Question

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

JEE Main 2020 (05 Sep Shift 1)

Options

• A: $e$
• B: $e^2$
• C: $1$
• D: $2$

Question

If $I_1={\int }_0^1{\left(1-x^{50}\right)}^{100}dx$ and $I_2={\int }_0^1{\left(1-x^{50}\right)}^{101}dx$ such that $I_2=\alpha I_1$ then $\alpha$ equals to :

JEE Main 2020 (06 Sep Shift 1)

Options

• A: $\frac{5049}{5050}$
• B: $\frac{5050}{5049}$
• C: $\frac{5050}{5051}$
• D: $\frac{5051}{5050}$

Question

The value of the integral $\int \frac{\sin \theta ·\sin 2\theta \left({\sin }^6\theta +{\sin }^4\theta +{\sin }^2\theta \right)\sqrt{2{\sin }^4\theta +3{\sin }^2\theta +6}}{1-\cos 2\theta }d\theta$ is (where $c$ is a constant of integration)

JEE Main 2021 (25 Feb Shift 1)

Options

• A: $\frac{1}{18}{\left[11-18{\sin }^2\theta +9{\sin }^4\theta -2{\sin }^6\theta \right]}^{\frac{3}{2}}+c$
• B: $\frac{1}{18}{\left[9-2{\sin }^6\theta -3{\sin }^4\theta -6{\sin }^2\theta \right]}^{\frac{3}{2}}+c$
• C: $\frac{1}{18}{\left[11-18{\cos }^2\theta +9{\cos }^4\theta -2{\cos }^6\theta \right]}^{\frac{3}{2}}+c$
• D: $\frac{1}{18}{\left[9-2{\cos }^6\theta -3{\cos }^4\theta -6{\cos }^2\theta \right]}^{-\frac{3}{2}}+c$

Question

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

JEE Main 2021 (18 Mar Shift 1)

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Question

$\int \frac{2e^x+3e^{-x}}{4e^x+7e^{-x}}dx=\frac{1}{14}\left(ux+v{\log }_e\left(4e^x+7e^{-x}\right)\right)+C$, where $C$ is a constant of integration, then $u+v$ is equal to

JEE Main 2021 (27 Aug Shift 2)

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Question

Let $g:\left(0,\infty \right)\to \mathit{R}$ be a differentiable function such that $\int \left(\frac{x\left(\cos x-\sin x\right)}{{\mathrm{e}}^x+1}+\frac{g\left(x\right)\left({\mathrm{e}}^x+1-x{\mathrm{e}}^x\right)}{{\left({\mathrm{e}}^x+1\right)}^2}\right)\mathrm{d}x=\frac{xg\left(x\right)}{{\mathrm{e}}^x+1}+C$, for all $x>0$, where $C$ is an arbitrary constant. Then

JEE Main 2022 (25 Jun Shift 1)

Options

• A: $g$ is decreasing in $\left(0,\frac{\pi }{4}\right)$
• B: $g-g^{\text{'}}$ is increasing in $\left(0,\frac{\pi }{2}\right)$
• C: $g^{\text{'}}$ is increasing in $\left(0,\frac{\pi }{4}\right)$
• D: $g+g^{\text{'}}$ is increasing in $\left(0,\frac{\pi }{2}\right)$

Question

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

JEE Main 2022 (27 Jun Shift 1)

Options

• A: $\frac{3}{4}$
• B: $\frac{3}{8}$
• C: $-\frac{3}{2}$
• D: $\frac{7}{8}$

Question

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

JEE Main 2022 (29 Jul Shift 2)

Options

• A: $3^{1010}I\left(\frac{\pi }{3}\right)-I\left(\frac{\pi }{6}\right)=0$
• B: $3^{1010}I\left(\frac{\pi }{6}\right)-I\left(\frac{\pi }{3}\right)=0$
• C: $3^{1011}I\left(\frac{\pi }{3}\right)-I\left(\frac{\pi }{6}\right)=0$
• D: $3^{1011}I\left(\frac{\pi }{6}\right)-I\left(\frac{\pi }{3}\right)=0$

Question

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

JEE Main 2023 (06 Apr Shift 1)

