \over 2}} \right) = 1$ and

let $g(x) = \int_x^{\pi /4} {(f'(t)\sec t + \tan t\sec t\,f(t))\,dt} $ for $x \in \left[ {{\pi \over 4},{\pi \over 2}} \right)$. Then $\mathop {\lim }\limits_{x \to {{\left( {{\pi \over 2}} \right)}^ - }} g(x)$ is equal to :

A

2

B

3

C

4

D

$-$3

Question 113: Palette 113/251

JEE Mains 2022 MCQ

Let f : R $\to$ R be a continuous function satisfying f(x) + f(x + k) = n, for all x $\in$ R where k > 0 and n is a positive integer. If ${I_1} = \int\limits_0^{4nk} {f(x)dx} $ and ${I_2} = \int\limits_{ - k}^{3k} {f(x)dx} $, then :

A

${I_1} + 2{I_2} = 4nk$

B

${I_1} + 2{I_2} = 2nk$

C

${I_1} + n{I_2} = 4{n^2}k$

D

${I_1} + n{I_2} = 6{n^2}k$

Question 114: Palette 114/251

JEE Mains 2022 MCQ

Let [t] denote the greatest integer less than or equal to t. Then, the value of the integral $\int\limits_0^1 {[ - 8{x^2} + 6x - 1]dx} $ is equal to :

A

$-$1

B

${{ - 5} \over 4}$

C

${{\sqrt {17} - 13} \over 8}$

D

${{\sqrt {17} - 16} \over 8}$

Question 115: Palette 115/251

JEE Mains 2022 MCQ

If m and n respectively are the number of local maximum and local minimum points of the function $f(x) = \int\limits_0^{{x^2}} {{{{t^2} - 5t + 4} \over {2 + {e^t}}}dt} $, then the ordered pair (m, n) is equal to

A

(3, 2)

B

(2, 3)

C

(2, 2)

D

(3, 4)

Question 116: Palette 116/251

JEE Mains 2022 MCQ

Let f be a differentiable function in $\left( {0,{\pi \over 2}} \right)$. If $\int\limits_{\cos x}^1 {{t^2}\,f(t)dt = {{\sin }^3}x + \cos x} $, then ${1 \over {\sqrt 3 }}f'\left( {{1 \over {\sqrt 3 }}} \right)$ is equal to

A

$6 - 9\sqrt 2 $

B

$6 - {9 \over {\sqrt 2 }}$

C

${9 \over 2} - 6\sqrt 2 $

D

${9 \over {\sqrt 2 }} - 6$

Question 117: Palette 117/251

JEE Mains 2022 MCQ

The integral $\int\limits_0^1 {{1 \over {{7^{\left[ {{1 \over x}} \right]}}}}dx} $, where [ . ] denotes the greatest integer function, is equal to

A

$1 + 6{\log _e}\left( {{6 \over 7}} \right)$

B

$1 - 6{\log _e}\left( {{6 \over 7}} \right)$

C

${\log _e}\left( {{7 \over 6}} \right)$

D

$1 - 7{\log _e}\left( {{6 \over 7}} \right)$

Question 118: Palette 118/251

JEE Mains 2022 MCQ

The value of the integral

$\int\limits_{ - 2}^2 {{{|{x^3} + x|} \over {({e^{x|x|}} + 1)}}dx} $ is equal to :

A

5e²

B

3e^$-$2

C

4

D

6

Question 119: Palette 119/251

JEE Mains 2022 MCQ

If ${b_n} = \int_0^{{\pi \over 2}} {{{{{\cos }^2}nx} \over {\sin x}}dx,\,n \in N} $, then

A

${b_3} - {b_2},\,{b_4} - {b_3},\,{b_5} - {b_4}$ are in A.P. with common difference $-$2

B

${1 \over {{b_3} - {b_2}}},{1 \over {{b_4} - {b_3}}},{1 \over {{b_5} - {b_4}}}$ are in an A.P. with common difference 2

C

${b_3} - {b_2},\,{b_4} - {b_3},\,{b_5} - {b_4}$ are in a G.P.

D

${1 \over {{b_3} - {b_2}}},{1 \over {{b_4} - {b_3}}},{1 \over {{b_5} - {b_4}}}$ are in an A.P. with common difference $-$2

Question 120: Palette 120/251

JEE Mains 2022 MCQ

The value of $\int\limits_0^\pi {{{{e^{\cos x}}\sin x} \over {(1 + {{\cos }^2}x)({e^{\cos x}} + {e^{ - \cos x}})}}dx} $ is equal to:

A

${{{\pi ^2}} \over 4}$

B

${{{\pi ^2}} \over 2}$

C

${\pi \over 4}$

D

${\pi \over 2}$

Question 121: Palette 121/251

JEE Mains 2022 MCQ

The value of the integral

$\int\limits_{ - \pi /2}^{\pi /2} {{{dx} \over {(1 + {e^x})({{\sin }^6}x + {{\cos }^6}x)}}} $ is equal to

A

2$\pi$

B

0

C

$\pi$

D

${\pi \over 2}$

Question 122: Palette 122/251

JEE Mains 2022 MCQ

$\mathop {\lim }\limits_{n \to \infty } \left( {{{{n^2}} \over {({n^2} + 1)(n + 1)}} + {{{n^2}} \over {({n^2} + 4)(n + 2)}} + {{{n^2}} \over {({n^2} + 9)(n + 3)}} + \,\,....\,\, + \,\,{{{n^2}} \over {({n^2} + {n^2})(n + n)}}} \right)$ is equal to :

A

${\pi \over 8} + {1 \over 4}{\log _e}2$

B

${\pi \over 4} + {1 \over 8}{\log _e}2$

C

${\pi \over 4} - {1 \over 8}{\log _e}2$

D

${\pi \over 8} + {\log _e}\sqrt 2 $

Question 123: Palette 123/251

JEE Mains 2021 MCQ

Let f : R $\to$ R be a continuous function. Then $\mathop {\lim }\limits_{x \to {\pi \over 4}} {{{\pi \over 4}\int\limits_2^{{{\sec }^2}x} {f(x)\,dx} } \over {{x^2} - {{{\pi ^2}} \over {16}}}}$ is equal to :

A

f (2)

B

2f (2)

C

2f $\left( {\sqrt 2 } \right)$

D

4f (2)

Question 124: Palette 124/251

JEE Mains 2021 MCQ

Let ${J_{n,m}} = \int\limits_0^{{1 \over 2}} {{{{x^n}} \over {{x^m} - 1}}dx} $, $\forall$ n > m and n, m $\in$ N. Consider a matrix $A = {[{a_{ij}}]_{3 \times 3}}$ where ${a_{ij}} = \left\{ {\matrix{ {{j_{6 + i,3}} - {j_{i + 3,3}},} & {i \le j} \cr {0,} & {i > j} \cr } } \right.$. Then $\left| {adj{A^{ - 1}}} \right|$ is :

A

(15)² $\times$ 2⁴²

B

(15)² $\times$ 2³⁴

C

(105)² $\times$ 2³⁸

D

(105)² $\times$ 2³⁶

Question 125: Palette 125/251

JEE Mains 2021 MCQ

The function f(x), that satisfies the condition
$f(x) = x + \int\limits_0^{\pi /2} {\sin x.\cos y\,f(y)\,dy} $, is :

A

$x + {2 \over 3}(\pi - 2)\sin x$

B

$x + (\pi + 2)\sin x$

C

$x + {\pi \over 2}\sin x$

D

$x + (\pi - 2)\sin x$

Question 126: Palette 126/251

JEE Mains 2021 MCQ

If [x] is the greatest integer $\le$ x, then

${\pi ^2}\int\limits_0^2 {\left( {\sin {{\pi x} \over 2}} \right)(x - [x]} {)^{[x]}}dx$ is equal to :

A

2($\pi$ $-$ 1)

B

4($\pi$ $-$ 1)

C

4($\pi$ + 1)

D

2($\pi$ + 1)

