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

Important Questions on Indefinite Integration

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

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

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

The value of the integral $\int \frac{e^x\left(x^2+1\right)}{{\left(x+1\right)}^2}dx=$

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

If $\int \frac{1}{\left(x^2+4\right)\left(x^2+9\right)}dx=A{\tan }^{-1}\left(\frac{x}{2}\right)+B{\tan }^{-1}\left(\frac{x}{3}\right)+C,$ then $A-B=$

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

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

The integral $\int \left(1+x-\frac{1}{x}\right)e^{x+\frac{1}{x}}dx$, 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

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 

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

$\int \frac{x{e}^x}{{\left(1+x\right)}^2}dx$ is equal to: (where $C$ is an arbitarary constant)

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

If $\int \frac{1}{\sqrt{9-16x^2}}dx=\alpha {\sin }^{-1}\left(\beta x\right)+c,$ then $\alpha +\frac{1}{\beta }=$

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

The integral $\int \sqrt{1+2\cot x\left(\mathrm{cosec}x+\cot x\right)}dx, \left(0<x<\frac{\pi }{2}\right)$is equal to

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

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

(where $C$ is the constant of integration).

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

If $m$ is a non-zero number and $\int \frac{x^{5m-1}+2x^{4m-1}}{{\left(x^{2m}+x^m+1\right)}^3}dx=f\left(x\right)+c,$ then $f\left(x\right)$ is equal to

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 

EASY

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

The integral $\int \mathrm{c}\mathrm{o}\mathrm{s}\left(\mathrm{lnx}\right)dx$, is equal to 

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