Question

If $[·]$ represents the greatest integer function, then the value of $\left|{\int }_0^{\sqrt{\frac{\pi }{2}}}\left[\left[x^2\right]-\cos x\right]dx\right|$ is ___________.

JEE Main 2021 (17 Mar Shift 1)

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Question

Let $f\left(x\right)=\min \left\{\left[x-1\right],\left[x-2\right],\dots ,\left[x-10\right]\right\}$ where $\left[t\right]$ denotes the greatest integer $\le t$. Then ${\int }_0^{10}f\left(x\right)dx+{\int }_0^{10}{\left(f\left(x\right)\right)}^{2}dx+{\int }_0^{10}\left|f\left(x\right)\right|dx$ is equal _______. to

JEE Main 2022 (27 Jul Shift 2)

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Question

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

JEE Main 2020 (04 Sep Shift 2)

Options

• A: $\frac{7}{18}$
• B: $-\frac{1}{9}$
• C: $-\frac{1}{18}$
• D: $\frac{9}{2}$

Question

The value of ${\int }_0^{\pi }\frac{{\mathrm{e}}^{\cos x}\sin x}{\left(1+{\cos }^{2}x\right)\left({\mathrm{e}}^{\cos x}+{\mathrm{e}}^{-\cos x}\right)}\mathrm{d}x$ is equal to

JEE Main 2022 (25 Jun Shift 1)

Options

• A: $\frac{{\pi }^2}{4}$
• B: $\frac{\pi }{4}$
• C: $\frac{\pi }{6}$
• D: $\frac{{\pi }^2}{2}$

Question

If $\left[x\right]$ is the greatest integer $\le x,$ then ${\pi }^2{\int }_0^2\left(\sin \frac{\pi x}{2}\right){\left(x-\left[x\right]\right)}^{\left[x\right]}\mathrm{d}x$ is equal to :

JEE Main 2021 (31 Aug Shift 2)

Options

• A: $2(\pi +1)$
• B: $4(\pi -1)$
• C: $2(\pi -1)$
• D: $4(\pi +1)$

Question

Let $\left[t\right]$ denote the greatest integer less than or equal to $t$. Then, the value of the integral ${\int }_0^1\left[-8x^2+6x-1\right]dx$ is equal to

JEE Main 2022 (28 Jun Shift 1)

Options

• A: $-1$
• B: $-\frac{5}{4}$
• C: $\frac{\sqrt{17}-13}{8}$
• D: $\frac{\sqrt{17}-16}{8}$

Question

If the integral ${\int }_0^{10}\frac{\left[\sin 2\pi x\right]}{{\mathrm{e}}^{x-\left[x\right]}}dx=\alpha e^{-1}+\beta e^{-\frac{1}{2}}+\gamma ,$ where $\alpha , \beta , \gamma$ are integers and $\left[x\right]$ denotes the greatest integer less than or equal to $x,$ then the value of $\alpha +\beta +\gamma$ is equal to:

JEE Main 2021 (17 Mar Shift 2)

Options

• A: $0$
• B: $20$
• C: $25$
• D: $10$

Question

Let $f$ be a real valued continuous function on $\left[0, 1\right]$ and $f\left(x\right)=x+{\int }_0^1\left(x-t\right)f\left(t\right)dt$. Then which of the following points $\left(x, y\right)$ lies on the curve $y=f\left(x\right)$?

JEE Main 2022 (29 Jun Shift 2)

Options

• A: $\left(2, 4\right)$
• B: $\left(1, 2\right)$
• C: $\left(4, 17\right)$
• D: $\left(6, 8\right)$

Question

The integral $16{\int }_1^2\frac{dx}{x^3{\left(x^2+2\right)}^2}$ is equal to

JEE Main 2023 (25 Jan Shift 2)

Options

• A: $\frac{11}{6}+{\log }_{e}4$
• B: $\frac{11}{12}+{\log }_{\mathrm{e}}4$
• C: $\frac{11}{12}-{\log }_{e}4$
• D: $\frac{11}{6}-{\log }_{e}4$

Question

Let $f\left(x\right)=\max \left\{\left|x+1\right|,\left|x+2\right|,\dots ,\left|x+5\right|\right\}$. Then ${\int }_{-6}^{0}f\left(x\right)dx$ is equal to ______.

