Integral Calculus & Area · JEE Main previous year questions

Let $I={\int }_{\pi /4}^{\pi /3}\left(\frac{8\sin x-\sin 2x}{x}\right)dx$. Then

1. A.$\frac{\pi }{2}<I<\frac{3\pi }{4}$
2. B.$\frac{\pi }{5}<I<\frac{5\pi }{12}$
3. C.$\frac{5\pi }{12}<I<\frac{\sqrt{2}}{3}\pi$
4. D.$\frac{3\pi }{4}<I<\pi$

The area of the region enclosed by $y\le 4x^2$, $x^2\le 9y$ and $y\le 4$, is equal to :

1. A.$\frac{40}{3}$
2. B.$\frac{56}{3}$
3. C.$\frac{112}{3}$
4. D.$\frac{80}{3}$

The odd natural number a, such that the area of the region bounded by $y=1$, $y=3$, $x=0$, $x=y^a$ is $\frac{364}{3}$, is equal to :

1. A.3
2. B.5
3. C.7
4. D.9

If $n(2n+1){\int }_0^1(1-x^n{)}^{2n}\mathrm{d}x=1177{\int }_0^1(1-x^n{)}^{2n+1}\mathrm{d}x$, then $n\in \mathbf{N}$ is equal to ____ .

Let a curve $y=y(x)$ pass through the point $(3,3)$ and the area of the region under this curve, above the $x$-axis and between the abscissae 3 and $x(>3)$ be ${\left(\frac{y}{x}\right)}^3$. If this curve also passes through the point $(\alpha ,6\sqrt{10})$ in the first quadrant, then $\alpha$ is equal to ____ .

Let $[t]$ denote the greatest integer less than or equal to $t$. Then the value of the integral ${\int }_{-3}^{101}\left([\sin (\pi x)]+e^{[\cos (2\pi x)]}\right)dx$ is equal to

1. A.$\frac{52(1-e)}{e}$
2. B.$\frac{52}{e}$
3. C.$\frac{52(2+e)}{e}$
4. D.$\frac{104}{e}$

Let $f$ be a twice differentiable function on $\mathbb{R}$. If $f^{\prime }(0)=4$ and $f(x)+{\int }_0^x(x-t)f^{\prime }(t)dt=(e^{2x}+e^{-2x})\cos 2x+\frac{2}{a}x$, then $(2a+1{)}^5{a}^2$ is equal to ___________.

Let $a_n={\int }_{-1}^n\left(1+\frac{x}{2}+\frac{x^2}{3}+\dots +\frac{x^{n-1}}{n}\right)dx$ for every $n\in \mathbb{N}$. Then the sum of all the elements of the set $\{n\in \mathbb{N}:a_n\in (2,30)\}$ is ___________.

If for some $\alpha >0$, the area of the region $\{(x,y):|x+\alpha |\le y\le 2-|x|\}$ is equal to $\frac{3}{2}$, then the area of the region $\{(x,y):0\le y\le x+2\alpha ,|x|\le 1\}$ is equal to ____ .

The area of the bounded region enclosed by the curve $y=3-\left|x-\frac{1}{2}\right|-|x+1|$ and the x-axis is :

1. A.$\frac{9}{4}$
2. B.$\frac{45}{16}$
3. C.$\frac{27}{8}$
4. D.$\frac{63}{16}$

Let $x=x(y)$ be the solution of the differential equation $2y{e}^{x/y^2}dx+\left(y^2-4x{e}^{x/y^2}\right)dy=0$ such that $x(1)=0$. Then, $x(e)$ is equal to :

1. A.$e{\log }_e(2)$
2. B.$-e{\log }_e(2)$
3. C.$e^2{\log }_e(2)$
4. D.$-e^2{\log }_e(2)$

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

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

The value of the integral $${\int }_{-2}^2\frac{|x^3+x|}{(e^{x|x|}+1)}dx$$ is equal to :

1. A.$5e^2$
2. B.$3e^{-2}$
3. C.4
4. D.6

The integral ${\int }_0^1\frac{1}{7^{\left[\frac{1}{x}\right]}}dx$, where $[.]$ denotes the greatest integer function, is equal to

