JEE MAIN Definite Integration Previous Year Question Paper and Solution with PDF

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Previous Year Questions

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Previous Year Questions
JEE MAIN

JEE MAIN Previous Year Questions (PYQs)

JEE MAIN Definite Integration PYQ

Qus : 1 JEE MAIN PYQ

If I₁ = $\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{100}}} dx$ andI₂ = $\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{101}}} dx$ such that I₂ = $\alpha $I₁then $\alpha $ equals to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 6 September 2020 (Morning) PYQ

Solution

Qus : 2 JEE MAIN PYQ

Let $a \in \left(0, \dfrac{\pi}{2}\right)$ be fixed. If $\displaystyle \int \dfrac{\tan x + \tan a}{\tan x - \tan a} , dx = A(x)\cos 2a + B(x)\sin 2a + C,$ where $C$ is a constant of integration, then the functions $A(x)$ and $B(x)$ are respectively:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 April Evening Shift) PYQ

Solution

Qus : 3 JEE MAIN PYQ

If $I=\displaystyle\int_{0}^{\pi/2}\frac{\sin^{3/2}x}{\sin^{3/2}x+\cos^{3/2}x}\,dx$, then $\displaystyle\int_{0}^{2I}\frac{x\sin x\cos x}{\sin^{4}x+\cos^{4}x}\,dx$ equals:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (23 January Evening Shift) PYQ

Solution

Qus : 4 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (27 July Morning Shift) PYQ

Solution

Qus : 5 JEE MAIN PYQ

Let $f(x)=\int_{0}^{x}\left(t+\sin(1-e^{t})\right)dt,\;x\in\mathbb{R}$. Then, $\displaystyle\lim_{x\to0}\dfrac{f(x)}{x^{3}}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (4 April Evening Shift) PYQ

Solution

Qus : 6 JEE MAIN PYQ

The value of $\displaystyle \int_{-\pi/2}^{\pi/2} \dfrac{dx}{[x] + [\sin x] + 4}$, where $[t]$ denotes the greatest integer less than or equal to $t$, is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 January Evening Shift) PYQ

Solution

Qus : 7 JEE MAIN PYQ

If the area of the bounded region $R = \left\{ {(x,y):\max \{ 0,{{\log }_e}x\} \le y \le {2^x},{1 \over 2} \le x \le 2} \right\}$ is , $\alpha {({\log _e}2)^{ - 1}} + \beta ({\log _e}2) + \gamma $, then the value of ${(\alpha + \beta - 2\lambda )^2}$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (27 July Morning Shift) PYQ

Solution

Qus : 8 JEE MAIN PYQ

$\int_{0}^{20\pi} (|\sin x| + |\cos x|)^{2} \, dx$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (26 July Evening Shift) PYQ

Solution

Qus : 9 JEE MAIN PYQ

The value of the integral $\displaystyle \int_{-\pi/4}^{\pi/4}\frac{x+\pi/4}{\,2-\cos 2x\,}\,dx$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (1 February Evening Shift) PYQ

Solution

Qus : 10 JEE MAIN PYQ

$ \text{The integral } \int \dfrac{\left(1-\tfrac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{1+\tfrac{2}{\sqrt{3}}\sin 2x},dx \text{ is equal to :}$

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (26 July Evening Shift) PYQ

Solution

Qus : 11 JEE MAIN PYQ

The area (in sq. units) of the region enclosedby the curves y = x² – 1 and y = 1 – x² is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 6 September 2020 (Evening) PYQ

Solution

Qus : 12 JEE MAIN PYQ

A value of $\alpha$ such that $\displaystyle \int_{\alpha}^{\alpha+1} \dfrac{dx}{(x+\alpha)(x+\alpha+1)} = \log_e\left(\dfrac{9}{8}\right)$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 April Evening Shift) PYQ

Solution

Qus : 13 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 6 September 2020 (Evening) PYQ

Solution

Qus : 14 JEE MAIN PYQ

If $\displaystyle \int_{-1}^{1}\frac{\cos(\alpha x)}{1+3x^{2}},dx=\frac{2}{\pi}$, then a value of $\alpha$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (4 April Evening Shift) PYQ

Solution

Qus : 15 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (4 April Evening Shift) PYQ

Solution

Qus : 16 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (27 July Morning Shift) PYQ

Solution

Qus : 17 JEE MAIN PYQ

Statement-1 : The value of the integral $\displaystyle \int_{\pi/6}^{\pi/3}\frac{dx}{1+\sqrt{\tan x}}$ is equal to $\pi/6$ Statement-2 : $\displaystyle \int_{a}^{b}f(x)\,dx=\int_{a}^{b}f(a+b-x)\,dx$.

