${\int }_0^1{e}^{e^x}\left(1+x{e}^x\right)dx$

Important Questions on Integrals

EASY

Mathematics>Integral Calculus>Integrals>Definite Integration

The value of $\underset{0}{\overset{2}{\int }}\frac{dx}{x^2-2x+2}$ is equal to

MEDIUM

Mathematics>Integral Calculus>Integrals>Definite Integration

The number of continuous function $f:\left[\mathrm{0,1}\right]\to \left[\mathrm{0,1}\right]$ such that $f\left(x\right)<x^2$ for all $x$ and $\underset{0}{\overset{1}{\int }}f\left(x\right)dx=\frac{1}{3}$ is:

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

$\underset{0}{\overset{1}{\int }}x{\tan }^{-1}xdx=$

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

For $x,t\in R \mathrm{l}\mathrm{e}\mathrm{t} p_t\left(x\right)=\left(\sin t\right)x^2-\left(2\cos t\right)x+\sin t$ be a family of quadratic polynomials in $x$ with variable coefficients. Let $A\left(t\right)={\int }_0^1{p}_t\left(x\right)dx$ . Which of the following statement are true?

(I) $A\left(t\right)<0$ for all $t$.

(II) $A\left(t\right)$ has infinitely many critical points.

(III) $A\left(t\right)=0$ for infinitely many $t$.

(IV) $A\text{'}\left(t\right)<0$ for all $t$

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

The value of $\underset{-\pi /2}{\overset{\pi /2}{\int }}\frac{dx}{\left[x\right]+\left[\sin x\right] + 4},$ where $\left[t\right]$ denotes the greatest integer less than or equal to $t,$ is

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

If $\underset{1}{\overset{2}{\int }}\frac{dx}{{\left(x^2-2x+4\right)}^{\frac{3}{2}}}=\frac{k}{k+5}$, then $k$ is equal to

EASY

Mathematics>Integral Calculus>Integrals>Definite Integration

The integral $\underset{\frac{\pi }{4}}{\overset{\frac{3\pi }{4}}{\int }}\frac{dx}{1+\cos x}$ is equal to

MEDIUM

Mathematics>Integral Calculus>Integrals>Definite Integration

A value of $\alpha$ such that $\underset{\alpha }{\overset{\alpha +1}{\int }}\frac{dx}{(x+\alpha )(x+\alpha +1)}=\mathrm{l}\mathrm{o}{\mathrm{g}}_e\left(\frac{9}{8}\right)$ is

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

The integral $\underset{1}{\overset{e}{\int }}\left\{{\left(\frac{x}{e}\right)}^{2x}-{\left(\frac{e}{x}\right)}^x\right\}\mathrm{l}\mathrm{o}{\mathrm{g}}_{e}xdx$ is equal to

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

For $x>0,$ let $f\left(x\right)={\int }_1^x\frac{\log t}{1+t} dt.$ Then $f\left(x\right)+f\left(\frac{1}{x}\right)$ is equal to 

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

Suppose that the earth is a sphere of radius $6400$ kilometers. The height from the earth's surface from where exactly a fourth of the earth's surface is visible, is

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

If $\underset{0}{\overset{k}{\int }}\frac{dx}{2+18x^2}=\frac{\pi }{24}$ , then the value of $k$ is

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

Let, the function $F$ be defined as $F\left(x\right)={\int }_1^x\frac{e^t}{t}dt, x>0,$ then the value of the integral ${\int }_1^x\frac{e^t}{t+a}dt,$ where $a>0,$ is

EASY

Mathematics>Integral Calculus>Integrals>Definite Integration

The integral $\underset{\frac{\pi }{12}}{\overset{\frac{\pi }{4}}{\int }}\frac{8\cos 2x}{{\left(\tan x+\cot x\right)}^3}dx$ equals

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

If $2\underset{0}{\overset{1}{\int }}{\tan }^{-1}xdx=\underset{0}{\overset{1}{\int }}{\cot }^{-1}\left(1-x+x^2\right)dx$, then $\underset{0}{\overset{1}{\int }}{\tan }^{-1}\left(1-x+x^2\right)dx$ is equal to

MEDIUM

Mathematics>Integral Calculus>Integrals>Definite Integration

The integral ${\int }_{\frac{\mathrm{\pi }}{6}}^{\frac{\mathrm{\pi }}{3}}{\mathrm{s}\mathrm{e}\mathrm{c}}^{\frac{2}{3}}x·\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}{\mathrm{c}}^{\frac{4}{3}}xdx$ is equal to

MEDIUM

Mathematics>Integral Calculus>Integrals>Definite Integration

If $\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\frac{\mathrm{c}\mathrm{o}\mathrm{t}x}{\mathrm{c}\mathrm{o}\mathrm{t}x+\mathrm{cosec}x}dx=m(\mathrm{\pi }+n),$ then $mn$ is equal to

MEDIUM

Mathematics>Integral Calculus>Integrals>Definite Integration

The number of continuous functions $f:\left[0,\mathrm{ }1\right]\to R$ that satisfy ${\int }_0^{1}xf\left(x\right)dx=\frac{1}{3}+\frac{1}{4}{\int }_0^1{\left(f\left(x\right)\right)}^{2}dx$ is

MEDIUM

Mathematics>Integral Calculus>Integrals>Definite Integration

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

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

$\underset{0}{\overset{1}{\int }}[f\left(x\right)g^{\text{'}\text{'}}\left(x\right)-f^{\text{'}\text{'}}\left(x\right)g\left(x\right)]dx$ is equal to 

[Given $f\left(0\right)=g\left(0\right)=0$]