Options

• A: ${\log }_{\mathrm{e}}\frac{(\pi +4{)}^2}{16}+\frac{{\pi }^2}{4(\pi +4)}$
• B: ${\log }_{\mathrm{e}}\frac{(\pi +4{)}^2}{16}-\frac{{\pi }^2}{4(\pi +4)}$
• C: ${\log }_{\mathrm{e}}\frac{(\pi +4{)}^2}{32}-\frac{{\pi }^2}{4(\pi +4)}$
• D: ${\log }_{\mathrm{e}}\frac{(\pi +4{)}^2}{32}+\frac{{\pi }^2}{4(\pi +4)}$

Question

For $\alpha ,\beta ,\gamma ,\delta \in \mathbb{N}$, if $\int \left({\left(\frac{x}{e}\right)}^{2x}+{\left(\frac{e}{x}\right)}^{2x}\right){\log }_{e}xdx=\frac{1}{\alpha }{\left(\frac{x}{e}\right)}^{\beta x}-\frac{1}{\gamma }{\left(\frac{e}{x}\right)}^{\delta x}+C$, where $e=\sum _{n=0}^{\infty }\frac{1}{n!}$ and $C$ is constant of integration, then $\alpha +2\beta +3\gamma -4\delta$ is equal to

JEE Main 2023 (10 Apr Shift 2)

Options

• A: $1$
• B: $4$
• C: $-4$
• D: $-8$

Question

If $\int \operatorname{cosec}^5 x d x=\alpha \cot x \operatorname{cosec} x\left(\operatorname{cossc}^2 x+\frac{3}{2}\right)+\beta \log _\epsilon\left|\tan \frac{x}{2}\right|+C$ where $\alpha, \beta \in \mathbb{R}$ and $\mathrm{C}$ is the constant of integration, then the value of $8(\alpha+\beta)$ equals _______

JEE Main 2024 (04 Apr Shift 2)

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Question

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

JEE Main 2024 (08 Apr Shift 1)

Options

• A: $2 \sqrt{3}$
• B: $\sqrt{3}$
• C: $3 \sqrt{3}$
• D: $6 \sqrt{3}$

Question

The integral $\int \frac{\left({\mathrm{x}}^8-{\mathrm{x}}^2\right)\mathrm{dx}}{\left({\mathrm{x}}^{12}+3{\mathrm{x}}^6+1\right){\tan }^{-1}\left({\mathrm{x}}^3+\frac{1}{{\mathrm{x}}^3}\right)}$ is equal to :

JEE Main 2024 (27 Jan Shift 2)

Options

• A: $\log {\left|{\tan }^{-1}\left({\mathrm{x}}^3+\frac{1}{{\mathrm{x}}^3}\right)\right|}^{\frac{1}{3}}+\mathrm{C}$
• B: ${\log }_{\mathrm{e}}{\left(\left|{\tan }^{-1}\left({\mathrm{x}}^3+\frac{1}{{\mathrm{x}}^3}\right)\right|\right)}^{\frac{1}{2}}+\mathrm{C}$
• C: ${\log }_{\mathrm{e}}\left(\left|{\tan }^{-1}\left({\mathrm{x}}^3+\frac{1}{{\mathrm{x}}^3}\right)\right|\right)+\mathrm{C}$
• D: ${\log }_{\mathrm{e}}{\left(\left|{\tan }^{-1}\left({\mathrm{x}}^3+\frac{1}{{\mathrm{x}}^3}\right)\right|\right)}^3+\mathrm{C}$

Question

If $\int \frac{{\sin }^{\frac{3}{2}}x+{\cos }^{\frac{3}{2}}x}{\sqrt{{\sin }^{3}x{\cos }^{3}x\sin (x-\theta )}}dx=A\sqrt{\cos \theta \tan x-\sin \theta }+B\sqrt{\cos \theta -\sin \theta \cot x}+C,$ where $C$ is the integration constant, then $AB$ is equal to

JEE Main 2024 (29 Jan Shift 2)

Options

• A: $4cosec\left(2\theta \right)$
• B: $4\sec \theta$
• C: $2\sec \theta$
• D: $8cosec\left(2\theta \right)$