Q JEE MAIN 2026

The area of the region $\mathrm{R}=\left\{(x, y): x y \leq 8,1 \leq y \leq x^2, x \geq 0\right\}$ is

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Q JEE MAIN 2026

If $\int\left(\frac{1-5 \cos ^2 x}{\sin ^5 x \cos ^2 x}\right) d x=f(x)+\mathrm{C}$, where C is the constant of integration, then...

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Q JEE MAIN 2026

Let $f(x)=\int \frac{\mathrm{d} x}{x^{\left(\frac{2}{3}\right)}+2 x^{\left(\frac{1}{2}\right)}}$ be such that $f(0)=-26+24 \log _{\mathrm{e}}(2)$. If $f(1)=\mathrm{a}+\mathrm{b} \log _{\mathrm{e}}(3)$, where $\mathrm{a}, \mathrm{b} \in \mathbf{Z}$,...

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Q JEE MAIN 2026

Let $\mathrm{I}(x)=\int \frac{3 d x}{(4 x+6)\left(\sqrt{4 x^2+8 x+3}\right)}$ and $\mathrm{I}(0)=\frac{\sqrt{3}}{4}+20$. If $\mathrm{I}\left(\frac{1}{2}\right)=\frac{a \sqrt{2}}{b}+\mathrm{c}$, where $...

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Q JEE MAIN 2026

Let $f(t)=\int\left(\frac{1-\sin \left(\log _e t\right)}{1-\cos \left(\log _e t\right)}\right) d t, t>1$. If $f\left(e^{\pi / 2}\right)=-e^{\pi / 2}$ and $...

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Q JEE MAIN_2026

Let $f(x)=\int \frac{\left(2-x^2\right) \cdot \mathrm{e}^x}{(\sqrt{1+x})(1-x)^{3 / 2}} \mathrm{~d} x$. If $f(0)=0$, then $f\left(\frac{1}{2}\right)$ is equal to :

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Q JEE MAIN_2026_

If $\int(\sin x)^{\frac{-11}{2}}(\cos x)^{\frac{-5}{2}} d x=-\frac{p_1}{q_1}(\cot x)^{\frac{9}{2}}-\frac{p_2}{q_2}(\cot x)^{\frac{5}{2}}-\frac{p_3}{q_3}(\cot x)^{\frac{1}{2}}+\frac{p_4}{q_4}(\cot x)^{\frac{-3}{2}}+\mathrm{C}$, where $p_i$ and $q_i$ are positive integers with $\operatorname{gcd}\left(p_i, q_i\right)=1$...

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Q JEE MAIN_2019

$$ \text { If } \int \frac{d x}{\left(x^2-2 x+10\right)^2}=A\left(\tan ^{-1}\left(\frac{x-1}{3}\right)+\frac{f(x)}{x^2-2 x+10}\right)+C $$ where C is a constant of integration, then...

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Q JEE MAIN 2019

The integral $\int \sec ^{2 / 3} x \operatorname{cosec}^{4 / 3}$ is equal to : (Here C is a constant...

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Q JEE MAIN 2019

Let $\alpha \in(0, \pi / 2)$ be fixed. If the integral $...

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Q JEE-MAIN 2019

The integral $\int \cos \left(\log _e x\right) d x$ is equal to (where $C$ is a constant of integration)

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Q JEE MAIN 2020

The integral $\int \frac{d x}{(x+4)^{\frac{8}{7}}(x-3)^{\frac{6}{7}}}$ is equal to (where C is a constant of integration)

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Q JEE MAIN 2019

The integral $\int \frac{3 x+2 x}{\left(2 x^4+3 x^2+1\right)^4} d x$ is equal to (where C is a constant of integration)

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Q JEE MAIN 2020

If $\quad \int\left(e^{2 x}+2 e^x-e^{-x}-1\right) e^{\left(e^x+e^{-x}\right)} d x=g(x) e^{\left(e^x+e^{-x}\right)}+c$ where c is a constant of integration, then $\mathrm{g}(0)$ is equal...

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Q JEE MAIN 2020

If $\int \frac{\cos \theta}{5+7 \sin \theta-2 \cos ^2 \theta} d \theta=A \log _e|B(\theta)|+C$, where $C$ is a constant of integration,...

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Q JEE MAIN 2020

If $\int \frac{\cos x d x}{\sin ^3 x\left(1+\sin ^6 x\right)^{2 / 3}}=f(x)\left(1+\sin ^6 x\right)^{1 / 2}+c$ where $c$ is a...

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Q JEE MAIN 2019

If $\int \frac{\sqrt{1-x^2}}{x^4} d x=A(x)\left(\sqrt{1-x^2}\right)^m+C$, for a suitable chosen integer $m$ and a function $A(x)$, where $C$ is a constant...

