Integral Calculus & Area · JEE Main previous year questions

Let for $f(x)=7{\tan }^{8}x+7{\tan }^{6}x-3{\tan }^{4}x-3{\tan }^{2}x$, ${\mathrm{I}}_1={\int }_0^{\pi /4}f(x)dx$ and ${\mathrm{I}}_2={\int }_0^{\pi /4}xf(x)dx$. Then $7{\mathrm{I}}_1+12{\mathrm{I}}_2$ is equal to :

1. A.$2\pi$
2. B.1
3. C.$\pi$
4. D.2

The solution of the differential equation $(x^2+y^2)\mathrm{d}x-5xy\mathrm{d}y=0$, $y(1)=0$, is :

1. A.$|x^2-4y^2{|}^5=x^2$
2. B.$|x^2-2y^2{|}^6=x$
3. C.$|x^2-2y^2{|}^5=x^2$
4. D.$|x^2-4y^2{|}^6=x$

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

1. A.3
2. B.7
3. C.1
4. D.4

Let $\underset{0}{\overset{x}{\int }}\sqrt{1-(y^{\prime }(t){)}^2}dt={\int }_0^{x}y(t)dt$, $0\le x\le 3$, $y\ge 0$, $y(0)=0$. Then at $x=2$, $y^{\prime \prime }+y+1$ is equal to

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

The integral $\underset{1/4}{\overset{3/4}{\int }}\cos \left(2{\cot }^{-1}\sqrt{\frac{1-x}{1+x}}\right)dx$ is equal to

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

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

1. A.$\sqrt{5}-\sqrt{2}+{\log }_e\left(\frac{7+4\sqrt{5}}{1+\sqrt{2}}\right)$
2. B.$\sqrt{2}-\sqrt{5}+{\log }_e\left(\frac{7+4\sqrt{5}}{1+\sqrt{2}}\right)$
3. C.$\sqrt{5}-\sqrt{2}+{\log }_e\left(\frac{9+4\sqrt{5}}{1+\sqrt{2}}\right)$
4. D.$\sqrt{2}-\sqrt{5}+{\log }_e\left(\frac{9+4\sqrt{5}}{1+\sqrt{2}}\right)$

The value of $k\in \mathbb{N}$ for which the integral $I_n={\int }_0^1(1-x^k{)}^{n}dx,n\in \mathbb{N}$, satisfies $147I_{20}=148I_{21}$ is

1. A.$8$
2. B.$14$
3. C.$7$
4. D.$10$

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

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

If $\int \frac{1}{\sqrt[5]{(x-1{)}^4(x+3{)}^6}}\mathrm{d}x=\mathrm{A}{\left(\frac{\alpha x-1}{\beta x+3}\right)}^{\mathrm{B}}+\mathrm{C}$, where $\mathrm{C}$ is the constant of integration, then the value of $\alpha +\beta +20\mathrm{AB}$ is ____.

Let $\alpha |x|=|y|e^{xy-\beta },\alpha ,\beta \in \mathbf{N}$ be the solution of the differential equation $x\mathrm{d}y-y\mathrm{d}x+xy(x\mathrm{d}y+y\mathrm{d}x)=0,y(1)=2$. Then $\alpha +\beta$ is equal to ____

If the area of the region $\{(x,y):\frac{a}{x^2}\le y\le \frac{1}{x},1\le x\le 2,0<a<1\}$ is $({\log }_{\mathrm{e}}2)-\frac{1}{7}$ then the value of $7a-3$ is equal to :

1. A.1
2. B.0
3. C.2
4. D.$-1$

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

1. A.$\sqrt{41}$
2. B.$\sqrt{39}$
3. C.$\sqrt{40}$
4. D.$\sqrt{42}$

If the locus of the point, whose distances from the point $(2,1)$ and $(1,3)$ are in the ratio $5:4$, is $a{x}^2+b{y}^2+cxy+dx+ey+170=0$, then the value of $a^2+2b+3c+4d+e$ is equal to :

1. A.37
2. B.$-27$
3. C.437
4. D.5

If the solution $y(x)$ of the given differential equation $(e^y+1)\cos xdx+e^y\sin xdy=0$ passes through the point $\left(\frac{\pi }{2},0\right)$, then the value of $e^{y\left(\frac{\pi }{6}\right)}$ is equal to ____.

