Integral Calculus & Area · JEE Main previous year questions

The number of integral values of 'k' for which the equation $3\sin x+4\cos x=k+1$ has a solution, $k\in \mathbb{R}$ is _________.

Let $f(x)={\int }_0^x{e}^{t}f(t)dt+e^x$ be a differentiable function for all $x\in \mathbb{R}$. Then f(x) equals :

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

For $x>0$, if $f(x)={\int }_1^x\frac{{\log }_{e}t}{(1+t)}dt$, then $f(e)+f\left(\frac{1}{e}\right)$ is equal to :

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

If $I_{m,n}={\int }_0^1{x}^{m-1}(1-x{)}^{n-1}dx$, for $m,n\ge 1$, and ${\int }_0^1\frac{x^{m-1}+x^{n-1}}{(1+x{)}^{m+n}}dx=\alpha I_{m,n},\alpha \in \mathbb{R}$, then $\alpha$ equals _________.

The integral $\int \frac{e^{3{\log }_{e}2x}+5e^{2{\log }_{e}2x}}{e^{4{\log }_{e}x}+5e^{3{\log }_{e}x}-7e^{2{\log }_{e}x}}dx$, $x>0$, is equal to : (where c is a constant of integration)

1. A.${\log }_e\sqrt{x^2+5x-7}+c$
2. B.$4{\log }_e|x^2+5x-7|+c$
3. C.$\frac{1}{4}{\log }_e|x^2+5x-7|+c$
4. D.${\log }_e|x^2+5x-7|+c$

If $I_n={\int }_{\frac{\pi }{4}}^{\frac{\pi }{2}}{\cot }^{n}xdx$, then :

1. A.$\frac{1}{I_2+I_4},\frac{1}{I_3+I_5},\frac{1}{I_4+I_6}$ are in A.P.
2. B.$I_2+I_4,I_3+I_5,I_4+I_6$ are in A.P.
3. C.$\frac{1}{I_2+I_4},\frac{1}{I_3+I_5},\frac{1}{I_4+I_6}$ are in G.P.
4. D.$I_2+I_4,(I_3+I_5{)}^2,I_4+I_6$ are in G.P.

The value of ${\int }_{-2}^2|3x^2-3x-6|dx$ is ______ .

A line 'l' passing through origin is perpendicular to the lines $l_1:\vec{r}=(3+t)\hat{i}+(-1+2t)\hat{j}+(4+2t)\hat{k}$ $l_2:\vec{r}=(3+2s)\hat{i}+(3+2s)\hat{j}+(2+s)\hat{k}$ If the co-ordinates of the point in the first octant on '$l_2$' at a distance of $\sqrt{17}$ from the point of intersection of 'l' and '$l_1$' are (a, b, c) then 18(a + b + c) is equal to ______ .

If the curve, $y=y(x)$ represented by the solution of the differential equation $(2x{y}^2-y)dx+xdy=0$, passes through the intersection of the lines, $2x-3y=1$ and $3x+2y=8$, then $|y(1)|$ is equal to ______ .

The general solution of the differential equation $\sqrt{1+x^2+y^2+x^2{y}^2}+xy\frac{dy}{dx}=0$ is : (where C is a constant of integration)

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

The area (in sq. units) of the region $\mathrm{A}=\{(x,y):|x|+|y|\le 1,2y^2\ge |x|\}$

1. A.$\frac{1}{6}$
2. B.$\frac{5}{6}$
3. C.$\frac{1}{3}$
4. D.$\frac{7}{6}$

The integral ${\int }_1^2{e}^x\cdot x^x(2+{\log }_{e}x)dx$ equals :

1. A.$e(4e+1)$
2. B.$e(2e-1)$
3. C.$e(4e-1)$
4. D.$4e^2-1$

The angle of elevation of the summit of a mountain from a point on the ground is $45^{\circ }$. After climbing up one km towards the summit at an inclination of $30^{\circ }$ from the ground, the angle of elevation of the summit is found to be $60^{\circ }$. Then the height (in km) of the summit from the ground is :

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

The area (in sq. units) of the region $\{(x,y):0\le y\le x^2+1,0\le y\le x+1,\frac{1}{2}\le x\le 2\}$ is :

1. A.$\frac{79}{16}$
2. B.$\frac{79}{24}$
3. C.$\frac{23}{6}$
4. D.$\frac{23}{16}$

${\int }_{-\pi }^{\pi }|\pi -|x||dx$ is equal to :

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

If the number of integral terms in the expansion of $(3^{1/2}+5^{1/8}{)}^n$ is exactly 33, then the least value of n is :

1. A.264
2. B.256
3. C.128
4. D.248

If the value of the integral ${\int }_0^{\frac{1}{2}}\frac{x^2}{(1-x^2{)}^{\frac{3}{2}}}dx$ is $\frac{k}{6}$, then k is equal to :

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

If $\int {\sin }^{-1}\left(\sqrt{\frac{x}{1+x}}\right)dx=\mathrm{A}(x){\tan }^{-1}(\sqrt{x})+\mathrm{B}(x)+\mathrm{C}$, where C is a constant of integration, then the ordered pair $(\mathrm{A}(x),\mathrm{B}(x))$ can be :

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

If $x^{3}dy+xydx=x^{2}dy+2ydx$; $y(2)=e$ and $x>1$, then $y(4)$ is equal to :

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

The integral ${\int }_0^2||x-1|-x|dx$ is equal to_______.

The number of integral values of k for which the line, $3x+4y=k$ intersects the circle, $x^2+y^2-2x-4y+4=0$ at two distinct points is _______.

Consider a region $R=\{(x,y)\in \mathbb{R}:x^2\le y\le 2x\}$. If a line $y=\alpha$ divides the area of region $R$ into two equal parts, then which of the following is true?

1. A.$3{\alpha }^2-8\alpha +8=0$
2. B.${\alpha }^3-6{\alpha }^{3/2}-16=0$
3. C.$3{\alpha }^2-8{\alpha }^{3/2}+8=0$
4. D.${\alpha }^3-6{\alpha }^2+16=0$

If a curve $y=f(x)$, passing through the point $(1,2)$, is the solution of the differential equation, $2x^{2}dy=(2xy+y^2)dx$, then $f\left(\frac{1}{2}\right)$ is equal to :

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

The value of $${\int }_0^{2\pi }\frac{x{\sin }^{8}x}{{\sin }^{8}x+{\cos }^{8}x}dx$$ is equal to :

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

The integral $\int \frac{dx}{(x+4{)}^{\frac{8}{7}}(x-3{)}^{\frac{6}{7}}}$ is equal to : (where C is a constant of integration)

1. A.$\frac{1}{2}{\left(\frac{x-3}{x+4}\right)}^{\frac{3}{7}}+C$
2. B.${\left(\frac{x-3}{x+4}\right)}^{\frac{1}{7}}+C$
3. C.$-\frac{1}{13}{\left(\frac{x-3}{x+4}\right)}^{\frac{13}{7}}+C$
4. D.$-{\left(\frac{x-3}{x+4}\right)}^{-\frac{1}{7}}+C$