Question 127: Palette 127/251

JEE Mains 2021 MCQ

Let f be a non-negative function in [0, 1] and twice differentiable in (0, 1). If $\int_0^x {\sqrt {1 - {{(f'(t))}^2}} dt = \int_0^x {f(t)dt} } $, $0 \le x \le 1$ and f(0) = 0, then $\mathop {\lim }\limits_{x \to 0} {1 \over {{x^2}}}\int_0^x {f(t)dt} $ :

A

equals 0

B

equals 1

C

does not exist

D

equals ${1 \over 2}$

Question 128: Palette 128/251

JEE Mains 2021 MCQ

The value of the integral $\int\limits_0^1 {{{\sqrt x dx} \over {(1 + x)(1 + 3x)(3 + x)}}} $ is :

A

${\pi \over 8}\left( {1 - {{\sqrt 3 } \over 2}} \right)$

B

${\pi \over 4}\left( {1 - {{\sqrt 3 } \over 6}} \right)$

C

${\pi \over 8}\left( {1 - {{\sqrt 3 } \over 6}} \right)$

D

${\pi \over 4}\left( {1 - {{\sqrt 3 } \over 2}} \right)$

Question 129: Palette 129/251

JEE Mains 2021 MCQ

If ${U_n} = \left( {1 + {1 \over {{n^2}}}} \right)\left( {1 + {{{2^2}} \over {{n^2}}}} \right)^2.....\left( {1 + {{{n^2}} \over {{n^2}}}} \right)^n$, then $\mathop {\lim }\limits_{n \to \infty } {({U_n})^{{{ - 4} \over {{n^2}}}}}$ is equal to :

A

${{{e^2}} \over {16}}$

B

${4 \over e}$

C

${{16} \over {{e^2}}}$

D

${4 \over {{e^2}}}$

Question 130: Palette 130/251

JEE Mains 2021 MCQ

$\int\limits_6^{16} {{{{{\log }_e}{x^2}} \over {{{\log }_e}{x^2} + {{\log }_e}({x^2} - 44x + 484)}}dx} $ is equal to :

A

6

B

8

C

5

D

10

Question 131: Palette 131/251

JEE Mains 2021 MCQ

If the value of the integral
$\int\limits_0^5 {{{x + [x]} \over {{e^{x - [x]}}}}dx = \alpha {e^{ - 1}} + \beta } $, where $\alpha$, $\beta$ $\in$ R, 5$\alpha$ + 6$\beta$ = 0, and [x] denotes the greatest integer less than or equal to x; then the value of ($\alpha$ + $\beta$)² is equal to :

A

100

B

25

C

16

D

36

Question 132: Palette 132/251

JEE Mains 2021 MCQ

The value of $\int\limits_{ - {\pi \over 2}}^{{\pi \over 2}} {\left( {{{1 + {{\sin }^2}x} \over {1 + {\pi ^{\sin x}}}}} \right)} \,dx$ is

A

${\pi \over 2}$

B

${{5\pi } \over 4}$

C

${{3\pi } \over 4}$

D

${{3\pi } \over 2}$

Question 133: Palette 133/251

JEE Mains 2021 MCQ

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

A

log_e 4

B

log_e 16

C

2log_e 16

D

4log_e (3 + 2${\sqrt 2 }$)

Question 134: Palette 134/251

JEE Mains 2021 MCQ

The value of

$\mathop {\lim }\limits_{n \to \infty } {1 \over n}\sum\limits_{r = 0}^{2n - 1} {{{{n^2}} \over {{n^2} + 4{r^2}}}} $ is :

A

${1 \over 2}{\tan ^{ - 1}}(2)$

B

${1 \over 2}{\tan ^{ - 1}}(4)$

C

${\tan ^{ - 1}}(4)$

D

${1 \over 4}{\tan ^{ - 1}}(4)$

Question 135: Palette 135/251

JEE Mains 2021 MCQ

Let f : (a, b) $\to$ R be twice differentiable function such that $f(x) = \int_a^x {g(t)dt} $ for a differentiable function g(x). If f(x) = 0 has exactly five distinct roots in (a, b), then g(x)g'(x) = 0 has at least :

A

twelve roots in (a, b)

B

five roots in (a, b)

C

seven roots in (a, b)

D

three roots in (a, b)

Question 136: Palette 136/251

JEE Mains 2021 MCQ

The value of $\mathop {\lim }\limits_{n \to \infty } {1 \over n}\sum\limits_{j = 1}^n {{{(2j - 1) + 8n} \over {(2j - 1) + 4n}}} $ is equal to :

A

$5 + {\log _e}\left( {{3 \over 2}} \right)$

B

$2 - {\log _e}\left( {{2 \over 3}} \right)$

C

$3 + 2{\log _e}\left( {{2 \over 3}} \right)$

D

$1 + 2{\log _e}\left( {{3 \over 2}} \right)$

Question 137: Palette 137/251

JEE Mains 2021 MCQ

The value of the definite integral

$\int\limits_{ - {\pi \over 4}}^{{\pi \over 4}} {{{dx} \over {(1 + {e^{x\cos x}})({{\sin }^4}x + {{\cos }^4}x)}}} $ is equal to :

A

$ - {\pi \over 2}$

B

${\pi \over {2\sqrt 2 }}$

C

$ - {\pi \over 4}$

D

${\pi \over {\sqrt 2 }}$

Question 138: Palette 138/251

JEE Mains 2021 MCQ

If $f(x) = \left\{ {\matrix{ {\int\limits_0^x {\left( {5 + \left| {1 - t} \right|} \right)dt,} } & {x > 2} \cr {5x + 1,} & {x \le 2} \cr } } \right.$, then

A

f(x) is not continuous at x = 2

B

f(x) is everywhere differentiable

C

f(x) is continuous but not differentiable at x = 2

D

f(x) is not differentiable at x = 1

Question 139: Palette 139/251

JEE Mains 2021 MCQ

The value of the

integral $\int\limits_{ - 1}^1 {\log \left( {x + \sqrt {{x^2} + 1} } \right)dx} $ is :

A

2

B

0

C

$-$1

D

1

Question 140: Palette 140/251

JEE Mains 2021 MCQ

The value of the definite integral $\int\limits_{\pi /24}^{5\pi /24} {{{dx} \over {1 + \root 3 \of {\tan 2x} }}} $ is :

A

${\pi \over 3}$

B

${\pi \over 6}$

C

${\pi \over {12}}$

D

${\pi \over {18}}$

Question 141: Palette 141/251

JEE Mains 2021 MCQ

Let $f:[0,\infty ) \to [0,\infty )$ be defined as $f(x) = \int_0^x {[y]dy} $

where [x] is the greatest integer less than or equal to x. Which of the following is true?

A

f is continuous at every point in $[0,\infty )$ and differentiable except at the integer points.

B

f is both continuous and differentiable except at the integer points in $[0,\infty )$.

C

f is continuous everywhere except at the integer points in $[0,\infty )$.

D

f is differentiable at every point in $[0,\infty )$.

Question 142: Palette 142/251

JEE Mains 2021 MCQ

If $\int\limits_0^{100\pi } {{{{{\sin }^2}x} \over {{e^{\left( {{x \over \pi } - \left[ {{x \over \pi }} \right]} \right)}}}}dx = {{\alpha {\pi ^3}} \over {1 + 4{\pi ^2}}},\alpha \in R} $ where [x] is the greatest integer less than or equal to x, then the value of $\alpha$ is :

A

200 (1 $-$ e^$-$1)

B

100 (1 $-$ e)

C

50 (e $-$ 1)

D

150 (e^$-$1 $-$ 1)

Question 143: Palette 143/251

JEE Mains 2021 MCQ

If [x] denotes the greatest integer less than or equal to x, then the value of the integral $\int_{ - \pi /2}^{\pi /2} {[[x] - \sin x]dx} $ is equal to :

A

$-$ $\pi$

B

$\pi$

C

0

D

1

Question 144: Palette 144/251

JEE Mains 2021 MCQ

If the real part of the complex number ${(1 - \cos \theta + 2i\sin \theta )^{ - 1}}$ is ${1 \over 5}$ for $\theta \in (0,\pi )$, then the value of the integral $\int_0^\theta {\sin x} dx$ is equal to:

A

1

B

2

C

$-$1

D

0

Question 145: Palette 145/251

JEE Mains 2021 MCQ

Let $g(t) = \int_{ - \pi /2}^{\pi /2} {\cos \left( {{\pi \over 4}t + f(x)} \right)} dx$, where $f(x) = {\log _e}\left( {x + \sqrt {{x^2} + 1} } \right),x \in R$. Then which one of the following is correct?