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2013 (Offline) PYQ

Solution

Qus : 18 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (24 February Evening Shift) PYQ

Solution

Qus : 19 JEE MAIN PYQ

Let $5f(x)+4f\!\left(\dfrac{1}{x}\right)=\dfrac{1}{x}+3,\; x>0.$ Then $18\displaystyle\int_{1}^{2} f(x)\,dx$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (6 April Morning Shift) PYQ

Solution

Qus : 20 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (24 February Evening Shift) PYQ

Solution

Qus : 21 JEE MAIN PYQ

The integral $\displaystyle \int_{0}^{\pi/4}\frac{136\sin x}{3\sin x+5\cos x},dx$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (5 April Morning Shift) PYQ

Solution

Qus : 22 JEE MAIN PYQ

If $\int_0^1 \frac{1}{\sqrt{3+x}+\sqrt{1+x}}\,dx = a + b\sqrt{2} + c\sqrt{3}$, where $a, b, c$ are rational numbers, then $2a + 3b - 4c$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (27 January Morning Shift) PYQ

Solution

Qus : 23 JEE MAIN PYQ

If the system of linear equations

$2x+2y+3z=a$

$3x-y+5z=b$

$x-3y+2z=c$

where $a,b,c$ are non-zero real numbers, has more than one solution, then :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (11 January Morning Shift) PYQ

Solution

Qus : 24 JEE MAIN PYQ

The value of the integral $\displaystyle \int_{-\pi/4}^{\pi/4} \sin^{4}x \left(1 + \log\left(\dfrac{2 + \sin 2x}{2 - \sin 2x}\right)\right),dx$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (15 April Morning Shift) PYQ

Solution

Qus : 25 JEE MAIN PYQ

The area of the region : $R = \{ (x,y):5{x^2} \le y \le 2{x^2} + 9\} $

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (24 February Evening Shift) PYQ

Solution

Qus : 26 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (24 January Morning Shift) PYQ

Solution

Qus : 27 JEE MAIN PYQ

The integral $\displaystyle \int_{0}^{\pi}\frac{(x+3)\sin x}{1+3\cos^{2}x}dx$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (7 April Morning Shift) PYQ

Solution

Qus : 28 JEE MAIN PYQ

The area of the region : $R = \{ (x,y):5{x^2} \le y \le 2{x^2} + 9\} $

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (24 February Evening Shift) PYQ

Solution

Qus : 29 JEE MAIN PYQ

The value of $\int_{-\pi}^{\pi}\dfrac{2y(1+\sin y)}{1+\cos^{2}y},dy$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (5 April Morning Shift) PYQ

Solution

Qus : 30 JEE MAIN PYQ

The value of $\displaystyle \int_{0}^{2}!\left(,|2x^{2}-3x|+\big[x-\tfrac{1}{2}\big]\right),dx$, where $[\cdot]$ is the greatest integer function, is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (27 July Evening Shift) PYQ

Solution

Qus : 31 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (6 April Evening Shift) PYQ

Solution

Qus : 32 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 February Morning Shift) PYQ

Solution

Qus : 33 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (26 August Morning Shift) PYQ

Solution

Qus : 34 JEE MAIN PYQ

The area (in sq. units) of the region, given by the set $\{ (x,y) \in R \times R|x \ge 0,2{x^2} \le y \le 4 - 2x\} $

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 February Morning Shift) PYQ

Solution

Qus : 35 JEE MAIN PYQ

The integral $\displaystyle \int_{\pi/6}^{\pi/4}\frac{dx}{\sin 2x\,(\tan^{5}x+\cot^{5}x)}$ equals :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (11 January Evening Shift) PYQ

Solution

Qus : 36 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (26 August Evening Shift) PYQ

Solution

Qus : 37 JEE MAIN PYQ

Let $\beta(m,n)=\displaystyle\int_{0}^{1}x^{m-1}(1-x)^{,n-1},dx,; m,n>0$. If $\displaystyle\int_{0}^{1}(1-x^{10})^{20},dx=a\times \beta(b,c)$, then $100(a+b+c)$ equals:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (5 April Evening Shift) PYQ

Solution

Qus : 38 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (24 June Evening Shift) PYQ

Solution

Qus : 39 JEE MAIN PYQ

If $I_1=\displaystyle\int_{0}^{1} e^{-x}\cos^{2}x,dx$; $I_2=\displaystyle\int_{0}^{1} e^{-x^{2}}\cos^{2}x,dx$ and $I_3=\displaystyle\int_{0}^{1} e^{-x^{3}},dx$; then

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (15 April Evening Shift) PYQ

