Integral Calculus & Area · JEE Main previous year questions

Let ${\sum }_{n=0}^{\infty }\frac{n^3((2n)!)+(2n-1)(n!)}{(n!)((2n)!)}=a\mathrm{e}+\frac{b}{\mathrm{e}}+c$, where $\mathrm{a},\mathrm{b},\mathrm{c}\in \mathbb{Z}$ and $\mathrm{e}={\sum }_{n=0}^{\infty }\frac{1}{n!}$ Then $a^2-b+c$ is equal to ____.

Let $\Delta$ be the area of the region $\left\{(x,y)\in R^2:x^2+y^2\le 21,y^2\le 4x,x\ge 1\right\}$. Then $\frac{1}{2}\left(\Delta -21{\sin }^{-1}\frac{2}{\sqrt{7}}\right)$ is equal to

1. A.$2\sqrt{3}-\frac{1}{3}$
2. B.$2\sqrt{3}-\frac{2}{3}$
3. C.$\sqrt{3}-\frac{4}{3}$
4. D.$\sqrt{3}-\frac{2}{3}$

The value of the integral ${\int }_1^2\left(\frac{t^4+1}{t^6+1}\right)dt$ is

1. A.${\tan }^{-1}\frac{1}{2}-\frac{1}{3}{\tan }^{-1}8+\frac{\pi }{3}$
2. B.${\tan }^{-1}2-\frac{1}{3}{\tan }^{-1}8+\frac{\pi }{3}$
3. C.${\tan }^{-1}2+\frac{1}{3}{\tan }^{-1}8-\frac{\pi }{3}$
4. D.${\tan }^{-1}\frac{1}{2}+\frac{1}{3}{\tan }^{-1}8-\frac{\pi }{3}$

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

1. A.$\frac{\pi }{2}{\log }_{e}2$
2. B.$\frac{\pi }{4}{\log }_{e}2$
3. C.$\frac{1}{2}{\log }_{e}2$
4. D.$\pi {\log }_{e}2$

Let $f(x)=\int \frac{2x}{(x^2+1)(x^2+3)}dx$. If $f(3)=\frac{1}{2}({\log }_{e}5-{\log }_{e}6)$, then $f(4)$ is equal to

1. A.${\log }_{e}19-{\log }_{e}20$
2. B.${\log }_{e}17-{\log }_{e}18$
3. C.$\frac{1}{2}({\log }_{e}19-{\log }_{e}17)$
4. D.$\frac{1}{2}({\log }_{e}17-{\log }_{e}19)$

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

1. A.$\frac{11}{12}+{\log }_{e}4$
2. B.$\frac{11}{6}+{\log }_{e}4$
3. C.$\frac{11}{12}-{\log }_{e}4$
4. D.$\frac{11}{6}-{\log }_{e}4$

If ${\int }_{\frac{1}{3}}^3|{\log }_{e}x|dx=\frac{m}{n}{\log }_e\left(\frac{n^2}{e}\right)$, where m and n are coprime natural numbers, then $m^2+n^2-5$ is equal to ____.

Let $y=y(x)$ be the solution of the differential equation $x^{3}dy+(xy-1)dx=0$, $x>0$, $y\left(\frac{1}{2}\right)=3-e$. Then $y(1)$ is equal to

1. A.$2-e$
2. B.3
3. C.1
4. D.e

The area enclosed by the curves $y^2+4x=4$ and $y-2x=2$ is :

1. A.$\frac{22}{3}$
2. B.9
3. C.$\frac{23}{3}$
4. D.$\frac{25}{3}$

The value of $\frac{8}{\pi }{\int }_0^{\frac{\pi }{2}}\frac{(\cos x{)}^{2023}}{(\sin x{)}^{2023}+(\cos x{)}^{2023}}dx$ is ____

Let $y=y(x)$ be the solution of the differential equation $(x^2-3y^2)dx+3xydy=0,y(1)=1$. Then $6y^2(e)$ is equal to

1. A.$\frac{3}{2}{\mathrm{e}}^2$
2. B.$3{\mathrm{e}}^2$
3. C.${\mathrm{e}}^2$
4. D.$2{\mathrm{e}}^2$

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

1. A.$\frac{\pi }{2}$
2. B.$\frac{\pi }{3}$
3. C.$\frac{\pi }{6}$
4. D.$2\pi$

Let $f$ be a differentiable function defined on $\left[0,\frac{\pi }{2}\right]$ such that $f(x)>0$ and $f(x)+{\int }_0^{x}f(t)\sqrt{1-({\log }_{e}f(t){)}^2}dt=e,\forall x\in \left[0,\frac{\pi }{2}\right]$. Then ${\left(6{\log }_{e}f\left(\frac{\pi }{6}\right)\right)}^2$ is equal to _______.