A

g(1) = g(0)

B

$\sqrt 2 g(1) = g(0)$

C

$g(1) = \sqrt 2 g(0)$

D

g(1) + g(0) = 0

Question 146: Palette 146/251

JEE Mains 2021 MCQ

Let a be a positive real number such that $\int_0^a {{e^{x - [x]}}} dx = 10e - 9$ where [ x ] is the greatest integer less than or equal to x. Then a is equal to:

A

$10 - {\log _e}(1 + e)$

B

$10 + {\log _e}2$

C

$10 + {\log _e}3$

D

$10 + {\log _e}(1 + e)$

Question 147: Palette 147/251

JEE Mains 2021 MCQ

The value of the integral $\int\limits_{ - 1}^1 {{{\log }_e}(\sqrt {1 - x} + \sqrt {1 + x} )dx} $ is equal to:

A

${1 \over 2}{\log _e}2 + {\pi \over 4} - {3 \over 2}$

B

$2{\log _e}2 + {\pi \over 4} - 1$

C

${\log _e}2 + {\pi \over 2} - 1$

D

$2{\log _e}2 + {\pi \over 2} - {1 \over 2}$

Question 148: Palette 148/251

JEE Mains 2021 MCQ

Let g(x) = $\int_0^x {f(t)dt} $, where f is continuous function in [ 0, 3 ] such that ${1 \over 3}$ $ \le $ f(t) $ \le $ 1 for all t$\in$ [0, 1] and 0 $ \le $ f(t) $ \le $ ${1 \over 2}$ for all t$\in$ (1, 3]. The largest possible interval in which g(3) lies is :

A

$\left[ { - 1, - {1 \over 2}} \right]$

B

$\left[ { - {3 \over 2}, - 1} \right]$

C

[1, 3]

D

$\left[ {{1 \over 3},2} \right]$

Question 149: Palette 149/251

JEE Mains 2021 MCQ

Let f : R $ \to $ R be defined as f(x) = e^$-$xsinx. If F : [0, 1] $ \to $ R is a differentiable function with that F(x) = $\int_0^x {f(t)dt} $, then the value of $\int_0^1 {(F'(x) + f(x)){e^x}dx} $ lies in the interval

A

$\left[ {{{331} \over {360}},{{334} \over {360}}} \right]$

B

$\left[ {{{330} \over {360}},{{331} \over {360}}} \right]$

C

$\left[ {{{335} \over {360}},{{336} \over {360}}} \right]$

D

$\left[ {{{327} \over {360}},{{329} \over {360}}} \right]$

Question 150: Palette 150/251

JEE Mains 2021 MCQ

If the integral

$\int_0^{10} {{{[\sin 2\pi x]} \over {{e^{x - [x]}}}}} dx = \alpha {e^{ - 1}} + \beta {e^{ - {1 \over 2}}} + \gamma $, where $\alpha$, $\beta$, $\gamma$ are integers and [x] denotes the greatest integer less than or equal to x, then the value of $\alpha$ + $\beta$ + $\gamma$ is equal to :

A

0

B

10

C

20

D

25

Question 151: Palette 151/251

JEE Mains 2021 MCQ

Which of the following statements is correct for the function g($\alpha$) for $\alpha$ $\in$ R such that

$g(\alpha ) = \int\limits_{{\pi \over 6}}^{{\pi \over 3}} {{{{{\sin }^\alpha }x} \over {{{\cos }^\alpha }x + {{\sin }^\alpha }x}}dx} $

A

$g(\alpha )$ is a strictly increasing function

B

$g(\alpha )$ is an even function

C

$g(\alpha )$ has an inflection point at $\alpha$ = $-$${1 \over 2}$

D

$g(\alpha )$ is a strictly decreasing function

Question 152: Palette 152/251

JEE Mains 2021 MCQ

Consider the integral
$I = \int_0^{10} {{{[x]{e^{[x]}}} \over {{e^{x - 1}}}}dx} $,
where [x] denotes the greatest integer less than or equal to x. Then the value of I is equal to :

A

45 (e $-$ 1)

B

45 (e + 1)

C

9 (e + 1)

D

9 (e $-$ 1)

Question 153: Palette 153/251

JEE Mains 2021 MCQ

Let P(x) = x² + bx + c be a quadratic polynomial with real coefficients such that $\int_0^1 {P(x)dx} $ = 1 and P(x) leaves remainder 5 when it is divided by (x $-$ 2). Then the value of 9(b + c) is equal to :

A

9

B

11

C

7

D

15

Question 154: Palette 154/251

JEE Mains 2021 MCQ

Let $f(x) = \int\limits_0^x {{e^t}f(t)dt + {e^x}} $ be a differentiable function for all x$\in$R. Then f(x) equals :

A

${e^{({e^{x - 1}})}}$

B

$2{e^{{e^x}}} - 1$

C

$2{e^{{e^x} - 1}} - 1$

D

${e^{{e^x}}} - 1$

Question 155: Palette 155/251

JEE Mains 2021 MCQ

For x > 0, if $f(x) = \int\limits_1^x {{{{{\log }_e}t} \over {(1 + t)}}dt} $, then $f(e) + f\left( {{1 \over e}} \right)$ is equal to :

A

${1 \over 2}$

B

$-$1

C

0

D

1

Question 156: Palette 156/251

JEE Mains 2021 MCQ

The value of $\int\limits_{ - \pi /2}^{\pi /2} {{{{{\cos }^2}x} \over {1 + {3^x}}}} dx$ is :

A

$2\pi $

B

${\pi \over 2}$

C

$4\pi $

D

${\pi \over 4}$

Question 157: Palette 157/251

JEE Mains 2021 MCQ

The value of $\sum\limits_{n = 1}^{100} {\int\limits_{n - 1}^n {{e^{x - [x]}}dx} } $, where [ x ] is the greatest integer $ \le $ x, is :

A

100e

B

100(e $-$ 1)

C

100(1 + e)

D

100(1 $-$ e)

Question 158: Palette 158/251

JEE Mains 2021 MCQ

If ${I_n} = \int\limits_{{\pi \over 4}}^{{\pi \over 2}} {{{\cot }^n}x\,dx} $, then :

A

${1 \over {{I_2} + {I_4}}},{1 \over {{I_3} + {I_5}}},{1 \over {{I_4} + {I_6}}}$ are in A.P.

B

I₂ + I₄, I₃ + I₅, I₄ + I₆ are in A.P.

C

${1 \over {{I_2} + {I_4}}},{1 \over {{I_3} + {I_5}}},{1 \over {{I_4} + {I_6}}}$ are in G.P.

D

I₂ + I₄, (I₃ + I₅)², I₄ + I₆ are in G.P.