Solution

Qus : 40 JEE MAIN PYQ

$ \text{Let } \alpha,\beta,\gamma \text{ be the three roots of } x^{3}+bx+c=0. \text{ If } \beta\gamma=1=-\alpha,\ \text{then } b^{3}+2c^{3}-3\alpha^{3}-6\beta^{3}-8\gamma^{3} \text{ is equal to:} $

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (8 April Morning Shift) PYQ

Solution

Qus : 41 JEE MAIN PYQ

The value of the integral $\displaystyle\int_{\pi/4}^{3\pi/4}\frac{x}{1+\sin x},dx$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (15 April Evening Shift) PYQ

Solution

Qus : 42 JEE MAIN PYQ

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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (25 June Morning Shift) PYQ

Solution

Qus : 43 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 February Evening Shift) PYQ

Solution

Qus : 44 JEE MAIN PYQ

Let $I_n(x) = \int_0^x \dfrac{1}{(t^2+5)^n} \, dt, \; n = 1, 2, 3, \dots$ Then :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (28 July Evening Shift) PYQ

Solution

Qus : 45 JEE MAIN PYQ

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 $\displaystyle \int_{0}^{a} f(x)\,g(x)\,dx$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 January Morning Shift) PYQ

Solution

Qus : 46 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (27 August Morning Shift) PYQ

Solution

Qus : 47 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (27 January Evening Shift) PYQ

Solution

Qus : 48 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (26 February Morning Shift) PYQ

Solution

Qus : 49 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (26 February Morning Shift) PYQ

Solution

Qus : 50 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (28 January Morning Shift) PYQ

Solution

Qus : 51 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (Offline) PYQ

Solution

Qus : 52 JEE MAIN PYQ

The integral $\int_{0}^{\tfrac{\pi}{2}} \dfrac{1}{3 + 2 \sin x + \cos x} , dx$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (29 July Morning Shift) PYQ

Solution

Qus : 53 JEE MAIN PYQ

The function $I(x)=\int e^{\sin^{2}x},(\cos x\sin 2x-\sin x),dx$ with $I(0)=1$. Then $I!\left(\dfrac{\pi}{3}\right)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (10 April Morning Shift) PYQ

Solution

Qus : 54 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (29 July Morning Shift) PYQ

Solution

Qus : 55 JEE MAIN PYQ

Let, for some function $y=f(x)$, $\displaystyle \int_{0}^{x} t,f(t),dt = x^{2}f(x)$ for $x>0$ and $f(2)=3$. Then $f(6)$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (28 January Morning Shift) PYQ

Solution

Qus : 56 JEE MAIN PYQ

The value of the integral $\displaystyle \int_{0}^{1} \frac{\sqrt{x}\,dx}{(1+x)(1+3x)(3+x)}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (27 August Evening Shift) PYQ

Solution

Qus : 57 JEE MAIN PYQ

For $x\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$, if $y(x)=\displaystyle\int \frac{\csc x+\sin x}{\csc x\sec x+\tan x\sin^2 x}\,dx$, and $\displaystyle\lim_{x\to \left(\frac{\pi}{2}\right)} y(x)=0$, then $y\!\left(\dfrac{\pi}{4}\right)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (29 January Morning Shift) PYQ

Solution

Qus : 58 JEE MAIN PYQ

$\displaystyle \int_{0}^{\pi/4}\frac{\cos^{2}x,\sin^{2}x}{\big(\cos^{3}x+\sin^{3}x\big)^{2}},dx$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (6 April Morning Shift) PYQ

Solution

Qus : 59 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (25 June Evening Shift) PYQ

Solution

Qus : 60 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 January Evening Shift) PYQ

Solution

Qus : 61 JEE MAIN PYQ

If the value of the integral $\displaystyle \int_{-\pi/2}^{\pi/2} \left( \dfrac{x^{2}\cos x}{1+x^{2}} +\dfrac{1+\sin^{2}x}{1+e^{\sin(2\tan^{-1}x)}} \right)\,dx = \dfrac{\pi}{4}\,(\pi+a)-2,$ then the value of $a$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (29 January Morning Shift) PYQ

Solution

Qus : 62 JEE MAIN PYQ

Let $f:\mathbb{R}\to\mathbb{R}$ be a twice differentiable function such that $f(2)=1$. If $F(x)=x f(x)$ for all $x\in\mathbb{R}$, $\displaystyle\int_{0}^{2} x F''(x),dx=6$ and $\displaystyle\int_{0}^{2} x^{2} F''(x),dx=40$, then $F'(2)+\displaystyle\int_{0}^{2} F(x),dx$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (28 January Evening Shift) PYQ

Solution

Qus : 63 JEE MAIN PYQ

Let A₁ be the area of the region bounded by the curves y = sinx, y = cosx and y-axis in the first quadrant. Also, let A₂ be the area of the region bounded by the curves y = sinx, y = cosx, x-axis and x = ${\pi \over 2}$ in the first quadrant. Then,

