Integral Calculus & Area · JEE Main previous year questions

The area (in sq. units) of the region $A=\{(x,y):x^2\le y\le x+2\}$ is

1. A.$\frac{31}{6}$
2. B.$\frac{10}{3}$
3. C.$\frac{13}{6}$
4. D.$\frac{9}{2}$

The value of the integral ${\int }_0^{1}x{\cot }^{-1}(1-x^2+x^4)dx$ is :-

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

If $\cos x\frac{dy}{dx}-y\sin x=6x$, $(0<x<\frac{\pi }{2})$ and $y\left(\frac{\pi }{3}\right)=0$ then $y\left(\frac{\pi }{6}\right)$ is equal to :-

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

$\int e^{\sec x}(\sec x\tan xf(x)+\sec x\tan x+{\sec }^{2}x)dx={\mathrm{e}}^{\sec x}f(x)+C$ then a possible choice of f(x) is :-

1. A.x sec x + tan x + 1/2
2. B.sec x + xtan x - 1/2
3. C.sec x - tan x - 1/2
4. D.sec x + tan x + 1/2

The area (in sq. units) of the region $\mathrm{A}=\{(x,y):\frac{y^2}{2}\le x\le y+4\}$ is :-

1. A.30
2. B.18
3. C.$\frac{53}{3}$
4. D.16

Let $S(\alpha )=\{(x,y):y^2\le x,0\le x\le \alpha \}$ and $A(\alpha )$ is area of the region $S(\alpha )$. If for a $\lambda$, $0<\lambda <4$, $A(\lambda ):A(4)=2:5$, then $\lambda$ equals

1. A.$2{\left(\frac{4}{25}\right)}^{\frac{1}{3}}$
2. B.$2{\left(\frac{2}{5}\right)}^{\frac{1}{3}}$
3. C.$4{\left(\frac{4}{25}\right)}^{\frac{1}{3}}$
4. D.$4{\left(\frac{2}{5}\right)}^{\frac{1}{3}}$

The number of integral values of m for which the equation $(1+m^2)x^2-2(1+3m)x+(1+8m)=0$ has no real root is :

1. A.2
2. B.infinitely many
3. C.1
4. D.3

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

1. A.$\frac{3}{x^2}$
2. B.$-\frac{1}{6x^3}$
3. C.$-\frac{1}{2x^3}$
4. D.$-\frac{1}{2x^2}$

The integral $\int \cos ({\log }_{e}x)dx$ is equal to : (where C is a constant of integration)

1. A.$\frac{x}{2}[\sin ({\log }_{e}x)-\cos ({\log }_{e}x)]+C$
2. B.$x[\cos ({\log }_{e}x)+\sin ({\log }_{e}x)]+C$
3. C.$\frac{x}{2}[\cos ({\log }_{e}x)+\sin ({\log }_{e}x)]+C$
4. D.$x[\cos ({\log }_{e}x)-\sin ({\log }_{e}x)]+C$

The integral ${\int }_1^e\left\{{\left(\frac{x}{e}\right)}^{2x}-{\left(\frac{e}{x}\right)}^x\right\}{\log }_{\mathrm{e}}xdx$ is equal to :

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

The integral $\int \frac{3x^{13}+2x^{11}}{(2x^4+3x^2+1{)}^4}dx$ is equal to : (where C is a constant of integration)

1. A.$\frac{x^{12}}{6(2x^4+3x^2+1{)}^3}+C$
2. B.$\frac{x^4}{6(2x^4+3x^2+1{)}^3}+C$
3. C.$\frac{x^{12}}{(2x^4+3x^2+1{)}^3}+C$
4. D.$\frac{x^4}{(2x^4+3x^2+1{)}^3}+C$

If $$\left|\begin{array}{ccc}a-b-c& 2a& 2a\\ 2b& b-c-a& 2b\\ 2c& 2c& c-a-b\end{array}\right|=(a+b+c)(x+a+b+c{)}^2,x\ne 0,$$ then x is equal to :

1. A.$-2(a+b+c)$
2. B.$2(a+b+c)$
3. C.abc
4. D.$-(a+b+c)$

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

1. A.$\frac{2}{3}(x-4)$
2. B.$\frac{1}{3}(x+4)$
3. C.$\frac{1}{3}(x+1)$
4. D.$\frac{2}{3}(x+2)$

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

1. A.$\frac{\pi }{40}$
2. B.$\frac{1}{20}{\tan }^{-1}\left(\frac{1}{9\sqrt{3}}\right)$
3. C.$\frac{1}{10}\left(\frac{\pi }{4}-{\tan }^{-1}\left(\frac{1}{9\sqrt{3}}\right)\right)$
4. D.$\frac{1}{5}\left(\frac{\pi }{4}-{\tan }^{-1}\left(\frac{1}{3\sqrt{3}}\right)\right)$