Question 159: Palette 159/251

JEE Mains 2021 MCQ

$\mathop {\lim }\limits_{n \to \infty } \left[ {{1 \over n} + {n \over {{{(n + 1)}^2}}} + {n \over {{{(n + 2)}^2}}} + ........ + {n \over {{{(2n + 1)}^2}}}} \right]$ is equal to :

A

${{1 \over 2}}$

B

${{1 \over 3}}$

C

1

D

${{1 \over 4}}$

Question 160: Palette 160/251

JEE Mains 2021 MCQ

The value of $\int\limits_{ - 1}^1 {{x^2}{e^{[{x^3}]}}} dx$, where [ t ] denotes the greatest integer $ \le $ t, is :

A

${{e + 1} \over 3}$

B

${{e - 1} \over {3e}}$

C

${1 \over {3e}}$

D

${{e + 1} \over {3e}}$

Question 161: Palette 161/251

JEE Mains 2021 MCQ

The value of the integral, $\int\limits_1^3 {[{x^2} - 2x - 2]dx} $, where [x] denotes the greatest integer less than or equal to x, is :

A

$-$ 5

B

$ - \sqrt 2 - \sqrt 3 + 1$

C

$-$ 4

D

$ - \sqrt 2 - \sqrt 3 - 1$

Question 162: Palette 162/251

JEE Mains 2021 MCQ

Let f(x) be a differentiable function defined on [0, 2] such that f'(x) = f'(2 $-$ x) for all x$ \in $ (0, 2), f(0) = 1 and f(2) = e². Then the value of $\int\limits_0^2 {f(x)} dx$ is :

A

1 + e²

B

2(1 + e²)

C

1 $-$ e²

D

2(1 $-$ e²)

Question 163: Palette 163/251

JEE Mains 2021 MCQ

Let f be a twice differentiable function defined on R such that f(0) = 1, f'(0) = 2 and f'(x) $ \ne $ 0 for all x $ \in $ R. If $\left| {\matrix{ {f(x)} & {f'(x)} \cr {f'(x)} & {f''(x)} \cr } } \right|$ = 0, for all x$ \in $R, then the value of f(1) lies in the interval :

A

(0, 3)

B

(9, 12)

C

(3, 6)

D

(6, 9)

Question 164: Palette 164/251

JEE Mains 2021 MCQ

$\mathop {\lim }\limits_{x \to 0} {{\int\limits_0^{{x^2}} {\left( {\sin \sqrt t } \right)dt} } \over {{x^3}}}$ is equal to :

A

${1 \over {15}}$

B

0

C

${2 \over 3}$

D

${3 \over 2}$

Question 165: Palette 165/251

JEE Mains 2020 MCQ

The integral $\int\limits_1^2 {{e^x}.{x^x}\left( {2 + {{\log }_e}x} \right)} dx$ equals :

A

e(4e + 1)

B

e(2e – 1)

C

e(4e – 1)

D

4e² – 1

Question 166: Palette 166/251

JEE Mains 2020 MCQ

If I₁ = $\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{100}}} dx$ and
I₂ = $\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{101}}} dx$ such
that I₂ = $\alpha $I₁ then $\alpha $ equals to :

A

${{5051} \over {5050}}$

B

${{5050} \over {5051}}$

C

${{5050} \over {5049}}$

D

${{5049} \over {5050}}$

Question 167: Palette 167/251

JEE Mains 2020 MCQ

$\mathop {\lim }\limits_{x \to 1} \left( {{{\int\limits_0^{{{\left( {x - 1} \right)}^2}} {t\cos \left( {{t^2}} \right)dt} } \over {\left( {x - 1} \right)\sin \left( {x - 1} \right)}}} \right)$

A

is equal to 0

B

is equal to ${1 \over 2}$

C

does not exist

D

is equal to $ - {1 \over 2}$

Question 168: Palette 168/251

JEE Mains 2020 MCQ

The value of $\int\limits_{{{ - \pi } \over 2}}^{{\pi \over 2}} {{1 \over {1 + {e^{\sin x}}}}dx} $ is:

A

$\pi $

B

${{3\pi \over 2}}$

C

${{\pi \over 2}}$

D

${{\pi \over 4}}$

Question 169: Palette 169/251

JEE Mains 2020 MCQ

The integral
$\int\limits_{{\pi \over 6}}^{{\pi \over 3}} {{{\tan }^3}x.{{\sin }^2}3x\left( {2{{\sec }^2}x.{{\sin }^2}3x + 3\tan x.\sin 6x} \right)dx} $
is equal to:

A

$ - {1 \over {9}}$

B

$ - {1 \over {18}}$

C

$ {7 \over {18}}$

D

${9 \over 2}$

Question 170: Palette 170/251

JEE Mains 2020 MCQ

Let $f(x) = \left| {x - 2} \right|$ and g(x) = f(f(x)), $x \in \left[ {0,4} \right]$. Then
$\int\limits_0^3 {\left( {g(x) - f(x)} \right)} dx$ is equal to:

A

1

B

0

C

${1 \over 2}$

D

${3 \over 2}$

Question 171: Palette 171/251

JEE Mains 2020 MCQ

If the value of the integral
$\int\limits_0^{{1 \over 2}} {{{{x^2}} \over {{{\left( {1 - {x^2}} \right)}^{{3 \over 2}}}}}} dx$

is ${k \over 6}$, then k is equal to :

A

$2\sqrt 3 + \pi $

B

$3\sqrt 2 - \pi $

C

$3\sqrt 2 + \pi $

D

$2\sqrt 3 - \pi $

Question 172: Palette 172/251

JEE Mains 2020 MCQ

Suppose f(x) is a polynomial of degree four, having critical points at –1, 0, 1. If
T = {x $ \in $ R | f(x) = f(0)}, then the sum of squares of all the elements of T is :

A

6

B

2

C

8

D

4

Question 173: Palette 173/251

JEE Mains 2020 MCQ

$\int\limits_{ - \pi }^\pi {\left| {\pi - \left| x \right|} \right|dx} $ is equal to :

A

${\pi ^2}$

B

2${\pi ^2}$

C

$\sqrt 2 {\pi ^2}$

D

${{{\pi ^2}} \over 2}$

Question 174: Palette 174/251

JEE Mains 2020 MCQ

Let a function ƒ : [0, 5] $ \to $ R be continuous, ƒ(1) = 3 and F be defined as :

$F(x) = \int\limits_1^x {{t^2}g(t)dt} $ , where $g(t) = \int\limits_1^t {f(u)du} $

Then for the function F, the point x = 1 is :

A

a point of inflection.

B

a point of local maxima.

C

a point of local minima.

D

not a critical point.

Question 175: Palette 175/251

JEE Mains 2020 MCQ

The value of
$\int\limits_0^{2\pi } {{{x{{\sin }^8}x} \over {{{\sin }^8}x + {{\cos }^8}x}}} dx$ is equal to :

A

4$\pi $

B

2$\pi $

C

$\pi $²

D

2$\pi $²

Question 176: Palette 176/251

JEE Mains 2020 MCQ

If for all real triplets (a, b, c), ƒ(x) = a + bx + cx²; then $\int\limits_0^1 {f(x)dx} $ is equal to :

A

${1 \over 6}\left\{ {f(0) + f(1) + 4f\left( {{1 \over 2}} \right)} \right\}$

B

$2\left\{ 3{f(1) + 2f\left( {{1 \over 2}} \right)} \right\}$

C

${1 \over 3}\left\{ {f(0) + f\left( {{1 \over 2}} \right)} \right\}$

D

${1 \over 2}\left\{ {f(1) + 3f\left( {{1 \over 2}} \right)} \right\}$

Question 177: Palette 177/251

JEE Mains 2020 MCQ

If $I = \int\limits_1^2 {{{dx} \over {\sqrt {2{x^3} - 9{x^2} + 12x + 4} }}} $, then :

A

${1 \over 16} < {I^2} < {1 \over 9}$

B

${1 \over 8} < {I^2} < {1 \over 4}$

C

${1 \over 9} < {I^2} < {1 \over 8}$

D

${1 \over 6} < {I^2} < {1 \over 2}$

Question 178: Palette 178/251

JEE Mains 2020 MCQ

$\mathop {\lim }\limits_{x \to 0} {{\int_0^x {t\sin \left( {10t} \right)dt} } \over x}$ is equal to

A

$ - {1 \over 5}$

B

$ - {1 \over 10}$

C

0

D

$ {1 \over 10}$

Question 179: Palette 179/251

JEE Mains 2020 MCQ

The value of $\alpha $ for which
$4\alpha \int\limits_{ - 1}^2 {{e^{ - \alpha \left| x \right|}}dx} = 5$, is:

A

${\log _e}2$

B

${\log _e}\sqrt 2 $

C

${\log _e}\left( {{4 \over 3}} \right)$

D

${\log _e}\left( {{3 \over 2}} \right)$

Question 180: Palette 180/251

JEE Mains 2020 MCQ

If $\theta $₁ and $\theta $₂ be respectively the smallest and the largest values of $\theta $ in (0, 2$\pi $) - {$\pi $} which satisfy the equation,
2cot²$\theta $ - ${5 \over {\sin \theta }}$ + 4 = 0, then
$\int\limits_{{\theta _1}}^{{\theta _2}} {{{\cos }^2}3\theta d\theta } $ is equal to :

