Mathematics

# Resolve into partial fraction $\displaystyle \frac{x^3-3x-2}{(x^2+x+1)(x+1)^2}$

$\displaystyle \frac{3x-1}{x^2+x+1}+\frac{2}{(x+1)^2} - \frac{3}{(x+1)}$

##### SOLUTION
Let $\displaystyle \frac { x^{ 3 }-3x-2 }{ \left( x^{ 2 }+x+1 \right) \left( x+1 \right) ^{ 2 } } =\frac { A }{ \left( x+1 \right) } +\frac { B }{ { \left( x+1 \right) }^{ 2 } } +\frac { Cx+D }{ \left( { x }^{ 2 }+x+1 \right) }$
$\Rightarrow x^{ 3 }-3x-2=A\left( x+1 \right) \left( { x }^{ 2 }+x+1 \right) +B\left( { x }^{ 2 }+x+1 \right) +\left( Cx+D \right) { \left( x+1 \right) }^{ 2 }\\ \Rightarrow x^{ 3 }-3x-2=A\left( { x }^{ 3 }+2{ x }^{ 2 }+2x+1 \right) +B\left( { x }^{ 2 }+x+1 \right) +C\left( { x }^{ 3 }+2{ x }^{ 2 }+x \right) +D\left( { x }^{ 2 }+2x+1 \right)$
On comparing coefficients
$A+C=1,2A+B+2C+D=0,2A+B+C+2D=-3,A+B+D=-2\\ \Rightarrow A=-3,B=2,C=3,D=-1$
Hence
$\displaystyle \frac { x^{ 3 }-3x-2 }{ \left( x^{ 2 }+x+1 \right) \left( x+1 \right) ^{ 2 } } =\frac { -3 }{ \left( x+1 \right) } +\frac { 2 }{ { \left( x+1 \right) }^{ 2 } } +\frac { 3x-1 }{ \left( { x }^{ 2 }+x+1 \right) }$
Hence, option 'C' is correct.

Its FREE, you're just one step away

Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Medium
For $x > 0$, let $f(x)=\displaystyle \int_{1}^{x}\dfrac {\log t}{1+t} \ dt$  then, the value of  $f(x)+f(1/x)$ will be
• A. $\dfrac {1}{4}\log x^{2}$
• B. $\log x$
• C. $\dfrac {1}{4}(\log x)^{2}$
• D. $\dfrac {1}{2}(\log x)^{2}$

1 Verified Answer | Published on 17th 09, 2020

Q2 One Word Medium
Evaluate $\displaystyle \int_{-1}^{1}\log \frac{2-x}{2+x}dx.$

1 Verified Answer | Published on 17th 09, 2020

Q3 Multiple Correct Hard
If $\displaystyle \int x\log \left ( 1+x^{2} \right )dx=\phi \left ( x \right ).\log \left ( 1+x^{2} \right )+\Psi \left ( x \right )+c$ then
• A. $\displaystyle \Psi \left ( x \right )=\frac{1+x^{2}}{2}$
• B. $\displaystyle \phi \left ( x \right )=-\frac{1+x^{2}}{2}$
• C. $\displaystyle \phi \left ( x \right )=\frac{1+x^{2}}{2}$
• D. $\displaystyle \Psi \left ( x \right )=-\frac{1+x^{2}}{2}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
$\displaystyle \int { \frac { \sin ^{ 3 }{ xdx } }{ \left( 1+\cos ^{ 2 }{ x } \right) \sqrt { 1+\cos ^{ 2 }{ x } +\cos ^{ 4 }{ x } } } dx }$ is equal to
• A. $\sec ^{ -1 }{ \left( \sec { x } -\cos { x } \right) } +c$
• B. $\sec ^{ -1 }{ \left( \cos { x } -\tan { x } \right) } +c$
• C. $\sec ^{ -1 }{ \left( \cos { x } +\tan { x } \right) } +c$
• D. $\sec ^{ -1 }{ \left( \sec { x } +\cos { x } \right) } +c$

Let $g(x)$ be a function defined on $[0, 7]$ and $g(x)=\int_0^x f(t) dt$, where $y=f(x)$ is the function whose graph is as shown in figure given below, then