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (26 February Evening Shift) PYQ

Solution

Qus : 64 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (29 July Evening Shift) PYQ

Solution

Qus : 65 JEE MAIN PYQ

For $ I(x) = \int \frac{\sec^{2}x - 2022}{\sin^{2022}x} \, dx, $ if $ I\!\left(\frac{\pi}{4}\right) = 2^{1011}, $ then

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (29 July Evening Shift) PYQ

Solution

Qus : 66 JEE MAIN PYQ

Let $f$ be a continuous function satisfying $\displaystyle \int_{0}^{t^2} \big(f(x) + x^2\big)\,dx = \dfrac{4}{3}t^3, \; \forall t > 0.$ Then $f\!\left(\dfrac{\pi^2}{4}\right)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (10 April Evening Shift) PYQ

Solution

Qus : 67 JEE MAIN PYQ

The area bounded by the curve y = |x² $-$ 9| and the line y = 3 is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (26 June Morning Shift) PYQ

Solution

Qus : 68 JEE MAIN PYQ

If $\sin\!\left(\dfrac{y}{x}\right)=\log_e|x|+\dfrac{\alpha}{2}$ is the solution of the differential equation $x\cos\!\left(\dfrac{y}{x}\right)\dfrac{dy}{dx}=y\cos\!\left(\dfrac{y}{x}\right)+x$ and $y(1)=\dfrac{\pi}{3}$, then $\alpha^{2}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (29 January Evening Shift) PYQ

Solution

Qus : 69 JEE MAIN PYQ

If $f(x)=\displaystyle\int \frac{1}{x^{1/4}\left(1+x^{1/4}\right)},dx,; f(0)=-6$, then $f(1)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (28 January Evening Shift) PYQ

Solution

Qus : 70 JEE MAIN PYQ

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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (11 April Morning Shift) PYQ

Solution

Qus : 71 JEE MAIN PYQ

If $\int {{1 \over x}\sqrt {{{1 - x} \over {1 + x}}} dx = g(x) + c} $, $g(1) = 0$, then $g\left( {{1 \over 2}} \right)$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (26 June Evening Shift) PYQ

Solution

Qus : 72 JEE MAIN PYQ

If $\displaystyle \int \frac{1}{a^{2}\sin^{2}x+b^{2}\cos^{2}x},dx=\frac{1}{12}\tan^{-1}(3\tan x)+\text{constant}$, then the maximum value of $a\sin x+b\cos x$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (6 April Evening Shift) PYQ

Solution

Qus : 73 JEE MAIN PYQ

The value of $k\in\mathbb{N}$ for which the integral $I_n=\displaystyle\int_{0}^{1}(1-x^{k})^{n},dx,\ n\in\mathbb{N}$, satisfies $147I_{20}=148I_{21}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (8 April Morning Shift) PYQ

Solution

Qus : 74 JEE MAIN PYQ

If $f(x)=\dfrac{2-x\cos x}{2+x\cos x}$ and $g(x)=\log_e x,\ (x>0)$, then the value of the integral $\displaystyle \int_{-\pi/4}^{\pi/4} g\big(f(x)\big),dx$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (8 April Morning Shift) PYQ

Solution

Qus : 75 JEE MAIN PYQ

The integral $\displaystyle \int_{-1}^{\tfrac{3}{2}} \left( |\pi^2 x \sin(\pi x)| \right) dx$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (8 April Evening Shift) PYQ

Solution

Qus : 76 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (16 March Evening Shift) PYQ

Solution

Qus : 77 JEE MAIN PYQ

Let $f(x)$ be a positive function and $I_{1}=\int_{-\tfrac{1}{2}}^{1} 2x,f\left(2x(1-2x)\right),dx$ and $I_{2}=\int_{-1}^{2} f\left(x(1-x)\right),dx$. Then the value of $\dfrac{I_{2}}{I_{1}}$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (8 April Evening Shift) PYQ

Solution

Qus : 78 JEE MAIN PYQ

The integral $\displaystyle \int_{\pi/4}^{3\pi/4}\dfrac{dx}{1+\cos x}$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (Offline) PYQ

Solution

Qus : 79 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (27 June Morning Shift) PYQ

Solution

Qus : 80 JEE MAIN PYQ

If $f:\mathbb{R}\to\mathbb{R}$ is a continuous function satisfying \[ \int_{0}^{\pi/2} f(\sin 2x)\,\sin x\,dx \;+\; \alpha \int_{0}^{\pi/4} f(\cos 2x)\,\cos x\,dx \;=\; 0, \] then the value of $\alpha$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (11 April Evening Shift) PYQ