A

${\pi \over 9}$

B

${{2\pi } \over 3}$

C

${{\pi } \over 3}$

D

${\pi \over 3} + {1 \over 6}$

Question 181: Palette 181/251

JEE Mains 2020 MCQ

If ƒ(a + b + 1 - x) = ƒ(x), for all x, where a and b are fixed positive real numbers, then

${1 \over {a + b}}\int_a^b {x\left( {f(x) + f(x + 1)} \right)} dx$ is equal to:

A

$\int_{a - 1}^{b - 1} {f(x+1)dx} $

B

$\int_{a + 1}^{b + 1} {f(x + 1)dx} $

C

$\int_{a - 1}^{b - 1} {f(x)dx} $

D

$\int_{a + 1}^{b + 1} {f(x)dx} $

Question 182: Palette 182/251

JEE Mains 2019 MCQ

A value of $\alpha $ such that
$\int\limits_\alpha ^{\alpha + 1} {{{dx} \over {\left( {x + \alpha } \right)\left( {x + \alpha + 1} \right)}}} = {\log _e}\left( {{9 \over 8}} \right)$ is :

A

2

B

- 2

C

${1 \over 2}$

D

$-{1 \over 2}$

Question 183: Palette 183/251

JEE Mains 2019 MCQ

If $\int\limits_0^{{\pi \over 2}} {{{\cot x} \over {\cot x + \cos ecx}}} dx$ = m($\pi $ + n), then m.n is equal to

A

- 1

B

1

C

$ - {1 \over 2}$

D

${1 \over 2}$

Question 184: Palette 184/251

JEE Mains 2019 MCQ

Let f : R $ \to $ R be a continuously differentiable function such that f(2) = 6 and f'(2) = ${1 \over {48}}$. If $\int\limits_6^{f\left( x \right)} {4{t^3}} dt$ = (x - 2)g(x), then $\mathop {\lim }\limits_{x \to 2} g\left( x \right)$ is equal to :

A

18

B

36

C

12

D

24

Question 185: Palette 185/251

JEE Mains 2019 MCQ

The integral $\int\limits_{\pi /6}^{\pi /3} {{{\sec }^{2/3}}} x\cos e{c^{4/3}}xdx$ is equal to :

A

${3^{{5 \over 3}}} - {3^{{1 \over 3}}}$

B

${3^{{5 \over 6}}} - {3^{{2 \over 3}}}$

C

${3^{{4 \over 3}}} - {3^{{1 \over 3}}}$

D

${3^{{7 \over 6}}} - {3^{{5 \over 6}}}$

Question 186: Palette 186/251

JEE Mains 2019 MCQ

The value of $\int\limits_0^{2\pi } {\left[ {\sin 2x\left( {1 + \cos 3x} \right)} \right]} dx$,
where [t] denotes the greatest integer function is :

A

2$\pi $

B

$\pi $

C

-2$\pi $

D

-$\pi $

Question 187: Palette 187/251

JEE Mains 2019 MCQ

$\mathop {\lim }\limits_{n \to \infty } \left( {{{{{(n + 1)}^{1/3}}} \over {{n^{4/3}}}} + {{{{(n + 2)}^{1/3}}} \over {{n^{4/3}}}} + ....... + {{{{(2n)}^{1/3}}} \over {{n^{4/3}}}}} \right)$
is equal to :

A

${4 \over 3}{\left( 2 \right)^{3/4}}$

B

${3 \over 4}{\left( 2 \right)^{4/3}} - {3 \over 4}$

C

${4 \over 3}{\left( 2 \right)^{4/3}}$

D

${3 \over 4}{\left( 2 \right)^{4/3}} - {4 \over 3}$

Question 188: Palette 188/251

JEE Mains 2019 MCQ

If f : R $ \to $ R is a differentiable function and f(2) = 6,
then $\mathop {\lim }\limits_{x \to 2} {{\int\limits_6^{f\left( x \right)} {2tdt} } \over {\left( {x - 2} \right)}}$ is :-

A

2f'(2)

B

24f'(2)

C

0

D

12f'(2)

Question 189: Palette 189/251

JEE Mains 2019 MCQ

The value of the integral $\int\limits_0^1 {x{{\cot }^{ - 1}}(1 - {x^2} + {x^4})dx} $ is :-

A

${\pi \over 2} - {1 \over 2}{\log _e}2$

B

${\pi \over 4} - {\log _e}2$

C

${\pi \over 4} - {1 \over 2}{\log _e}2$

D

${\pi \over 2} - {\log _e}2$

Question 190: Palette 190/251

JEE Mains 2019 MCQ

The value of $\int\limits_0^{\pi /2} {{{{{\sin }^3}x} \over {\sin x + \cos x}}dx} $ is

A

${{\pi - 2} \over 8}$

B

${{\pi - 2} \over 4}$

C

${{\pi - 1} \over 2}$

D

${{\pi - 1} \over 4}$

Question 191: Palette 191/251

JEE Mains 2019 MCQ

Let $f(x) = \int\limits_0^x {g(t)dt} $ where g is a non-zero even function. If ƒ(x + 5) = g(x), then $ \int\limits_0^x {f(t)dt} $ equals-

A

5$\int\limits_{x + 5}^5 {g(t)dt} $

B

$\int\limits_{x + 5}^5 {g(t)dt} $

C

$\int\limits_{5}^{x+5} {g(t)dt} $

D

2$\int\limits_{5}^{x+5} {g(t)dt} $

Question 192: Palette 192/251

JEE Mains 2019 MCQ

If $f(x) = {{2 - x\cos x} \over {2 + x\cos x}}$ and g(x) = log_ex, (x > 0) then the value of integral

$\int\limits_{ - {\pi \over 4}}^{{\pi \over 4}} {g\left( {f\left( x \right)} \right)dx{\rm{ }}} $ is

A

log_e3

B

log_e2

C

log_e1

D

log_ee

Question 193: Palette 193/251

JEE Mains 2019 MCQ

$\mathop {\lim }\limits_{x \to \infty } \left( {{n \over {{n^2} + {1^2}}} + {n \over {{n^2} + {2^2}}} + {n \over {{n^2} + {3^2}}} + ..... + {1 \over {5n}}} \right)$ is equal to :

A

tan^–1 (2)

B

tan^–1 (3)

C

${\pi \over 4}$

D

${\pi \over 2}$

Question 194: Palette 194/251

JEE Mains 2019 MCQ

The integral $\int\limits_1^e {\left\{ {{{\left( {{x \over e}} \right)}^{2x}} - {{\left( {{e \over x}} \right)}^x}} \right\}} \,$ log_e x dx is equal to :

A

$ - {1 \over 2} + {1 \over e} - {1 \over {2{e^2}}}$

B

${3 \over 2} - e - {1 \over {2{e^2}}}$

C

${1 \over 2} - e - {1 \over {{e^2}}}$

D

${3 \over 2} - {1 \over e} - {1 \over {2{x^2}}}$

Question 195: Palette 195/251

JEE Mains 2019 MCQ

Let f and g be continuous functions on [0, a] such that f(x) = f(a – x) and g(x) + g(a – x) = 4, then $\int\limits_0^a \, $f(x) g(x) dx is equal to :

A

4$\int\limits_0^a \, $f(x)dx

B

$-$ 3$\int\limits_0^a \, $f(x)dx

C

$\int\limits_0^a \, $f(x)dx

D

2$\int\limits_0^a \, $f(x)dx

Question 196: Palette 196/251

JEE Mains 2019 MCQ

The integral  $\int\limits_{\pi /6}^{\pi /4} {{{dx} \over {\sin 2x\left( {{{\tan }^5}x + {{\cot }^5}x} \right)}}} $  equals :

A

${\pi \over {40}}$

B

${1 \over {20}}{\tan ^{ - 1}}\left( {{1 \over {9\sqrt 3 }}} \right)$

C

${1 \over {10}}\left( {{\pi \over 4} - {{\tan }^{ - 1}}\left( {{1 \over {9\sqrt 3 }}} \right)} \right)$