For $x^2\ne n\pi +1$, $n\in \mathbb{N}$ (the set of natural numbers), the integral $$\int x\sqrt{\frac{2\sin (x^2-1)-\sin 2(x^2-1)}{2\sin (x^2-1)+\sin 2(x^2-1)}}dx$$ is equal to : (where c is a constant of integration)

1. A.${\log }_e\left|\frac{1}{2}{\sec }^2(x^2-1)\right|+c$
2. B.$\frac{1}{2}{\log }_e\left|\sec (x^2-1)\right|+c$
3. C.$\frac{1}{2}{\log }_e\left|{\sec }^2\left(\frac{x^2-1}{2}\right)\right|+c$
4. D.${\log }_e\left|\sec \left(\frac{x^2-1}{2}\right)\right|+c$

If $\int \frac{\tan x}{1+\tan x+{\tan }^{2}x}dx=x-\frac{K}{\sqrt{A}}{\tan }^{-1}\left(\frac{K\tan x+1}{\sqrt{A}}\right)+C$, (C is a constant of integration) then the ordered pair (K, A) is equal to :

1. A.(2, 1)
2. B.$(-2,3)$
3. C.(2, 3)
4. D.$(-2,1)$

If $f(x)={\int }_0^{x}t(\sin x-\sin t)dt$ then :

1. A.$f^{\prime \prime \prime }(x)+f^{\prime \prime }(x)=\sin x$
2. B.$f^{\prime \prime \prime }(x)+f^{\prime \prime }(x)-f^{\prime }(x)=\cos x$
3. C.$f^{\prime \prime \prime }(x)+f^{\prime }(x)=\cos x-2x\sin x$
4. D.$f^{\prime \prime \prime }(x)-f^{\prime \prime }(x)=\cos x-2x\sin x$

If the area of the region bounded by the curves, $y=x^2,y=\frac{1}{x}$ and the lines $y=0$ and $x=t$ ($t>1$) is 1 sq. unit, then t is equal to :

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

If $f\left(\frac{x-4}{x+2}\right)=2x+1$, $(x\in \mathbf{R}-\{1,-2\})$, then $\int f(x)dx$ is equal to : (where C is a constant of integration)

1. A.$12{\log }_e|1-x|+3x+\text{C}$
2. B.$-12{\log }_e|1-x|-3x+\text{C}$
3. C.$12{\log }_e|1-x|-3x+\text{C}$
4. D.$-12{\log }_e|1-x|+3x+\text{C}$

The area (in sq. units) of the region $\{x\in \mathbf{R}:x\ge 0,y\ge 0,y\ge x-2\text{ and }y\le \sqrt{x}\}$, is :

1. A.$\frac{13}{3}$
2. B.$\frac{8}{3}$
3. C.$\frac{10}{3}$
4. D.$\frac{5}{3}$

The value of the integral ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{\sin }^{4}x\left(1+\log \left(\frac{2+\sin x}{2-\sin x}\right)\right)dx$ is :

1. A.0
2. B.$\frac{3}{4}$
3. C.$\frac{3}{8}\pi$
4. D.$\frac{3}{16}\pi$

The value of ${\int }_{-\pi /2}^{\pi /2}\frac{{\sin }^{2}x}{1+2^x}dx$ is

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

The integral $$\int \frac{{\sin }^{2}x{\cos }^{2}x}{{\left({\sin }^{5}x+{\cos }^{3}x{\sin }^{2}x+{\sin }^{3}x{\cos }^{2}x+{\cos }^{5}x\right)}^2}dx$$ is equal to

1. A.$\frac{-1}{1+{\cot }^{3}x}+C$
2. B.$\frac{1}{3(1+{\tan }^{3}x)}+C$
3. C.$\frac{-1}{3(1+{\tan }^{3}x)}+C$
4. D.$\frac{1}{1+{\cot }^{3}x}+C$

Let $\mathrm{y}=\mathrm{y}(\mathrm{x})$ be the solution of the differential equation $\sin x\frac{dy}{dx}+y\cos x=4x,x\in (0,\pi )$. If $y\left(\frac{\pi }{2}\right)=0$, then $y\left(\frac{\pi }{6}\right)$ is equal to :

1. A.$-\frac{4}{9}{\pi }^2$
2. B.$\frac{4}{9\sqrt{3}}{\pi }^2$
3. C.$-\frac{8}{9\sqrt{3}}{\pi }^2$
4. D.$-\frac{8}{9}{\pi }^2$

If $f\left(\frac{3x-4}{3x+4}\right)=x+2$, $x\ne -\frac{4}{3}$, and $\int f(x)\mathrm{d}x=\mathrm{A}\log |1-x|+\mathrm{B}x+\mathrm{C}$, then the ordered pair (A, B) is equal to : (where C is a constant of integration)

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