1. A.$1+6{\log }_e\left(\frac{6}{7}\right)$
2. B.$1-6{\log }_e\left(\frac{6}{7}\right)$
3. C.${\log }_e\left(\frac{7}{6}\right)$
4. D.$1-7{\log }_e\left(\frac{6}{7}\right)$

If the area of the region $\left\{(x,y):x^{\frac{2}{3}}+y^{\frac{2}{3}}\le 1,x+y\ge 0,y\ge 0\right\}$ is A, then $\frac{256A}{\pi }$ is equal to ________.

The value of the integral $$\frac{48}{{\pi }^4}{\int }_0^{\pi }\left(\frac{3\pi x^2}{2}-x^3\right)\frac{\sin x}{1+{\cos }^{2}x}dx$$ is equal to ____.

The area of the region bounded by $y^2=8x$ and $y^2=16(3-x)$ is equal to:

1. A.$\frac{32}{3}$
2. B.$\frac{40}{3}$
3. C.16
4. D.19

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

1. A.${\log }_e\left(\frac{\sqrt{3}-1}{\sqrt{3}+1}\right)+\frac{\pi }{3}$
2. B.${\log }_e\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right)+\frac{\pi }{3}$
3. C.${\log }_e\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right)-\frac{\pi }{3}$
4. D.$\frac{1}{2}{\log }_e\left(\frac{\sqrt{3}-1}{\sqrt{3}+1}\right)-\frac{\pi }{6}$

The integral $\frac{24}{\pi }{\int }_0^{\sqrt{2}}\frac{(2-x^2)dx}{(2+x^2)\sqrt{4+x^4}}$ is equal to ________.

The value of $\underset{0}{\overset{\pi }{\int }}\frac{e^{\cos x}\sin x}{(1+{\cos }^{2}x)(e^{\cos x}+e^{-\cos x})}dx$ is equal to:

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

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

1. A.g is decreasing in $\left(0,\frac{\pi }{4}\right)$
2. B.g' is increasing in $\left(0,\frac{\pi }{4}\right)$
3. C.g + g' is increasing in $\left(0,\frac{\pi }{2}\right)$
4. D.g - g' is increasing in $\left(0,\frac{\pi }{2}\right)$

If $b_n={\int }_0^{\frac{\pi }{2}}\frac{{\cos }^{2}nx}{\sin x}dx$, $n\in \mathbb{N}$, then

1. A.$b_3-b_2$, $b_4-b_3$, $b_5-b_4$ are in A.P. with common difference $-2$
2. B.$\frac{1}{b_3-b_2}$, $\frac{1}{b_4-b_3}$, $\frac{1}{b_5-b_4}$ are in an A.P. with common difference $2$
3. C.$b_3-b_2$, $b_4-b_3$, $b_5-b_4$ are in a G.P.
4. D.$\frac{1}{b_3-b_2}$, $\frac{1}{b_4-b_3}$, $\frac{1}{b_5-b_4}$ are in an A.P. with common difference $-2$

The value of the integral $${\int }_{-\pi /2}^{\pi /2}\frac{dx}{(1+e^x)({\sin }^{6}x+{\cos }^{6}x)}$$ is equal to

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

If the value of the integral $\underset{0}{\overset{5}{\int }}\frac{x+[x]}{e^{x-[x]}}dx=\alpha e^{-1}+\beta$, where $\alpha ,\beta \in \mathbb{R}$, $5\alpha +6\beta =0$, and $[x]$ denotes the greatest integer less than or equal to x; then the value of $(\alpha +\beta {)}^2$ is equal to :

1. A.100
2. B.25
3. C.16
4. D.36

Let y(x) be the solution of the differential equation $2x^{2}dy+(e^y-2x)dx=0$, $x>0$. If $y(e)=1$, then $y(1)$ is equal to :

1. A.0
2. B.2
3. C.${\log }_{e}2$
4. D.${\log }_e(2e)$