The value of ${\int }_0^{1}x\sqrt{1-x}dx$ is equal to

Important Questions on Indefinite Integration

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Methods of 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>Indefinite Integration>Methods of Integration

If $\int \frac{\log \left(t+\sqrt{1+t^2}\right)}{\sqrt{1+t^2}}dt=\frac{1}{2}{\left(g\left(t\right)\right)}^2+c$, where $c$ is a constant, then $g\left(2\right)$, is equal to

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Methods of 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

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

If $\int \frac{\tan x}{1+\tan x+{\tan }^{2}x}dx=x-\frac{K}{\sqrt{A}}{\tan }^{-1}\left(\frac{K\tan x+1}{\sqrt{A}}\right)+C$, ($C$ is a constant of integration), then the ordered pair $\left(K,A\right)$ is equal to

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

If $\int x^5{e}^{-4x^3}dx=\frac{1}{48}{e}^{-4x^3}f\left(x\right)+C$, where $C$ is a constant of integration, then $f\left(x\right)$ is equal to

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

Let, $I_n=\int {\tan }^{n}xdx\left(n>1\right)$ . If $I_4+I_6=a{\tan }^{5}x+b{x}^5+c$, then the ordered pair $\left(a,b\right)$, is equal to

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

The integral $\int \frac{dx}{{\left(x+1\right)}^{\frac{3}{4}}{\left(x-2\right)}^{\frac{5}{4}}}$, is equal to

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

If $f\left(\frac{3x-4}{3x+4}\right)=x+2, x\ne -\frac{4}{3}$, and $\int f\left(x\right)dx=A\log \left|1-x\right|+Bx+C$ , then the ordered pair $\left(A, B\right)$ is equal to

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

The integral $\int \frac{3x^{13}+2x^{11}}{{\left(2x^4+3x^2+1\right)}^4}dx$, is equal to 

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

If $f\left(x\right)=\int \frac{\left(5x^8+7x^6\right)}{{\left(x^2+1+2x^7\right)}^2}dx$, $\left(x\ge 0\right),$ and $f\left(0\right)=0,$ then the value of $f\left(1\right)$ is

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

$\int \frac{{\sec }^{8}x}{\mathrm{cosec}x}dx=$

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

Let, $n\ge 2$ be a natural number and $0<\theta <\frac{\pi }{2}.$ Then $\int \frac{{\left(\mathrm{s}\mathrm{i}{\mathrm{n}}^n\theta -\mathrm{s}\mathrm{i}\mathrm{n}\theta \right)}^{\frac{1}{n}}\mathrm{c}\mathrm{o}\mathrm{s}\theta }{\mathrm{s}\mathrm{i}{\mathrm{n}}^{n+1}\theta }d\theta ,$ is equal to

EASY

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

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

EASY

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

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

EASY

Mathematics>Integral Calculus>Indefinite Integration>Methods of 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>Indefinite Integration>Methods of 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>Indefinite Integration>Methods of 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 $t$

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

If $\int f\left(x\right)dx=\psi \left(x\right)$, then $\int x^{5}f\left(x^3\right)dx$, is equal to 

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of 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

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

The integral $\int \frac{dx}{x^2{\left(x^4+1\right)}^{\frac{3}{4}}}$ equals to