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Q JEE MAIN 2019

If $\int x^5 e^{-x^2} d x=g(x) e^{-x^2}+c$, where $c$ is a constant of integration, then $\mathrm{g}(-1)$ is equal to :

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Q JEE Main 2019

Let $\mathrm{n} \geq 2$ be a nutural number and $0

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Q JEE MAIN 2020

If $\int \frac{d \theta}{\cos ^2 \theta(\tan 2 \theta+\sec 2 \theta)}=\lambda \tan \theta+2 \log _e|f(\theta)|+C$ where C is a constant of...

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Q JEE MAIN 2020

Let $f(x)=\int \frac{\sqrt{x}}{(1+x)^2} d x(x \geq 0)$. Then $f(3)-f(1)$ is equal to:

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Q JEE MAIN 2020

The integral $\int\left(\frac{x}{x \sin x+\cos x}\right)^2 d x$ is equal to (where C is a constant of integration):

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Q JEE MAIN 2019

If $\int \frac{x+1}{\sqrt{2 x-1}} d x=f(x) \sqrt{2 x-1}+C$, where $C$ is a constant of integration, then $f(x)$ is equal to...

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Q JEE MAIN 2019

If $f(x)=\int \frac{5 x^8+7 x^6}{\left(x^2+1+2 x\right)^2} d x,(x \geq 0)$, and $f(0)=0$, then the value of $f(1)$ is

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Q JEE MAIN 2019

If $\int e^{\sec x}\left(\sec x \tan x f(x)+\sec x \tan x+\sec ^2 x\right) d x \quad=e^{\sec x} f(x)+C$, then a...

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Q JEE Main 2021

If $...

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Q JEE MAIN 2021

The integral $\int \frac{1}{\sqrt[4]{(x-1)^3(x+2)^5}} d x$ is equal to :

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Q JEE MAIN 2019

If $\int x^5 e^{-4 x^3} d x=\frac{1}{48} e^{-4 x^3} f(x)+C$ where $C$ is a constant of integration, then $f(x)$ is...

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Q JEE MAIN 2021

If $\int \frac{d x}{\left(x^2+x+1\right)^2}=a \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)+b\left(\frac{2 x+1}{x^2+x+1}\right)+C, x>0$ where $C$ is the constant of integration, then the value of...

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Q JEE MAIN_2021_

If $\int \frac{2 e^x+3 e^{-x}}{4 e^x+7 e^{-x}} d x=\frac{1}{14}\left(u x+v \log _e\left(4 e^x+7 e^{-x}\right)\right)+C$, where $C$ is a constant of...

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Q JEE Main 2019

For $x^2 \neq n \pi+1, n \in N$ (the set of natural numbers), the integral $...

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Q JEE MAIN 2019

$\int \frac{\sin \frac{5 x}{2}}{\sin \frac{x}{2}} d x$ is equal to: (where c is a constant of integration)

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Q JEE MAIN 2019

If $\int \frac{d x}{x^3\left(1+x^6\right)^{2 / 3}}=x f(x)\left(1+x^6\right)^{\frac{1}{3}}+C$ where $C$ is a constant of integration, then the function $f(x)$ is equal...

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Q JEE MAIN 2021

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Q JEE MAIN 2021

The integral $...

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Q JEE MAIN 2021

If $f(x)=\int \frac{5 x^8+7 x^6}{\left(x^2+1+2 x^7\right)^2} d x,(x \geq 0), f(0)=0$ and $f(1) =\frac{1}{\mathrm{~K}}$, then the value of K is

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Q JEE MAIN 2021

The integral $\int \frac{(2 x-1) \cos \sqrt{(2 x-1)^2+5}}{\sqrt{4 x^2-4 x+6}} d x$ is equal to (where c is a constant...

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Q JEE MAIN 2021

If $\int \frac{\cos x-\sin x}{\sqrt{8-\sin 2 x}} d x=a \sin ^{-1}\left(\frac{\sin x+\cos x}{b}\right)+c$, where $c$ is a constant of integration,...

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Q JEE MAIN 2021

The value of the integral $...

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Q JEE MAIN 2022

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

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Q JEE MAIN 2022

$\int \frac{\left(x^2+1\right) e^x}{(x+1)^2} d x=f(x) e^x+C$, Where $C$ is a constant, then $\frac{d^3 f}{d x^3}$ at $x=1$ is equal to...

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Q JEE MAIN 2022

Let $\mathrm{g}:(0, \infty) \rightarrow \mathrm{R}$ be a differentiable function such that $\int\left(\frac{x(\cos x-\sin x)}{e^x+1}+\frac{g(x)\left(e^x+1-x e^x\right)}{\left(e^x+1\right)^2}\right) d x=\frac{x g(x)}{e^x+1}+c$, for all...

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Q JEE MAIN_2025

If $\int\left(\frac{1}{x}+\frac{1}{x^3}\right)\left(\sqrt[7]{3 x^{-24}+x^{-26}}\right) \mathrm{d} x=-\frac{\alpha}{3(\alpha+1)}\left(3 x^\beta+x^7\right)^{\frac{\alpha+4}{\alpha}}+C, x>0,(\alpha, \beta, \gamma \in \mathbf{Z})$, where C is the constant of integration, then $\alpha+\beta+\gamma$...

