Integral Calculus & Area · JEE Main previous year questions

The value of integral, ${\int }_3^6\frac{\sqrt{x}}{\sqrt{9-x}+\sqrt{x}}dx$ is

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

${\int }_{-\frac{3\pi }{2}}^{-\frac{\pi }{2}}[(x+\pi {)}^3+{\cos }^2(x+3\pi )]dx$ is equal to

1. A.$\frac{{\pi }^4}{32}$
2. B.$\frac{{\pi }^4}{32}+\frac{\pi }{2}$
3. C.$\frac{\pi }{2}$
4. D.$\frac{\pi }{4}-1$

The value of ${\int }_1^a[x]f^{\prime }(x)dx,a>1$ where $[x]$ denotes the greatest integer not exceeding $x$ is

1. A.$af(a)-\{f(1)+f(2)+\cdots +f([a])\}$
2. B.$[a]f(a)-\{f(1)+f(2)+\cdots +f([a])\}$
3. C.$[a]f([a])-\{f(1)+f(2)+\cdots +f(a)\}$
4. D.$af([a])-\{f(1)+f(2)+\cdots +f(a)\}$

If $a^2+b^2+c^2=-2$ and $$f(x)=\left|\begin{array}{ccc}1+a^{2}x& (1+b^2)x& (1+c^2)x\\ (1+a^2)x& 1+b^{2}x& (1+c^2)x\\ (1+a^2)x& (1+b^2)x& 1+c^{2}x\end{array}\right|,$$ then $f(x)$ is a polynomial of degree :

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

$\int {\left\{\frac{\log x-1}{1+(\log x{)}^2}\right\}}^{2}dx$ is equal to

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

The area enclosed between the curve $y={\log }_e(x+e)$ and the coordinate axes is :

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

The value of ${\int }_{-\pi }^{\pi }\frac{{\cos }^{2}x}{1+a^x}dx$, $a>0$, is

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

If $x=e^{y+e^{y+\dots \infty }},x>0$, then $\frac{dy}{dx}$ is

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

If $\int \frac{\sin x}{\sin (x-\alpha )}dx=Ax+B\log \sin (x-\alpha ),+C$, then value of $(A,B)$ is

1. A.$(-\cos \alpha ,\sin \alpha )$
2. B.$(\cos \alpha ,\sin \alpha )$
3. C.$(-\sin \alpha ,\cos \alpha )$
4. D.$(\sin \alpha ,\cos \alpha )$

If $f(x)=\frac{e^x}{1+e^x},I_1={\int }_{f(-a)}^{f(a)}xg\{x(1-x)\}dx$ and $I_2={\int }_{f(-a)}^{f(a)}g\{x(1-x)\}dx$, then the value of $\frac{I_2}{I_1}$ is

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

The area of the region bounded by the curves $y=|x-2|,x=1,x=3$ and the $x$-axis is :

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

If $2a+3b+6c=0$, then at least one root of the equation $a{x}^2+bx+c=0$ lies in the interval

1. A.$(1,3)$
2. B.$(1,2)$
3. C.$(2,3)$
4. D.$(0,1)$

${\int }_{-\pi }^{\pi }\frac{2x(1+\sin x)}{1+{\cos }^{2}x}dx$ is

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

The area bounded by the curves $y=\ln x$, $y=\ln |x|$, $y=|\ln x|$ and $y=|\ln |x||$ is :

1. A.$4$ sq. units
2. B.$6$ sq. units
3. C.$10$ sq. units
4. D.none of these

If $|\vec{a}|=5,|\vec{b}|=4,|\vec{c}|=3$ thus what will be the value of $|\vec{a}.\vec{b}+\vec{b}.\vec{c}+\vec{c}.\vec{a}|$, given that $\vec{a}+\vec{b}+\vec{c}=0$ :

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

If the pair of lines $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ intersect on the $y$-axis then :

1. A.$2fgh=b{g}^2+c{h}^2$
2. B.$b{g}^2\ne c{h}^2$
3. C.$abc=2fgh$
4. D.none of these