Let $S_1=\{z\in \mathbf{C}:|z|\le 5\}$, $S_2=\left\{z\in \mathbf{C}:\mathrm{Im}\left(\frac{z+1-\sqrt{3}i}{1-\sqrt{3}i}\right)\ge 0\right\}$ and $S_3=\{z\in \mathbf{C}:\mathrm{Re}(z)\ge 0\}$. Then the area of the region $S_1\cap S_2\cap S_3$ is :

1. A.$\frac{125\pi }{24}$
2. B.$\frac{125\pi }{6}$
3. C.$\frac{125\pi }{12}$
4. D.$\frac{125\pi }{4}$

Let $\beta (m,n)=\underset{0}{\overset{1}{\int }}{x}^{m-1}(1-x{)}^{n-1}dx,m,n>0$. If $\underset{0}{\overset{1}{\int }}(1-x^{10}{)}^{20}dx=a\times \beta (b,c)$, then $100(a+b+c)$ equals _________.

1. A.2012
2. B.1021
3. C.1120
4. D.2120

The area enclosed between the curves $y=x|x|$ and $y=x-|x|$ is :

1. A.$\frac{8}{3}$
2. B.$\frac{2}{3}$
3. C.$\frac{4}{3}$
4. D.1

One of the points of intersection of the curves $y=1+3x-2x^2$ and $y=\frac{1}{x}$ is $\left(\frac{1}{2},2\right)$. Let the area of the region enclosed by these curves be $\frac{1}{24}(l\sqrt{5}+m)-n{\log }_e(1+\sqrt{5})$, where $l,m,n\in \mathbf{N}$. Then $l+m+n$ is equal to

1. A.30
2. B.29
3. C.31
4. D.32

If the shortest distance between the lines $\frac{x+2}{2}=\frac{y+3}{3}=\frac{z-5}{4}$ and $\frac{x-3}{1}=\frac{y-2}{-3}=\frac{z+4}{2}$ is $\frac{38}{3\sqrt{5}}k$, and $\underset{0}{\overset{k}{\int }}[x^2]\mathrm{d}x=\alpha -\sqrt{\alpha }$, where $[x]$ denotes the greatest integer function, then $6{\alpha }^3$ is equal to ____.

If $\underset{0}{\overset{\frac{\pi }{4}}{\int }}\frac{{\sin }^{2}x}{1+\sin x\cos x}\mathrm{d}x=\frac{1}{\mathrm{a}}{\log }_e\left(\frac{\mathrm{a}}{3}\right)+\frac{\pi }{\mathrm{b}\sqrt{3}}$, where $\mathrm{a},\mathrm{b}\in \mathrm{N}$, then $\mathrm{a}+\mathrm{b}$ is equal to ____.

Let $a=1+\frac{{}^2{\mathrm{C}}_2}{3!}+\frac{{}^3{\mathrm{C}}_2}{4!}+\frac{{}^4{\mathrm{C}}_2}{5!}+\dots$, $\mathrm{b}=1+\frac{{}^1{\mathrm{C}}_0+{}^1{\mathrm{C}}_1}{1!}+\frac{{}^2{\mathrm{C}}_0+{}^2{\mathrm{C}}_1+{}^2{\mathrm{C}}_2}{2!}+\frac{{}^3{\mathrm{C}}_0+{}^3{\mathrm{C}}_1+{}^3{\mathrm{C}}_2+{}^3{\mathrm{C}}_3}{3!}+\dots$. Then $\frac{2b}{a^2}$ is equal to ____.

The area (in sq. units) of the region described by $\left\{(x,y):y^2\le 2x\text{, and }y\ge 4x-1\right\}$ is

1. A.$\frac{9}{32}$
2. B.$\frac{11}{12}$
3. C.$\frac{8}{9}$
4. D.$\frac{11}{32}$

If the value of the integral ${\int }_{-1}^1\frac{\cos \alpha x}{1+3^x}dx$ is $\frac{2}{\pi }$, then a value of $\alpha$ is

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

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

The value of the integral ${\int }_0^{\pi /4}\frac{xdx}{{\sin }^4(2x)+{\cos }^4(2x)}$ equals :

1. A.$\frac{\sqrt{2}{\pi }^2}{8}$
2. B.$\frac{\sqrt{2}{\pi }^2}{16}$
3. C.$\frac{\sqrt{2}{\pi }^2}{32}$
4. D.$\frac{\sqrt{2}{\pi }^2}{64}$