D

${1 \over 5}\left( {{\pi \over 4}{{-\tan }^{ - 1}}\left( {{1 \over {3\sqrt 3 }}} \right)} \right)$

Question 197: Palette 197/251

JEE Mains 2019 MCQ

The value of the integral $\int\limits_{ - 2}^2 {{{{{\sin }^2}x} \over { \left[ {{x \over \pi }} \right] + {1 \over 2}}}} \,dx$ (where [x] denotes the greatest integer less than or equal to x) is

A

0

B

4

C

4$-$ sin 4

D

sin 4

Question 198: Palette 198/251

JEE Mains 2019 MCQ

The value of   $\int\limits_{ - \pi /2}^{\pi /2} {{{dx} \over {\left[ x \right] + \left[ {\sin x} \right] + 4}}} ,$  where [t] denotes the greatest integer less than or equal to t, is

A

${1 \over {12}}\left( {7\pi - 5} \right)$

B

${1 \over {12}}\left( {7\pi + 5} \right)$

C

${3 \over {10}}\left( {4\pi - 3} \right)$

D

${3 \over {20}}\left( {4\pi - 3} \right)$

Question 199: Palette 199/251

JEE Mains 2019 MCQ

If  $\int\limits_0^x \, $f(t) dt = x² + $\int\limits_x^1 \, $ t²f(t) dt then f '$\left( {{1 \over 2}} \right)$ is -

A

${{18} \over {25}}$

B

${{6} \over {25}}$

C

${{24} \over {25}}$

D

${{4} \over {5}}$

Question 200: Palette 200/251

JEE Mains 2019 MCQ

Let  ${\rm I} = \int\limits_a^b {\left( {{x^4} - 2{x^2}} \right)} dx.$  If I is minimum then the ordered pair (a, b) is -

A

$\left( {\sqrt 2 , - \sqrt 2 } \right)$

B

$\left( {0,\sqrt 2 } \right)$

C

$\left( { - \sqrt 2 ,\sqrt 2 } \right)$

D

$\left( { - \sqrt 2 ,0} \right)$

Question 201: Palette 201/251

JEE Mains 2019 MCQ

If   $\int\limits_0^{{\pi \over 3}} {{{\tan \theta } \over {\sqrt {2k\,\sec \theta } }}} \,d\theta = 1 - {1 \over {\sqrt 2 }},\left( {k > 0} \right),$ then value of k is :

A

4

B

${1 \over 2}$

C

1

D

2

Question 202: Palette 202/251

JEE Mains 2019 MCQ

Let f be a differentiable function from

R to R such that $\left| {f\left( x \right) - f\left( y \right)} \right| \le 2{\left| {x - y} \right|^{{3 \over 2}}},$   

for all  $x,y \in $ R.

If   $f\left( 0 \right) = 1$  

then   $\int\limits_0^1 {{f^2}} \left( x \right)dx$  is equal to :

A

1

B

2

C

${1 \over 2}$

D

0

Question 203: Palette 203/251

JEE Mains 2019 MCQ

The value of $\int\limits_0^\pi {{{\left| {\cos x} \right|}^3}} \,dx$ is :

A

$4 \over 3$

B

$-$ $4 \over 3$

C

0

D

$2 \over 3$

Question 204: Palette 204/251

JEE Mains 2018 MCQ

If $f(x) = \int\limits_0^x {t\left( {\sin x - \sin t} \right)dt\,\,\,} $ then :

A

f'''(x) + f''(x) = sinx

B

f'''(x) + f''(x) $-$ f'(x) = cosx

C

f'''(x) + f'(x) = cosx $-$ 2x sinx

D

f'''(x) $-$ f''(x) = cosx $-$ 2x sinx

Question 205: Palette 205/251

JEE Mains 2018 MCQ

The value of $\int\limits_{ - \pi /2}^{\pi /2} {{{{{\sin }^2}x} \over {1 + {2^x}}}} dx$ is

A

${\pi \over 4}$

B

${\pi \over 8}$

C

${\pi \over 2}$

D

${4\pi }$

Question 206: Palette 206/251

JEE Mains 2018 MCQ

If   ${I_1} = \int_0^1 {{e^{ - x}}} {\cos ^2}x{\mkern 1mu} dx;$

   ${I_2} = \int_0^1 {{e^{ - {x^2}}}} {\cos ^2}x{\mkern 1mu} dx$  and

${I_3} = \int_0^1 {{e^{ - {x^3}}}} dx;$ then

A

I₂  >  I₃  >  I₁

B

I₂  >  I₁  >  I₃

C

I₃  >  I₂  >  I₁

D

I₃  >  I₁  >  I₂

Question 207: Palette 207/251

JEE Mains 2018 MCQ

The value of integral $\int_{{\pi \over 4}}^{{{3\pi } \over 4}} {{x \over {1 + \sin x}}dx} $ is :

A

$\pi \sqrt 2 $

B

$\pi \left( {\sqrt 2 - 1} \right)$

C

${\pi \over 2}\left( {\sqrt 2 + 1} \right)$

D

$2\pi \left( {\sqrt 2 - 1} \right)$

Question 208: Palette 208/251

JEE Mains 2018 MCQ

The value of the integral

$\int\limits_{ - {\pi \over 2}}^{{\pi \over 2}} {{{\sin }^4}} x\left( {1 + \log \left( {{{2 + \sin x} \over {2 - \sin x}}} \right)} \right)dx$ is :

A

0

B

${3 \over 4}$

C

${3 \over 8}$ $\pi $

D

${3 \over 16}$ $\pi $

Question 209: Palette 209/251

JEE Mains 2017 MCQ

If    $\mathop {\lim }\limits_{n \to \infty } \,\,{{{1^a} + {2^a} + ...... + {n^a}} \over {{{(n + 1)}^{a - 1}}\left[ {\left( {na + 1} \right) + \left( {na + 2} \right) + ..... + \left( {na + n} \right)} \right]}} = {1 \over {60}}$

for some positive real number a, then a is equal to :

A

7

B

8

C

${{15} \over 2}$

D

${{17} \over 2}$

Question 210: Palette 210/251

JEE Mains 2017 MCQ

If    $\int\limits_1^2 {{{dx} \over {{{\left( {{x^2} - 2x + 4} \right)}^{{3 \over 2}}}}}} = {k \over {k + 5}},$ then k is equal to :

A

1

B

2

C

3

D

4

Question 211: Palette 211/251

JEE Mains 2017 MCQ

The integral $\int_{{\pi \over {12}}}^{{\pi \over 4}} {\,\,{{8\cos 2x} \over {{{\left( {\tan x + \cot x} \right)}^3}}}} \,dx$ equals :

A

${{15} \over {128}}$

B

${{15} \over {64}}$

C

${{13} \over {32}}$

D

${{13} \over {256}}$

Question 212: Palette 212/251

JEE Mains 2017 MCQ

The integral $\int\limits_{{\pi \over 4}}^{{{3\pi } \over 4}} {{{dx} \over {1 + \cos x}}} $ is equal to

A

2

B

4

C

$-$ 1

D

$-$ 2

Question 213: Palette 213/251

JEE Mains 2016 MCQ

For x $ \in $ R, x $ \ne $ 0, if y(x) is a differentiable function such that

x $\int\limits_1^x y $ (t) dt = (x + 1) $\int\limits_1^x ty $ (t) dt,  then y (x) equals :

(where C is a constant.)