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Q JEE MAIN 2023

For, $\alpha, \beta, \gamma, \delta \in \mathrm{N}$, if $\int\left(\left(\frac{\mathrm{x}}{\mathrm{e}}\right)^{2 \mathrm{x}}+\left(\frac{\mathrm{e}}{\mathrm{x}}\right)^{2 \mathrm{x}}\right) \log _{\mathrm{e}} \mathrm{xdx}=\frac{1}{\alpha}\left(\frac{\mathrm{x}}{\mathrm{e}}\right)^{\beta \mathrm{x}}-\frac{1}{\gamma}\left(\frac{\mathrm{x}}{\mathrm{e}}\right)^{\delta \mathrm{x}}+\mathrm{C}$ Where $e=\sum_{n=0}^{\infty} \frac{1}{n!}$ and...

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Q JEE MAIN 2025

If $\int \frac{\left(\sqrt{1+x^2}+x\right)^{10}}{\left(\sqrt{1+x^2}-x\right)^9} \mathrm{~d} x=\frac{1}{\mathrm{~m}}\left(\left(\sqrt{1+x^2}+x\right)^0\left(\pi \sqrt{1+x^2}-x\right)\right)+\mathrm{C}$ where C is the constant of integration and $m, n \in \mathbf{N}$, then $m+n$...

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Q JEE MAIN 2023

Let $f(x)=\int \frac{d x}{\left(3+4 x^2\right) \sqrt{4-3 x^2}},|x|0$, then $\alpha^2+\beta^2$ is equal to. $\_\_\_\_$ .

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Q JEE-Main 2023

Let $I(x)=\int \frac{x^2\left(x \sec ^2 x+\tan x\right)}{(x \tan x+1)^2} d x$. If $I(0)=0$ the $I\left(\frac{\pi}{4}\right)$ is equal to

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Q JEE MAIN 2023

Let $I(x)=\int \sqrt{\frac{x+7}{x}} d x$ and $I(9)=12+7 \log _e 7$. If $I(1)=\alpha+7 \log _e(1+2 \sqrt{2})$, then $\alpha^4$ is equal to...

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Q JEE MAIN 2023

If $I(x)=\int e^{\sin 2} x(\cos x \sin 2 x-\sin x) d x$ and $I(0)=1$, then $I\left(\frac{\pi}{3}\right)$ is equal to

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Q JEE-Main 2023

$\operatorname{Let} I(x)=\int \frac{x^2\left(x \sec ^2 x+\tan x\right)}{(x \tan x+1)^2} d x$. If $I(0)=0$ the $I\left(\frac{\pi}{4}\right)$ is equal to

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Q JEE MAIN

Let $\int \frac{2-\tan x}{3+\tan x} d x=\frac{1}{2}\left(\alpha x+\log _e|\beta \sin x+\gamma \cos x|\right)+C$, where C is the constant of integration....

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Q JEE MAIN 2024

If the domain of the function $\sin ^{-1}\left(\frac{3 x-22}{2 x-19}\right)+\log _e\left(\frac{3 x^2-8 x+5}{x^2-3 x-10}\right)$ is $(\alpha, \beta]$, then $...

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Q JEE MAIN 2024

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

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Q JEE MAIN 2024

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 $147 I_{20}=148 I_{21}$...

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Q JEE MAIN

For $\mathrm{x} \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, if $y(x)=\int \frac{\operatorname{cosec} x+\sin x}{\operatorname{cosec} x \sec x+\tan x \sin ^2 x} d x$ and $...

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Q JEE MAIN 2025

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

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Q JEE MAIN 2025

Let $f(x) = \int {{x^3}} \sqrt {3 - {x^2}} dx$. If $5f(\sqrt 2 ) = - 4$, then $f(1)$ is...

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Q JEE MAIN 2025

If $\int \frac{2 x^{2}+5 x+9}{\sqrt{x^{2}+x+1}} \mathrm{~d} x=x \sqrt{x^{2}+x+1}+\alpha \sqrt{x^{2}+x+1}+\beta \log _{\mathrm{e}}\left|x+\frac{1}{2}+\sqrt{x^{2}+x+1}\right|+\mathrm{C}$, where $C$ is the constant of integration, then $...

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Q JEE MAIN 2025

Let $\int \mathrm{x}^{3} \sin x \mathrm{~d} x=\mathrm{g}(x)+\mathrm{C}$, where C is the constant of integration. If $...

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Q JEE MAIN 2025

Let ${\rm{I}}(x) = \int {\frac{{dx}}{{{{(x - 11)}^{\frac{{11}}{{13}}}}{{(x + 15)}^{\frac{{15}}{{13}}}}}}} \cdot $. If $...

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Q JEE MAIN 2025

If $\int \mathrm{e}^x\left(\frac{x \sin ^{-1} x}{\sqrt{1-x^2}}+\frac{\sin ^{-1} x}{\left(1-x^2\right)^{3 / 2}}+\frac{x}{1-x^2}\right) \mathrm{d} x=\mathrm{g}(x)+\mathrm{C}$, where C is the constant of integration, then...

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