JEE Main 2022 (26 Jun Shift 1)

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Question

Let $f\left(x\right)$ and $g\left(x\right)$ be two functions satisfying $f\left(x^2\right)+g(4-x)=4x^3$ and $g\left(4-x\right)+g\left(x\right)=0,$ then the value of ${\int }_{-4}^{4}f\left(x^2\right)dx$ is

JEE Main 2021 (18 Mar Shift 1)

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Question

The integral ${\int }_0^1\frac{1}{7^{\left[\frac{1}{x}\right]}}dx$, where $\left[·\right]$ denotes the greatest integer function, is equal to

JEE Main 2022 (27 Jun Shift 2)

Options

• A: $1-6\ln \left(\frac{6}{7}\right)$
• B: $1+6\ln \left(\frac{6}{7}\right)$
• C: $1-7\ln \left(\frac{6}{7}\right)$
• D: $1+7\ln \left(\frac{6}{7}\right)$

Question

If $\left[t\right]$ denotes the greatest integer $\le \mathrm{t}$, then the value of ${\int }_0^1\left[2x-\left|3x^2-5x+2\right|+1\right]dx$ is

JEE Main 2022 (29 Jul Shift 2)

Options

• 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

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

JEE Main 2022 (25 Jul Shift 2)

Options

• A: $\frac{52\left(1-e\right)}{e}$
• B: $\frac{52}{e}$
• C: $\frac{52\left(2+e\right)}{e}$
• D: $\frac{104}{e}$

Question

The minimum value of the twice differentiable function $f\left(x\right)={\int }_0^x{e}^{x-t}{f}^{\text{'}}\left(t\right)dt-\left(x^2-x+1\right)e^x,x\in R$, is

JEE Main 2022 (28 Jul Shift 1)

Options

• A: $-\frac{2}{\sqrt{e}}$
• B: $-2\sqrt{e}$
• C: $-\sqrt{e}$
• D: $\frac{2}{\sqrt{e}}$

Question

If $I=\underset{1}{\overset{2}{\int }}\frac{dx}{\sqrt{2x^3-9x^2+12x+4}}$, then

JEE Main 2020 (08 Jan Shift 2)

Options

• A: $\frac{1}{8}<I^2<\frac{1}{4}$
• B: $\frac{1}{9}<I^2<\frac{1}{8}$
• C: $\frac{1}{16}<I^2<\frac{1}{9}$
• D: $\frac{1}{6}<I^2<\frac{1}{2}$

Question

If [ $t$ denotes the greatest integer $\le 1$, then the value of $\frac{3\left(e-1\right)}{e}{\int }_1^2{x}^2{e}^{\left[x\right]+\left[x^3\right]}dx$ is :

JEE Main 2023 (30 Jan Shift 1)

Options

• A: $e^9-e$
• B: $e^8-e$
• C: $e^7-1$
• D: $e^8-1$

Question

The value of the integral ${\int }_{-{\log }_{e}2}^{{\log }_{e}2}{e}^x\left({\log }_e\left(e^x+\sqrt{1+e^{2x}}\right)\right)dx$ is equal to

JEE Main 2023 (11 Apr Shift 1)

Options

• A: ${\log }_e\left(\frac{\sqrt{2}(2+\sqrt{5}{)}^2}{\sqrt{1+\sqrt{5}}}\right)-\frac{\sqrt{5}}{2}$
• B: ${\log }_e\left(\frac{(2+\sqrt{5}{)}^2}{\sqrt{1+\sqrt{5}}}\right)+\frac{\sqrt{5}}{2}$
• 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

