Integral Calculus & Area · JEE Main previous year questions

The area enclosed by the curves $xy+4y=16$ and $x+y=6$ is equal to :

1. A.$28-30{\log }_{e}2$
2. B.$30-28{\log }_{e}2$
3. C.$30-32{\log }_{e}2$
4. D.$32-30{\log }_{e}2$

If $x=x(t)$ is the solution of the differential equation $(t+1)dx=\left(2x+(t+1{)}^4\right)dt,x(0)=2$, then, $x(1)$ equals ____.

Let S be the set of positive integral values of $a$ for which $\frac{a{x}^2+2(a+1)x+9a+4}{x^2-8x+32}<0,\forall x\in \mathbb{R}$. Then, the number of elements in S is :

1. A.0
2. B.$\infty$
3. C.3
4. D.1

The area of the region $\left\{(x,y):y^2\le 4x,x<4,\frac{xy(x-1)(x-2)}{(x-3)(x-4)}>0,x\ne 3\right\}$ is

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

If the integral $525{\int }_0^{\frac{\pi }{2}}\sin 2x{\cos }^{\frac{11}{2}}x{\left(1+{\mathrm{Cos}}^{\frac{5}{2}}x\right)}^{\frac{1}{2}}dx$ is equal to $(n\sqrt{2}-64)$, then $n$ is equal to ____.

Let $f:\mathbb{R}\to \mathbb{R}$ be a function defined by $f(x)=\frac{4^x}{4^x+2}$ and $M={\int }_{f(a)}^{f(1-a)}x{\sin }^4(x(1-x))dx,N={\int }_{f(a)}^{f(1-a)}{\sin }^4(x(1-x))dx;a\ne \frac{1}{2}$. If $\alpha M=\beta N,\alpha ,\beta \in \mathbb{N}$, then the least value of ${\alpha }^2+{\beta }^2$ is equal to ____.

$\left|\frac{120}{{\pi }^3}{\int }_0^{\pi }\frac{x^2\sin x\cos x}{{\sin }^{4}x+{\cos }^{4}x}dx\right|$ is equal to ____.

The value of $9{\int }_0^9\left[\sqrt{\frac{10x}{x+1}}\right]dx$, where $[t]$ denotes the greatest integer less than or equal to $t$, is _________.

Number of integral terms in the expansion of ${\left\{7^{\left(\frac{1}{2}\right)}+11^{\left(\frac{1}{6}\right)}\right\}}^{824}$ is equal to _________.

If the value of the integral ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\left(\frac{x^2\cos x}{1+{\pi }^x}+\frac{1+{\sin }^{2}x}{1+e^{{\sin }^{2023}x}}\right)\mathrm{d}x=\frac{\pi }{4}(\pi +a)-2$, then the value of $a$ is

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

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

1. A.$2\sec \theta$
2. B.$8\mathrm{cosec}(2\theta )$
3. C.$4\mathrm{cosec}(2\theta )$
4. D.$4\sec \theta$

Let the area of the region $\{(x,y):0\le x\le 3,0\le y\le \min \{x^2+2,2x+2\}\}$ be A. Then 12 A is equal to _________.

If ${\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}\sqrt{1-\sin 2x}dx=\alpha +\beta \sqrt{2}+\gamma \sqrt{3}$, where $\alpha ,\beta$ and $\gamma$ are rational numbers, then $3\alpha +4\beta -\gamma$ is equal to _________.

If $\underset{0}{\overset{1}{\int }}\frac{1}{\sqrt{3+x}+\sqrt{1+x}}\mathrm{d}x=a+b\sqrt{2}+c\sqrt{3}$, where a, b, c are rational numbers, then $2a+3b-4c$ is equal to :

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

Let the area of the region $\{(x,y):x-2y+4\ge 0,x+2y^2\ge 0,x+4y^2\le 8,y\ge 0\}$ be $\frac{m}{n}$, where $m$ and $n$ are coprime numbers. Then $m+n$ is equal to ________.

The least positive integral value of $\alpha$, for which the angle between the vectors $\alpha \hat{i}-2\hat{j}+2\hat{k}$ and $\alpha \hat{i}+2\alpha \hat{j}-2\hat{k}$ is acute, is ________.

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

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

For $0<a<1$, the value of the integral ${\int }_0^{\pi }\frac{\mathrm{d}x}{1-2a\cos x+a^2}$ is :

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

If the area of the region $\left\{(x,y):0\le y\le \min \left\{2x,6x-x^2\right\}\right\}$ is A, then $12\text{ A}$ is equal to _______.

Let for $x\in \mathbb{R}$, $S_0(x)=x$, $S_k(x)=C_{k}x+k\underset{0}{\overset{x}{\int }}{S}_{k-1}(t)dt$, where $C_0=1$, $C_k=1-\underset{0}{\overset{1}{\int }}{S}_{k-1}(x)dx$, $k=1,2,3,\dots$ Then $S_2(3)+6C_3$ is equal to ___________.

The area of the region $\left\{(x,y):x^2\le y\le |x^2-4|,y\ge 1\right\}$ is

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

The value of $\frac{e^{-\frac{\pi }{4}}+\underset{0}{\overset{\frac{\pi }{4}}{\int }}{e}^{-x}{\tan }^{50}xdx}{\underset{0}{\overset{\frac{\pi }{4}}{\int }}{e}^{-x}({\tan }^{49}x+{\tan }^{51}x)dx}$ is

1. A.$51$
2. B.$50$
3. C.$25$
4. D.$49$

Let the lines $l_1:\frac{x+5}{3}=\frac{y+4}{1}=\frac{z-\alpha }{-2}$ and $l_2:3x+2y+z-2=0=x-3y+2z-13$ be coplanar. If the point $\mathrm{P}(a,b,c)$ on $l_1$ is nearest to the point $\mathrm{Q}(-4,-3,2)$, then $|a|+|b|+|c|$ is equal to

1. A.$12$
2. B.$14$
3. C.$10$
4. D.$8$

Let ${\mathrm{D}}_k=\left|\begin{array}{ccc}1& 2k& 2k-1\\ n& n^2+n+2& n^2\\ n& n^2+n& n^2+n+2\end{array}\right|$. If ${\sum }_{k=1}^n{\mathrm{D}}_k=96$, then $n$ is equal to _________.

If ${\int }_{-0.15}^{0.15}|100x^2-1|dx=\frac{k}{3000}$, then $k$ is equal to _________.