Solution

Qus : 81 JEE MAIN PYQ

$\displaystyle \int_{\tfrac{3\sqrt{2}}{4}}^{\tfrac{3\sqrt{3}}{4}} \dfrac{48}{\sqrt{9-4x^{2}}}\,dx$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (24 January Evening Shift) PYQ

Solution

Qus : 82 JEE MAIN PYQ

Let $I(x)=\displaystyle\int \frac{6}{\sin^{2}x,(1-\cot x)^{2}},dx$. If $I(0)=3$, then $I!\left(\tfrac{\pi}{12}\right)$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (8 April Morning Shift) PYQ

Solution

Qus : 83 JEE MAIN PYQ

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} $

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (17 March Morning Shift) PYQ

Solution

Qus : 84 JEE MAIN PYQ

Let the function $f:[0,2]\to\mathbb{R}$ be defined as \[ f(x)= \begin{cases} e^{\min\{x^2,\; x-[x]\}}, & x\in[0,1),\\[4pt] e^{[\,x-\log_e x\,]}, & x\in[1,2], \end{cases} \] where $[t]$ denotes the greatest integer less than or equal to $t$. Then the value of the integral $\displaystyle \int_{0}^{2} x f(x)\,dx$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (11 April Evening Shift) PYQ

Solution

Qus : 85 JEE MAIN PYQ

Let $f:\mathbb{R}\to\mathbb{R}$ be defined by $f(x)=\dfrac{x}{(1+2x^{4})^{1/4}}$, and $g(x)=f(f(f(f(x))))$. Then $18\displaystyle\int_{0}^{\sqrt{2\sqrt{5}}} x^{2}g(x),dx$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (30 January Morning Shift) PYQ

Solution

Qus : 86 JEE MAIN PYQ

For $x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, if $y(x)=\int \frac{\operatorname{cosec} x+\sin x}{\operatorname{cosec} x \sec x+\tan x \sin ^2 x} d x$, and $\lim _\limits{x \rightarrow\left(\frac{\pi}{2}\right)^{-}} y(x)=0$ then $y\left(\frac{\pi}{4}\right)$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (29 January Morning Shift) PYQ

Solution

Qus : 87 JEE MAIN PYQ

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 $\dfrac{\pi}{6}$ and $\dfrac{\pi}{4}$ respectively with the positive $x$-axis. If $27\displaystyle\int_{1}^{3}\big((f'(t))^{2}+1\big)f'''(t),dt=\alpha+\beta\sqrt{3}$, where $\alpha,\beta$ are integers, then the value of $\alpha+\beta$ equals:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (30 January Evening Shift) PYQ

Solution

Qus : 88 JEE MAIN PYQ

The area of the region in the first quadrant inside the circle $x^{2}+y^{2}=8$ and outside the parabola $y^{2}=2x$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (8 April Evening Shift) PYQ

Solution

Qus : 89 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (27 June Evening Shift) PYQ

Solution

Qus : 90 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (27 June Evening Shift) PYQ

Solution

Qus : 91 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 3 September 2020 (Morning) PYQ

Solution

Qus : 92 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (8 April Evening Shift) PYQ

Solution

Qus : 93 JEE MAIN PYQ

If $\displaystyle \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\dfrac{x^2\cos x}{1+x^2}+\dfrac{1+\sin^2 x}{1+e^{\sin 2x}}\right)dx = \dfrac{\pi}{4}(\pi+a)-2$, then the value of $a$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (29 January Morning Shift) PYQ

Solution

Qus : 94 JEE MAIN PYQ

$a$ 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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (30 January Evening Shift) PYQ

Solution

Qus : 95 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (8 April Morning Shift) PYQ

Solution

Qus : 96 JEE MAIN PYQ

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 $|a+b+c|$

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (30 January Evening Shift) PYQ

Solution

Qus : 97 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (17 March Evening Shift) PYQ

Solution

Qus : 98 JEE MAIN PYQ

$ \text{If } \displaystyle \int_{0}^{10}\frac{[\sin 2\pi x]}{e^{,x-[x]}},dx ;=; \alpha e^{-1}+\beta e^{-1/2}+\gamma,\ \text{ where } \alpha,\beta,\gamma \text{ are integers and } [x] \text{ is the greatest integer } \le x,\ \text{then the value of } \alpha+\beta+\gamma \text{ is:} $

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (17 March Evening Shift) PYQ

Solution

Qus : 99 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (28 June Morning Shift) PYQ

Solution

Qus : 100 JEE MAIN PYQ

Evaluate the integral $ \displaystyle \int_{0}^{\infty}\frac{6}{e^{3x}+6e^{2x}+11e^{x}+6},dx $ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (13 April Morning Shift) PYQ