Let the function $f:\left[0,2\right]\to \mathrm{ℝ}$ be defined as $f\left(x\right)=\left\{\begin{array}{cc}{e}^{\min \left\{x^2,x-\left[x\right]\right\}},& x\in [0,1)\\ e^{\left[x-{\log }_{e}x\right]},& x\in \left[1,2\right]\end{array}\right.$, where $\left[t\right]$ denotes the greatest integer less than or equal to $t$. Then the value of the integral ${\int }_0^{2}xf\left(x\right)dx$ is

JEE Main 2023 (11 Apr Shift 2)

Options

• A: $1+\frac{3e}{2}$
• B: $\left(e-1\right)\left(e^2+\frac{1}{2}\right)$
• C: $2e-1$
• D: $2e-\frac{1}{2}$
   

Question

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

JEE Main 2024 (04 Apr Shift 1)

Options

• A: 1
• B: 6
• C: 4
• D: 2

Question

Let $\alpha \in \left(0,1\right)$ and $\beta ={\log }_e\left(1-\alpha \right)$. Let $P_n\left(x\right)=x+\frac{x^2}{2}+\frac{x^3}{3}+\dots .+\frac{x^n}{n},x\in \left(0,1\right)$. Then the integral ${\int }_0^{\alpha }\frac{t^{50}}{1-t}dt$ is equal to

JEE Main 2023 (31 Jan Shift 1)

Options

• A: $\beta -P_{50}\left(\alpha \right)$
• B: $-\left(\beta +P_{50}\left(\alpha \right)\right)$
• C: $P_{50}\left(\alpha \right)-\beta$
• D: $\beta +P_{50}\left(\alpha \right)$

Question

The value of ${\int }_{\frac{\pi }{3}}^{\frac{\pi }{2}}\frac{\left(2+3\sin x\right)}{\sin x\left(1+\cos x\right)}dx$ is equal to

JEE Main 2023 (31 Jan Shift 1)

Options

• A: $\frac{7}{2}-\sqrt{3}-{\log }_e\sqrt{3}$
• B: $-2+3\sqrt{3}+{\log }_e\sqrt{3}$
• C: $\frac{10}{3}-\sqrt{3}+{\log }_e\sqrt{3}$
• D: $\frac{10}{3}-\sqrt{3}-{\log }_e\sqrt{3}$

Question

The minimum value of the function $f\left(x\right)={\int }_0^2{e}^{\left|x-t\right|}dt$ is

JEE Main 2023 (25 Jan Shift 1)

Options

• A: $2\left(e-1\right)$
• B: $2e-1$
• C: $2$
• D: $e\left(e-1\right)$

Question

Let $\beta(\mathrm{m}, \mathrm{n})=\int_0^1 x^{\mathrm{m}-1}(1-x)^{\mathrm{n}-1} \mathrm{~d} x, \mathrm{~m}, \mathrm{n}>0$. If $\int_0^1\left(1-x^{10}\right)^{20} \mathrm{~d} x=\mathrm{a} \times \beta(\mathrm{b}, \mathrm{c})$, then $100(\mathrm{a}+\mathrm{b}+\mathrm{c})$ equals____

JEE Main 2024 (05 Apr Shift 2)

Options

• A: 1021
• B: 2120
• C: 2012
• D: 1120

Question

Let $[t]$ denote the largest integer less than or equal to $t$. If $\int_0^3\left(\left[x^2\right]+\left[\frac{x^2}{2}\right]\right) \mathrm{d} x=\mathrm{a}+\mathrm{b} \sqrt{2}-\sqrt{3}-\sqrt{5}+\mathrm{c} \sqrt{6}-\sqrt{7}$, where $\mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathbf{Z}$, then $\mathrm{a}+\mathrm{b}+\mathrm{c}$ is equal to_______

JEE Main 2024 (06 Apr Shift 2)

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