A

${C \over x}{e^{ - {1 \over x}}}$

B

${C \over {{x^2}}}{e^{ - {1 \over x}}}$

C

${C \over {{x^3}}}{e^{ - {1 \over x}}}$

D

$C{x^3}\,{1 \over {{e^x}}}$

Question 214: Palette 214/251

JEE Mains 2016 MCQ

The value of the integral

$\int\limits_4^{10} {{{\left[ {{x^2}} \right]dx} \over {\left[ {{x^2} - 28x + 196} \right] + \left[ {{x^2}} \right]}}} ,$

where [x] denotes the greatest integer less than or equal to x, is :

A

6

B

3

C

7

D

${1 \over 3}$

Question 215: Palette 215/251

JEE Mains 2016 MCQ

If   $2\int\limits_0^1 {{{\tan }^{ - 1}}xdx = \int\limits_0^1 {{{\cot }^{ - 1}}} } \left( {1 - x + {x^2}} \right)dx,$

then $\int\limits_0^1 {{{\tan }^{ - 1}}} \left( {1 - x + {x^2}} \right)dx$ is equalto :

A

log4

B

${\pi \over 2}$ + log2

C

log2

D

${\pi \over 2}$ $-$ log4

Question 216: Palette 216/251

JEE Mains 2016 MCQ

$\mathop {\lim }\limits_{n \to \infty } {\left( {{{\left( {n + 1} \right)\left( {n + 2} \right)...3n} \over {{n^{2n}}}}} \right)^{{1 \over n}}}$ is equal to:

A

${9 \over {{e^2}}}$

B

$3\,\log \,3 - 2$

C

${{18} \over {{e^4}}}$

D

${{27} \over {{e^2}}}$

Question 217: Palette 217/251

JEE Mains 2015 MCQ

The integral
$\int\limits_2^4 {{{\log \,{x^2}} \over {\log {x^2} + \log \left( {36 - 12x + {x^2}} \right)}}dx} $ is equal to :

A

$1$

B

$6$

C

$2$

D

$4$

Question 218: Palette 218/251

JEE Mains 2014 MCQ

The integral $\int\limits_0^\pi {\sqrt {1 + 4{{\sin }^2}{x \over 2} - 4\sin {x \over 2}{\mkern 1mu} } } dx$ equals:

A

$4\sqrt 3 - 4$

B

$4\sqrt 3 - 4 - {\pi \over 3}$

C

$\pi - 4$

D

${{2\pi } \over 3} - 4 - 4\sqrt 3 $

Question 219: Palette 219/251

JEE Mains 2013 MCQ

Statement-1 : The value of the integral
$\int\limits_{\pi /6}^{\pi /3} {{{dx} \over {1 + \sqrt {\tan \,x} }}} $ is equal to $\pi /6$

Statement-2 : $\int\limits_a^b {f\left( x \right)} dx = \int\limits_a^b {f\left( {a + b - x} \right)} dx.$

A

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

B

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

C

Statement- 1 is true; Statement-2 is False.

D

Statement-1 is false; Statement-2 is true.

Question 220: Palette 220/251

JEE Mains 2012 MSQ

If $g\left( x \right) = \int\limits_0^x {\cos 4t\,dt,} $ then $g\left( {x + \pi } \right)$ equals

A

${{g\left( x \right)} \over {8\left( \pi \right)}}$

B

$g\left( x \right) + g\left( \pi \right)$

C

$g\left( x \right) - g\left( \pi \right)$

D

$g\left( x \right) . g\left( \pi \right)$

Question 221: Palette 221/251

JEE Mains 2011 MCQ

The value of $\int\limits_0^1 {{{8\log \left( {1 + x} \right)} \over {1 + {x^2}}}} dx$ is

A

${\pi \over 8}\log 2$

B

${\pi \over 2}\log 2$

C

$\log 2$

D

$\pi \log 2$

Question 222: Palette 222/251

JEE Mains 2010 MCQ

Let $p(x)$ be a function defined on $R$ such that $p'(x)=p'(1-x),$ for all $x \in \left[ {0,1} \right],p\left( 0 \right) = 1$ and $p(1)=41.$ Then $\int\limits_0^1 {p\left( x \right)dx} $ equals :

A

$21$

B

$41$

C

$42$

D

$\sqrt {41} $

Question 223: Palette 223/251

JEE Mains 2009 MCQ

$\int\limits_0^\pi {\left[ {\cot x} \right]dx,} $ where $\left[ . \right]$ denotes the greatest integer function, is equal to:

A

$1$

B

$-1$

C

$ - {\pi \over 2}$

D

$ {\pi \over 2}$

Question 224: Palette 224/251

JEE Mains 2007 MCQ

Let $I = \int\limits_0^1 {{{\sin x} \over {\sqrt x }}dx} $ and $J = \int\limits_0^1 {{{\cos x} \over {\sqrt x }}dx} .$ Then which one of the following is true?

A

$1 > {2 \over 3}$ and $J > 2$

B

$1 < {2 \over 3}$ and $J < 2$

C

$1 < {2 \over 3}$ and $J > 2$

D

$1 > {2 \over 3}$ and $J < 2$

Question 225: Palette 225/251

JEE Mains 2007 MCQ

Let $F\left( x \right) = f\left( x \right) + f\left( {{1 \over x}} \right),$ where $f\left( x \right) = \int\limits_l^x {{{\log t} \over {1 + t}}dt,} $ Then $F(e)$ equals

A

$1$

B

$2$

C

$1/2$

D

$0$

Question 226: Palette 226/251

JEE Mains 2007 MCQ

The solution for $x$ of the equation $\int\limits_{\sqrt 2 }^x {{{dt} \over {t\sqrt {{t^2} - 1} }} = {\pi \over 2}} $ is

A

${{\sqrt 3 } \over 2}$

B

$2\sqrt 2 $

C

$2$

D

None

Question 227: Palette 227/251

JEE Mains 2006 MCQ

The value of $\int\limits_1^a {\left[ x \right]} f'\left( x \right)dx,a > 1$ where ${\left[ x \right]}$ denotes the greatest integer not exceeding $x$ is

A

$af\left( a \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + .............f\left( {\left[ a \right]} \right)} \right\}$

B

$\left[ a \right]f\left( a \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + ...........f\left( {\left[ a \right]} \right)} \right\}$

C

$\left[ a \right]f\left( {\left[ a \right]} \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + ...........f\left( a \right)} \right\}$

D

$af\left( {\left[ a \right]} \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + .............f\left( a \right)} \right\}$

Question 228: Palette 228/251

JEE Mains 2006 MCQ

$\int\limits_0^\pi {xf\left( {\sin x} \right)dx} $ is equal to

A

$\pi \int\limits_0^\pi {f\left( {\cos x} \right)dx} $

B

$\,\pi \int\limits_0^\pi {f\left( {sinx} \right)dx} $

C

${\pi \over 2}\int\limits_0^{\pi /2} {f\left( {sinx} \right)dx} $

D

$\pi \int\limits_0^{\pi /2} {f\left( {\cos x} \right)dx} $

Question 229: Palette 229/251

JEE Mains 2006 MCQ

$\int\limits_{ - {{3\pi } \over 2}}^{ - {\pi \over 2}} {\left[ {{{\left( {x + \pi } \right)}^3} + {{\cos }^2}\left( {x + 3\pi } \right)} \right]} dx$ is equal to

A

${{{\pi ^4}} \over {32}}$

B

${{{\pi ^4}} \over {32}} + {\pi \over 2}$

C

${\pi \over 2}$

D

${\pi \over 4} - 1$

Question 230: Palette 230/251

JEE Mains 2005 MCQ

If ${I_1} = \int\limits_0^1 {{2^{{x^2}}}dx,{I_2} = \int\limits_0^1 {{2^{{x^3}}}dx,\,{I_3} = \int\limits_1^2 {{2^{{x^2}}}dx} } } $ and ${I_4} = \int\limits_1^2 {{2^{{x^3}}}dx} $ then

A

${I_2} > {I_1}$

B

${I_1} > {I_2}$

C

${I_3} = {I_4}$

D

${I_3} > {I_4}$

Question 231: Palette 231/251

JEE Mains 2005 MCQ

$\mathop {\lim }\limits_{n \to \infty } \left[ {{1 \over {{n^2}}}{{\sec }^2}{1 \over {{n^2}}} + {2 \over {{n^2}}}{{\sec }^2}{4 \over {{n^2}}}.... + {1 \over n}{{\sec }^2}1} \right]$
equals