Solution

Qus : 101 JEE MAIN PYQ

The value of $\displaystyle \int_{0}^{\pi/2}\frac{\sin^{3}x}{\sin x+\cos x},dx$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 April Morning Shift) PYQ

Solution

Qus : 102 JEE MAIN PYQ

$ \text{The integral } 16 \int_{1}^{2} \frac{dx}{x^{3}(x^{2}+2)^{2}} \text{ is equal to:}$

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (25 January Evening Shift) PYQ

Solution

Qus : 103 JEE MAIN PYQ

The integral $\int {{{(2x - 1)\cos \sqrt {{{(2x - 1)}^2} + 5} } \over {\sqrt {4{x^2} - 4x + 6} }}} dx$ is equal to (where c is a constant of integration)

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (18 March Morning Shift) PYQ

Solution

Qus : 104 JEE MAIN PYQ

If $\displaystyle \int_{1}^{2} \frac{dx}{(x^{2} - 2x + 4)^{\tfrac{3}{2}}} = \frac{k}{k+5}$, then $k$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (9 April Morning Shift) PYQ

Solution

Qus : 105 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (28 June Evening Shift) PYQ

Solution

Qus : 106 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (28 June Evening Shift) PYQ

Solution

Qus : 107 JEE MAIN PYQ

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} $ :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (31 August Morning Shift) PYQ

Solution

Qus : 108 JEE MAIN PYQ

Let $f(x)=x+\dfrac{a}{\pi^{2}-4}\sin x+\dfrac{b}{\pi^{2}-4}\cos x,\ x\in\mathbb{R}$ be a function which satisfies $\displaystyle f(x)=x+\int_{0}^{\pi/2}\sin(x+y)\,f(y)\,dy.$ Then $(a+b)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (29 January Morning Shift) PYQ

Solution

Qus : 109 JEE MAIN PYQ

The value of the integral $\displaystyle \int_{0}^{1} x\cot^{-1}\left(1 - x^{2} + x^{4}\right),dx$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 April Evening Shift) PYQ

Solution

Qus : 110 JEE MAIN PYQ

The value of $\dfrac{e^{-\pi/4}+\displaystyle\int_{0}^{\pi/4} e^{-x}\tan^{50}x\,dx}{\displaystyle\int_{0}^{\pi/4} e^{-x}\big(\tan^{49}x+\tan^{51}x\big)\,dx}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (13 April Evening Shift) PYQ

Solution

Qus : 111 JEE MAIN PYQ

The area bounded by the curve 4y² = x²(4 $-$ x)(x $-$ 2) is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (18 March Evening Shift) PYQ

Solution

Qus : 112 JEE MAIN PYQ

Let $[x]$ denote the greatest integer $\le x$. Consider the function $$f(x)=\max\{x^{2},\,1+[x]\}.$$ Then the value of the integral $\displaystyle \int_{0}^{2} f(x)\,dx$ i

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (29 January Morning Shift) PYQ

Solution

Qus : 113 JEE MAIN PYQ

If $\displaystyle \int_{0}^{1} \frac{1}{(5+2x-2x^2)\,(1+e^{\,2-4x})}\,dx=\frac{1}{\alpha}\log_e\!\left(\frac{\alpha+1}{\beta}\right),\ \alpha,\beta>0,$ then $\alpha^4-\beta^4$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (15 April Morning Shift) PYQ

Solution

Qus : 114 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (9 April Evening Shift) PYQ

Solution

Qus : 115 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 3 September 2020 (Evening) PYQ

Solution

Qus : 116 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (18 March Evening Shift) PYQ

Solution

Qus : 117 JEE MAIN PYQ

The value of the integral $\displaystyle \int_{-1}^{2} \log_e \big(x + \sqrt{x^2 + 1}\big),dx$ is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (9 April Evening Shift) PYQ

Solution

Qus : 118 JEE MAIN PYQ

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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (20 July Morning Shift) PYQ

Solution

Qus : 119 JEE MAIN PYQ

$\displaystyle \int e^{\sec x},\big(\sec x\tan x,f(x)+\sec x\tan x+\sec^{2}x\big),dx ;=; e^{\sec x}f(x)+C$ Then a possible choice of $f(x)$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 April Evening Shift) PYQ

Solution

Qus : 120 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (20 July Morning Shift) PYQ

Solution

Qus : 121 JEE MAIN PYQ

Let $f:\mathbb{R}\to(0,\infty)$ be a strictly increasing function such that $\displaystyle \lim_{x\to\infty}\frac{f(7x)}{f(x)}=1$. Then the value of $\displaystyle \lim_{x\to\infty}\Big[\frac{f(5x)}{f(x)}-1\Big]$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (31 January Evening Shift) PYQ

Solution

Qus : 122 JEE MAIN PYQ

$\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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (29 June Morning Shift) PYQ