If the area of the region bounded by the curves $y^2-2y=-x,x+y=0$ is A, then 8 A is equal to _______

The integral ${\int }_0^{\frac{\pi }{2}}\frac{1}{3+2\sin x+\cos x}\mathrm{d}x$ is equal to :

1. A.${\tan }^{-1}(2)$
2. B.${\tan }^{-1}(2)-\frac{\pi }{4}$
3. C.$\frac{1}{2}{\tan }^{-1}(2)-\frac{\pi }{8}$
4. D.$\frac{1}{2}$

If $f(\alpha )={\int }_1^{\alpha }\frac{{\log }_{10}t}{1+t}\mathrm{d}t,\alpha >0$, then $f\left({\mathrm{e}}^3\right)+f\left({\mathrm{e}}^{-3}\right)$ is equal to :

1. A.9
2. B.$\frac{9}{2}$
3. C.$\frac{9}{{\log }_{\mathrm{e}}(10)}$
4. D.$\frac{9}{2{\log }_{\mathrm{e}}(10)}$

The area of the region $\left\{(x,y):|x-1|\le y\le \sqrt{5-x^2}\right\}$ is equal to :

1. A.$\frac{5}{2}{\sin }^{-1}\left(\frac{3}{5}\right)-\frac{1}{2}$
2. B.$\frac{5\pi }{4}-\frac{3}{2}$
3. C.$\frac{3\pi }{4}+\frac{3}{2}$
4. D.$\frac{5\pi }{4}-\frac{1}{2}$

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

1. A.$\frac{\sqrt{37}+\sqrt{13}-4}{6}$
2. B.$\frac{\sqrt{37}-\sqrt{13}-4}{6}$
3. C.$\frac{-\sqrt{37}-\sqrt{13}+4}{6}$
4. D.$\frac{-\sqrt{37}+\sqrt{13}+4}{6}$

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

1. A.$3^{1010}I\left(\frac{\pi }{3}\right)-I\left(\frac{\pi }{6}\right)=0$
2. B.$3^{1010}I\left(\frac{\pi }{6}\right)-I\left(\frac{\pi }{3}\right)=0$
3. C.$3^{1011}I\left(\frac{\pi }{3}\right)-I\left(\frac{\pi }{6}\right)=0$
4. D.$3^{1011}I\left(\frac{\pi }{6}\right)-I\left(\frac{\pi }{3}\right)=0$

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

1. A.$-\frac{2}{\sqrt{\mathrm{e}}}$
2. B.$-2\sqrt{\mathrm{e}}$
3. C.$-\sqrt{\mathrm{e}}$
4. D.$\frac{2}{\sqrt{\mathrm{e}}}$

If $\underset{0}{\overset{\sqrt{3}}{\int }}\frac{15x^3}{\sqrt{1+x^2+\sqrt{(1+x^2{)}^3}}}dx=\alpha \sqrt{2}+\beta \sqrt{3}$, where $\alpha ,\beta$ are integers, then $\alpha +\beta$ is equal to ____.

Let $f(x)=a{x}^2+bx+c$ be such that $f(1)=3$, $f(-2)=\lambda$ and $f(3)=4$. If $f(0)+f(1)+f(-2)+f(3)=14$, then $\lambda$ is equal to :

1. A.$-4$
2. B.$\frac{13}{2}$
3. C.$\frac{23}{2}$
4. D.4

Let $I_n(x)={\int }_0^x\frac{1}{(t^2+5{)}^n}dt$, $n=1,2,3,\dots$. Then :

1. A.$50I_6-9I_5=x{I}_5^{\prime }$
2. B.$50I_6-11I_5=x{I}_5^{\prime }$
3. C.$50I_6-9I_5=I_5^{\prime }$
4. D.$50I_6-11I_5=I_5^{\prime }$

The area enclosed by the curves $y={\log }_e(x+e^2)$, $x={\log }_e\left(\frac{2}{y}\right)$ and $x={\log }_{e}2$, above the line $y=1$ is:

1. A.$2+e-{\log }_{e}2$
2. B.$1+e-{\log }_{e}2$
3. C.$e-{\log }_{e}2$
4. D.$1+{\log }_{e}2$

The value of the integral ${\int }_0^{\frac{\pi }{2}}60\frac{\sin (6x)}{\sin x}dx$ is equal to _________.