A

${1 \over 2}\sec 1$

B

${1 \over 2}$cosec 1

C

tan 1

D

${1 \over 2}$tan 1

Question 232: Palette 232/251

JEE Mains 2005 MCQ

The value of integral, $\int\limits_3^6 {{{\sqrt x } \over {\sqrt {9 - x} + \sqrt x }}} dx $ is

A

${1 \over 2}$

B

${3 \over 2}$

C

$2$

D

$1$

Question 233: Palette 233/251

JEE Mains 2005 MCQ

Let $f:R \to R$ be a differentiable function having $f\left( 2 \right) = 6$,
$f'\left( 2 \right) = \left( {{1 \over {48}}} \right)$. Then $\mathop {\lim }\limits_{x \to 2} \int\limits_6^{f\left( x \right)} {{{4{t^3}} \over {x - 2}}dt} $ equals :

A

$24$

B

$36$

C

$12$

D

$18$

Question 234: Palette 234/251

JEE Mains 2005 MCQ

The value of $\int\limits_{ - \pi }^\pi {{{{{\cos }^2}} \over {1 + {a^x}}}dx,\,\,a > 0,} $ is

A

$a\,\pi $

B

${\pi \over 2}$

C

${\pi \over a}$

D

${2\pi }$

Question 235: Palette 235/251

JEE Mains 2004 MCQ

$\mathop {Lim}\limits_{n \to \infty } \sum\limits_{r = 1}^n {{1 \over n}{e^{{r \over n}}}} $ is

A

$e+1$

B

$e-1$

C

$1-e$

D

$e$

Question 236: Palette 236/251

JEE Mains 2004 MCQ

If $\int\limits_0^\pi {xf\left( {\sin x} \right)dx = A\int\limits_0^{\pi /2} {f\left( {\sin x} \right)dx,} } $ then $A$ is

A

$2\pi $

B

$\pi $

C

${\pi \over 4}$

D

$0$

Question 237: Palette 237/251

JEE Mains 2004 MCQ

The value of $I = \int\limits_0^{\pi /2} {{{{{\left( {\sin x + \cos x} \right)}^2}} \over {\sqrt {1 + \sin 2x} }}dx} $ is

A

$3$

B

$1$

C

$2$

D

$0$

Question 238: Palette 238/251

JEE Mains 2004 MCQ

The value of $\int\limits_{ - 2}^3 {\left| {1 - {x^2}} \right|dx} $ is

A

${1 \over 3}$

B

${14 \over 3}$

C

${7 \over 3}$

D

${28 \over 3}$

Question 239: Palette 239/251

JEE Mains 2004 MCQ

If $f\left( x \right) = {{{e^x}} \over {1 + {e^x}}},{I_1} = \int\limits_{f\left( { - a} \right)}^{f\left( a \right)} {xg\left\{ {x\left( {1 - x} \right)} \right\}dx} $
and ${I_2} = \int\limits_{f\left( { - a} \right)}^{f\left( a \right)} {g\left\{ {x\left( {1 - x} \right)} \right\}dx} ,$ then the value of ${{{I_2}} \over {{I_1}}}$ is

A

$1$

B

$-3$

C

$-1$

D

$2$

Question 240: Palette 240/251

JEE Mains 2003 MCQ

If $f\left( y \right) = {e^y},$ $g\left( y \right) = y;y > 0$ and

$F\left( t \right) = \int\limits_0^t {f\left( {t - y} \right)g\left( y \right)dy,} $ then :

A

$F\left( t \right) = t{e^{ - t}}$

B

$F\left( t \right) = 1t - t{e^{ - 1}}\left( {1 + t} \right)$

C

$F\left( t \right) = {e^t} - \left( {1 + t} \right)$

D

$F\left( t \right) = t{e^t}$.

Question 241: Palette 241/251

JEE Mains 2003 MCQ

Let $f(x)$ be a function satisfying $f'(x)=f(x)$ with $f(0)=1$ and $g(x)$ be a function that satisfies $f\left( x \right) + g\left( x \right) = {x^2}$. Then the value of the integral $\int\limits_0^1 {f\left( x \right)g\left( x \right)dx,} $ is

A

$e + {{{e^2}} \over 2} + {5 \over 2}$

B

$e - {{{e^2}} \over 2} - {5 \over 2}$

C

$e + {{{e^2}} \over 2} - {3 \over 2}$

D

$e - {{{e^2}} \over 2} - {3 \over 2}$

Question 242: Palette 242/251

JEE Mains 2003 MCQ

The value of the integral $I = \int\limits_0^1 {x{{\left( {1 - x} \right)}^n}dx} $ is

A

${1 \over {n + 1}} + {1 \over {n + 2}}$

B

${1 \over {n + 1}}$

C

${1 \over {n + 2}}$

D

${1 \over {n + 1}} - {1 \over {n + 2}}$

Question 243: Palette 243/251

JEE Mains 2003 MCQ

$\mathop {\lim }\limits_{n \to \infty } {{1 + {2^4} + {3^4} + .... + {n^4}} \over {{n^5}}}$ - $\mathop {\lim }\limits_{n \to \infty } {{1 + {2^3} + {3^3} + .... + {n^3}} \over {{n^5}}}$

A

${1 \over 5}$

B

${1 \over 30}$

C

zero

D

${1 \over 4}$

Question 244: Palette 244/251

JEE Mains 2003 MCQ

If $f\left( {a + b - x} \right) = f\left( x \right)$ then $\int\limits_a^b {xf\left( x \right)dx} $ is equal to

A

${{a + b} \over 2}\int\limits_a^b {f\left( {a + b + x} \right)dx} $

B

${{a + b} \over 2}\int\limits_a^b {f\left( {b - x} \right)dx} $

C

${{a + b} \over 2}\int\limits_a^b {f\left( x \right)dx} $

D

$\,{{b - a} \over 2}\int\limits_a^b {f\left( x \right)dx} $

Question 245: Palette 245/251

JEE Mains 2003 MCQ

The value of $\mathop {\lim }\limits_{x \to 0} {{\int\limits_0^{{x^2}} {{{\sec }^2}tdt} } \over xsinx}$ is

A

0

B

3

C

2

D

1

Question 246: Palette 246/251

JEE Mains 2002 MCQ

If $y=f(x)$ makes +$ve$ intercept of $2$ and $0$ unit on $x$ and $y$ axes and encloses an area of $3/4$ square unit with the axes then $\int\limits_0^2 {xf'\left( x \right)dx} $ is

A

$3/2$

B

$1$

C

$5/4$

D

$-3/4$

Question 247: Palette 247/251

JEE Mains 2002 MCQ

$\int\limits_0^2 {\left[ {{x^2}} \right]dx} $ is

A

$2 - \sqrt 2 $

B

$2 + \sqrt 2 $

C

$\,\sqrt 2 - 1$

D

$ - \sqrt 2 - \sqrt 3 + 5$

Question 248: Palette 248/251

JEE Mains 2002 MCQ

${I_n} = \int\limits_0^{\pi /4} {{{\tan }^n}x\,dx} $ then $\,\mathop {\lim }\limits_{n \to \infty } \,n\left[ {{I_n} + {I_{n + 2}}} \right]$ equals

A

${1 \over 2}$

B

$1$

C

$\infty $

D

zero

Question 249: Palette 249/251

JEE Mains 2002 MCQ

$\int\limits_0^{10\pi } {\left| {\sin x} \right|dx} $ is

A

$20$

B

$8$

C

$10$

D

$18$

Question 250: Palette 250/251

JEE Mains 2002 MCQ

$\int_{ - \pi }^\pi {{{2x\left( {1 + \sin x} \right)} \over {1 + {{\cos }^2}x}}} dx$ is

A

${{{\pi ^2}} \over 4}$

B

${{\pi ^2}}$

C

zero

D

${\pi \over 2}$

Question 251: Palette 251/251

JEE Mains 2002 MCQ

$\mathop {\lim }\limits_{n \to \infty } {{{1^p} + {2^p} + {3^p} + ..... + {n^p}} \over {{n^{p + 1}}}}$ is

A

${1 \over {p + 1}}$

B

${1 \over {1 - p}}$

C

${1 \over p} - {1 \over {p - 1}}$

D

${1 \over {p + 2}}$