Solution

Qus : 123 JEE MAIN PYQ

The value of the integral $\displaystyle \int_{1}^{2} \left(\frac{t^{4}+1}{t^{6}+1}\right) dt$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (29 January Evening Shift) PYQ

Solution

Qus : 124 JEE MAIN PYQ

Let $(a, b)$ be the point of intersection of the curve $x^2 = 2y$ and the straight line $y - 2x - 6 = 0$ in the second quadrant. Then the integral $I = \int_a^b \dfrac{9x^2}{1 + 5x^4},dx$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (2 April Evening Shift) PYQ

Solution

Qus : 125 JEE MAIN PYQ

The value of the integral $\displaystyle \int_{1/2}^{2} \frac{\tan^{-1}x}{x}\,dx$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (29 January Evening Shift) PYQ

Solution

Qus : 126 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (31 August Evening Shift) PYQ

Solution

Qus : 127 JEE MAIN PYQ

$\displaystyle 4 \int_0^1 \left(\dfrac{1}{\sqrt{3 + x^2} + \sqrt{1 + x^2}}\right) dx - 3 \log_e(\sqrt{3})$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (2 April Evening Shift) PYQ

Solution

Qus : 128 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (29 June Evening Shift) PYQ

Solution

Qus : 129 JEE MAIN PYQ

Let $f\left( x \right) = \int {{{\sqrt x } \over {{{\left( {1 + x} \right)}^2}}}dx\left( {x \ge 0} \right)} $. Then f(3) – f(1) is eqaul to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 4 September 2020 (Morning) PYQ

Solution

Qus : 130 JEE MAIN PYQ

$ \text{If [t] denotes the greatest integer } \le t, \text{ then the value of } \frac{3(e-1)}{e} \int_{1}^{2} x^2 e^{\lfloor x \rfloor + \lfloor x^3 \rfloor} dx \text{ is:} $

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (30 January Morning Shift) PYQ

Solution

Qus : 131 JEE MAIN PYQ

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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 4 September 2020 (Morning) PYQ

Solution

Qus : 132 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (1 February Morning Shift) PYQ

Solution

Qus : 133 JEE MAIN PYQ

For x² $ \ne $ n$\pi $ + 1, n $ \in $ N (the set of natural numbers), the integral

$\int {x\sqrt {{{2\sin ({x^2} - 1) - \sin 2({x^2} - 1)} \over {2\sin ({x^2} - 1) + \sin 2({x^2} - 1)}}} dx} $ is equal to : (where c is a constant of integration)

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 January Morning Shift) PYQ

Solution

Qus : 134 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (20 July Evening Shift) PYQ

Solution

Qus : 135 JEE MAIN PYQ

The value of $\displaystyle\int_{0}^{\pi}\!\lvert\cos x\rvert^{3}\,dx$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 January Morning Shift) PYQ

Solution

Qus : 136 JEE MAIN PYQ

Let the domain of the function $f(x) = \log_2 \log_4 \log_6 (3 + 4x - x^2)$ be $(a, b)$. If $\int_0^{b - a} [x^2] , dx = p - \sqrt{q - \sqrt{r}}, ; p, q, r \in \mathbb{N}, ; \gcd(p, q, r) = 1$, where $[,]$ is the greatest integer function, then $p + q + r$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (3 April Morning Shift) PYQ

Solution

Qus : 137 JEE MAIN PYQ

If $2\displaystyle\int_{0}^{1} \tan^{-1} x , dx = \displaystyle\int_{0}^{1} \cot^{-1} (1 - x + x^{2}) , dx,$ then $\displaystyle\int_{0}^{1} \tan^{-1} (1 - x + x^{2}) , dx$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (9 April Morning Shift) PYQ

Solution

Qus : 138 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (1 September Evening Shift) PYQ

Solution

Qus : 139 JEE MAIN PYQ

If $\displaystyle \int_{0}^{\pi/8}\frac{\tan\theta}{\sqrt{2k\,\sec\theta}}\;d\theta =1-\frac{1}{\sqrt{2}},\ (k>0)$, then the value of $k$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 January Evening Shift) PYQ

Solution

Qus : 140 JEE MAIN PYQ

The value of $\displaystyle \int_{0}^{2\pi}\big\lfloor \sin 2x,(1+\cos 3x)\big\rfloor,dx$, where $[\cdot]$ denotes the greatest integer function, is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 April Morning Shift) PYQ

Solution

Qus : 141 JEE MAIN PYQ

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?

