Definite Integrals — JEE Main PYQs

If ${\int }_0^{\frac{\pi }{3}}\frac{\tan \theta }{\sqrt{2\mathrm{k}\sec \theta }}\mathrm{d}\theta =1-\frac{1}{\sqrt{2}},(\mathrm{k}>0)$, then the value of k is:

The value of ${\int }_0^{\pi }|\cos x{|}^{3}dx$ is:

The value of ${\int }_0^{2\pi }[\sin 2x(1+\cos 3x)]dx$ where $[t]$ denotes the greatest integer function, is:

The value of ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\frac{{\sin }^{2}x}{1+2^x}\mathrm{d}x$ is :

The integral ${\int }_{\frac{\pi }{4}}^{\frac{3\pi }{4}}\frac{\mathrm{d}x}{1+\cos x}$ is equal to :

The integral ${\int }_2^4\frac{\log {\mathrm{x}}^2}{\log {\mathrm{x}}^2+\log \left(36-12\mathrm{x}+{\mathrm{x}}^2\right)}\mathrm{dx}$ is equal to :

The integral ${\int }_0^{\pi }\sqrt{1+4{\sin }^2\frac{x}{2}-4\sin \frac{x}{2}}dx$ equals

At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production $P$ w.r.t. additional number of workers $x$ is given by $\frac{dP}{dx}=100-12\sqrt{x}$. If the firm employs 25 more workers, then the new level of production of items is

Statement - I : The value of the integral ${\int }_{\pi /6}^{\pi /3}\frac{dx}{1+\sqrt{\tan x}}$ is equal to $\frac{\pi }{6}$.

Statement - II : ${\int }_a^{b}f(x)dx={\int }_a^{b}f(a+b-x)dx$.

If $g(x)={\int }_0^x\cos 4tdt$, then $g(x+\pi )$ equals

The value of ${\int }_0^1\frac{8\log (1+x)}{1+x^2}dx$ is

Let $p(x)$ be a function defined on $\mathbb{R}$ such that $p^{\prime }(x)=p^{\prime }(1-x)$, for all $x\in [0,1]$, $p(0)=1$ and $p(1)=41$. Then ${\int }_0^{1}p(x)dx$ equals

${\int }_0^{\pi }[\cot x]dx$, $[\bullet ]$ denotes the greatest integer function, is equal to

Let $\mathrm{I}={\int }_0^1\frac{\sin \mathrm{x}}{\sqrt{\mathrm{x}}}\mathrm{dx}$ and $\mathrm{J}={\int }_0^1\frac{\cos \mathrm{x}}{\sqrt{\mathrm{x}}}\mathrm{dx}$. Then which one of the following is true?

Let $F(x)=f(x)+f\left(\frac{1}{x}\right)$, where $f(x)={\int }_1^x\frac{\log t}{1+t}dt$. Then $F(e)$ equals

The solution for x of the equation
is

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

${\int }_0^{\pi }xf(\sin x)dx$ is equal to

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

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

If ${\mathrm{I}}_1={\int }_0^1{2}^{{\mathrm{x}}^2}\mathrm{dx},{\mathrm{I}}_2={\int }_0^1{2}^{{\mathrm{x}}^3}\mathrm{dx},{\mathrm{I}}_3={\int }_1^2{2}^{{\mathrm{x}}^2}\mathrm{dx}$ and ${\mathrm{I}}_4={\int }_1^2{2}^{{\mathrm{x}}^3}\mathrm{dx}$ then

Let $f:R\to R$ be a differentiable function having $f(2)=6,f^{\prime }(2)=\left(\frac{1}{48}\right)$. Then ${\lim }_{x\to 2}{\int }_6^{f(x)}\frac{4t^3}{x-2}dt$ equals

Let $f(x)$ be a non-negative continuous function such that the area bounded by the curve $y=f(x)$, $x$-axis and the ordinates $x=\frac{\pi }{4}$ and $x=\beta >\frac{\pi }{4}$ is $\left(\beta \sin \beta +\frac{\pi }{4}\cos \beta +\sqrt{2}\beta \right)$. Then …

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

${\lim }_{n\to \infty }{\sum }_{r=1}^n\frac{1}{n}{e}^{\frac{r}{n}}$ is

The value of ${\int }_{-2}^3\left|1-x^2\right|dx$ is

The value of $\mathrm{I}={\int }_0^{\pi /2}\frac{(\sin \mathrm{x}+\cos \mathrm{x}{)}^2}{\sqrt{1+\sin 2\mathrm{x}}}\mathrm{dx}$ is

If ${\int }_0^{\pi }xf(\sin x)dx=A{\int }_0^{\pi /2}f(\sin x)dx$, then $A$ is

If $f(x)=\frac{e^x}{1+e^x},{\mathrm{I}}_1={\int }_{\mathrm{f}(-\mathrm{a})}^{\mathrm{f}(\mathrm{a})}\mathrm{xg}\{\mathrm{x}(1-\mathrm{x})\}\mathrm{dx}$ and …