Mathematics

# Solve : $\displaystyle \int \dfrac{x^4dx}{(1+x^2)^3}$ ?

##### SOLUTION
Given the integral,
$\int { \dfrac { { x }^{ 4 }dx }{ { ({ x }^{ 2 }+1) }^{ 3 } } }$
using partial fraction we get,
$=\int { \left( \dfrac { 1 }{ { x }^{ 2 }+1 } -\dfrac { 2 }{ { ({ x }^{ 2 }+1) }^{ 2 } } +\dfrac { 1 }{ { ({ x }^{ 2 }+1) }^{ 3 } } \right) } \\ =\int { \dfrac { 1 }{ { x }^{ 2 }+1 } } dx-2\int { \dfrac { 1 }{ { ({ x }^{ 2 }+1) }^{ 2 } } } dx+\int { \dfrac { 1 }{ { ({ x }^{ 2 }+1) }^{ 3 } } } dx$
Here,
$\int { \dfrac { 1 }{ { x }^{ 2 }+1 } } dx\\ =\tan^{-1} { (x) }$
For,
$\int { \dfrac { 1 }{ { ({ x }^{ 2 }+1) }^{ 2 } } } dx$
applying reduction formula we get,
$=\dfrac { x }{ 2({ x }^{ 2 }+1) } +\dfrac { 1 }{ 2 } \int { \dfrac { 1 }{ { x }^{ 2 }+1 } } dx\\ =\dfrac { \tan^{-1} { (x) } }{ 2 } +\dfrac { x }{ 2({ x }^{ 2 }+1) }$
Again for,
$\int { \dfrac { 1 }{ { ({ x }^{ 2 }+1) }^{ 3 } } } dx$
applying reduction formula we get,
$=\dfrac { x }{ 4{ ({ x }^{ 2 }+1) }^{ 2 } } +\dfrac { 3 }{ 4 } \int { \dfrac { 1 }{ { ({ x }^{ 2 }+1) }^{ 2 } } } dx\\ =\dfrac { 3\tan^{-1} { (x) } }{ 8 } +\dfrac { 3x }{ 8({ x }^{ 2 }+1) } +\dfrac { x }{ 4{ ({ x }^{ 2 }+1) }^{ 2 } } \\ \therefore \int { \dfrac { 1 }{ { x }^{ 2 }+1 } } dx-2\int { \dfrac { 1 }{ { ({ x }^{ 2 }+1) }^{ 2 } } } dx+\int { \dfrac { 1 }{ { ({ x }^{ 2 }+1) }^{ 3 } } } dx\\ =\dfrac { 3\tan^{-1} { (x) } }{ 8 } -\dfrac { 5x }{ 8({ x }^{ 2 }+1) } +\dfrac { x }{ 4{ ({ x }^{ 2 }+1) }^{ 2 } }$
Hence,
$\int { \dfrac { { x }^{ 4 }dx }{ { ({ x }^{ 2 }+1) }^{ 3 } } } \\ =\dfrac { 3\tan^{-1} { (x) } }{ 8 } -\dfrac { 5x }{ 8({ x }^{ 2 }+1) } +\dfrac { x }{ 4{ ({ x }^{ 2 }+1) }^{ 2 } } +C\\ =\dfrac { 3\tan^{-1} { (x) } }{ 8 } +\dfrac { -5{ x }^{ 3 }-3x }{ 8{ ({ x }^{ 2 }+1) }^{ 2 } } +C.$

Its FREE, you're just one step away

Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Medium
If $f(x)$ is an even function in $[-a, a]$ then $\displaystyle \int_{-a}^{+a}f(x) dx = ?$
• A. $0$
• B. $\displaystyle \int_{0}^{2a}[f(x)+f(a-x)]dx$
• C. $2$
• D. $2\displaystyle \int_{0}^{a}f(x)dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate:
$\displaystyle\int \dfrac{1}{n\sqrt{n^3-1}}dx$.

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Prove that $\displaystyle\int^{3\pi/4}_{\pi/4}\dfrac{dx}{(1+\cos x)}=2$.

1 Verified Answer | Published on 17th 09, 2020

Q4 Multiple Correct Hard
If $\varphi (t)$ = $\begin{cases}1,\,\,\,\,\, 0\le t < 1 \\ 0 \,\,\,\,\,\,\, \text{otherwise} \end{cases}$ then
$\displaystyle \int ^{3000}_{-3000} \begin{pmatrix} \displaystyle \sum_{r' = 2014}^{2016} \varphi (t-r')\varphi(t-2016)\end{pmatrix} dt =$
• A. $0$
• B. Does not exist
• C. A real number
• D. $1$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
$\int {\dfrac{{dx}}{{2\sqrt x (1 + x)}} = }$
• A. $\dfrac{1}{2}\tan (\sqrt x ) + C$
• B. $2\tan{^{ - 1}}(\sqrt x ) + C$
• C. $None\,of\,these$
• D. $\tan{^{ - 1}}(\sqrt x ) + C$