The function $f\left(x\right)={\int }_0^x\sqrt{1-t^2}dt$ is such that

Important Questions on Definite Integration

HARD

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integrals

Let $f:\left[0, \infty \right)\to R$ be a continuous function such that $f\left(x\right)=1-2x+{\int }_0^x{e}^{x-t}f\left(t\right)dt$ for all $x\in \left[0,\infty \right).$ Then, which of the following statement(s) is (are) TRUE?

HARD

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integrals

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

EASY

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integrals

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>Definite Integration>Basics of Definite Integrals

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>Definite Integration>Basics of Definite Integrals

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>Definite Integration>Basics of Definite Integrals

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>Definite Integration>Basics of Definite Integrals

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

HARD

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integrals

Let $f^{\prime }\left(x\right)= \frac{192x^3}{2+{\sin }^4\pi x}$ for all $x\in \mathbb{ }R$ with $f\left(\frac{1}{2}\right)=0.$ If $m \le \underset{\frac{1}{2}}{\overset{1}{\int }}f\left(x\right)dx \le M,$ then the possible values of $m$ and $M$ are

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integrals

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>Definite Integration>Basics of Definite Integrals

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>Definite Integration>Basics of Definite Integrals

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>Definite Integration>Basics of Definite Integrals

The value of the integral $\underset{0}{\overset{\frac{1}{2}}{\int }}\frac{1+\sqrt{3}}{{\left({\left(x+1\right)}^2{\left(1-x\right)}^6\right)}^{\frac{1}{4}}} dx$ is

HARD

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integrals

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

HARD

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integrals

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>Definite Integration>Basics of Definite Integrals

Let $F:R\to R$ be a thrice differentiable function. Suppose that $F\left(1\right)=0, F\left(3\right)=-4$ and $F^{\prime }\left(x\right)<0$ for all $x\in \left(\frac{1}{2}, 3\right).$ Let $f\left(x\right)=xF\left(x\right)$ for all $x\in R$. If $\underset{1}{\overset{3}{\int }}{x}^2{F}^{\prime }\left(x\right)dx=-12$ and $\underset{1}{\overset{3}{\int }}{x}^3 F^{\prime \prime }\left(x\right)dx=40,$ then the correct expression(s) is(are)

HARD

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integrals

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

HARD

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integrals

Let $f\left(x\right)=7{\tan }^{8}x+7{\tan }^{6}x-3{\tan }^{4}x-3{\tan }^{2}x$ for all $x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right).$ Then the correct expression(s) is(are)

EASY

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integrals

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

HARD

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integrals

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 

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integrals

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