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (20 July Evening Shift) PYQ

Solution

Qus : 142 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (22 January Morning Shift) PYQ

Solution

Qus : 143 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (22 July Evening Shift) PYQ

Solution

Qus : 144 JEE MAIN PYQ

If $\displaystyle \int_{0}^{\pi/3}\!\cos^{4}x\,dx=a\pi+b\sqrt{3}$, where $a$ and $b$ are rational numbers, then $9a+8b$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (1 February Evening Shift) PYQ

Solution

Qus : 145 JEE MAIN PYQ

Let $y=y(x)$ be the solution of the differential equation $\dfrac{dy}{dx}+3\tan^2 x,y+3y=\sec^2 x$, $y(0)=\dfrac{1}{3}+e^3$. Then $y!\left(\dfrac{\pi}{4}\right)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (3 April Evening Shift) PYQ

Solution

Qus : 146 JEE MAIN PYQ

Let $\alpha\in(0,1)$ and $\beta=\log_{e}(1-\alpha)$. Let $P_{n}(x)=x+\dfrac{x^{2}}{2}+\dfrac{x^{3}}{3}+\cdots+\dfrac{x^{n}}{n},\ x\in(0,1)$. Then the integral $\displaystyle \int_{0}^{\alpha}\frac{t^{50}}{1-t}\,dt$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (31 January Morning Shift) PYQ

Solution

Qus : 147 JEE MAIN PYQ

$\displaystyle \int_{\pi/6}^{\pi/3}\sec^{\tfrac{2}{3}}x;\csc^{\tfrac{4}{3}}x,dx$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 April Evening Shift) PYQ

Solution

Qus : 148 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (10 April Morning Shift) PYQ

Solution

Qus : 149 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 5 September 2020 (Morning) PYQ

Solution

Qus : 150 JEE MAIN PYQ

The integral $\displaystyle \int_{0}^{\pi}\frac{8x,dx}{4\cos^{2}x+\sin^{2}x}$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (3 April Evening Shift) PYQ

Solution

Qus : 151 JEE MAIN PYQ

$ \text{The value of } \displaystyle \int_{\pi/3}^{\pi/2} \frac{2+3\sin x}{\sin x\,(1+\cos x)}\,dx \text{ is equal to:} $

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (31 January Morning Shift) PYQ

Solution

Qus : 152 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 July Morning Shift) PYQ

Solution

Qus : 153 JEE MAIN PYQ

The value of $\int_{0}^{1} (2x^{3} - 3x^{2} - x + 1)^{\frac{1}{3}} \, dx$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (1 February Evening Shift) PYQ

Solution

Qus : 154 JEE MAIN PYQ

The area (in sq. units) of the region, given by the set $\{ (x,y) \in R \times R|x \ge 0,2{x^2} \le y \le 4 - 2x\} $ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 July Morning Shift) PYQ

Solution

Qus : 155 JEE MAIN PYQ

Let $[t]$ denote the greatest integer less than or equal to $t$.

Then the value of the integral

$\int_{-3}^{101} \left( [\sin(\pi x)] + e^{[\cos(2\pi x)]} \right) dx$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (25 July Evening Shift) PYQ

Solution

Qus : 156 JEE MAIN PYQ

Let $\alpha>0$. If $\displaystyle \int_{0}^{\alpha}\frac{x}{\sqrt{x+\alpha}-\sqrt{x}}\,dx=\dfrac{16+20\sqrt{2}}{15}$, then $\alpha$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (31 January Evening Shift) PYQ

Solution

Qus : 157 JEE MAIN PYQ

If $\displaystyle \int_{0}^{\pi/2} \dfrac{\cot x}{\cot x + \cos \csc x} , dx = m(\pi + n)$, then $m \cdot n$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 April Morning Shift) PYQ

Solution

Qus : 158 JEE MAIN PYQ

Let $f(x)= \begin{cases} -2, & -2 \le x \le 0,\\[4pt] x-2, & 0 < x \le 2, \end{cases}$ and $h(x)=f(|x|)+|f(x)|.$ Then $\displaystyle \int_{-2}^{2} h(x)\,dx$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (4 April Morning Shift) PYQ

Solution

Qus : 159 JEE MAIN PYQ

The integral $\displaystyle \int_{2}^{4}\dfrac{\log x^{2}}{\log x^{2}+\log(36-12x+x^{2})}\,dx$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2015 (Offline) PYQ

Solution

Qus : 160 JEE MAIN PYQ

If $f(x) = \begin{cases} \int_{0}^{x} \left( 5 + |1 - t| \right) dt, & x > 2 \\ 5x + 1, & x \leq 2 \end{cases}$, then

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 July Evening Shift) PYQ

Solution

Qus : 161 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (4 April Morning Shift) PYQ

Solution

Qus : 162 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 April Morning Shift) PYQ

Solution

Qus : 163 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 July Evening Shift) PYQ

Solution

Qus : 164 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (23 January Morning Shift) PYQ

Solution

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