Mathematics

# Resolve $\displaystyle \frac{x^4-x^2+1}{x^2(x^2+1)^2}$ into partial fractions.

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

##### SOLUTION
The given fraction has $x^2$ everywhere then put $x^2=t$ for the sake of partial fractions.
$\therefore \displaystyle \frac{x^4-x^2+1}{x^2(x^2+1)^2}=\frac{t^2-t+1}{t(t+1)^2}$

Now let         $\displaystyle \frac{t^2-t+1}{t(t+1)^2} = \frac{A}{t}+\frac{B}{t+1}=\frac{C}{(t+1)^2}$                       ...(i)

$\Longrightarrow t^2-t+1=A(t+1)^2+Bt(t+1)+Ct$               ...(ii)
Substituting $t+1=0$ or $t=-1$ in (ii), we ontain
$\displaystyle (-1)^2-(-1)+1=0+0+C(-1)$
$\Longrightarrow C=-3$
Substituting $t=0$ in (ii) we obtain
$\displaystyle 0-0+1=A(0+1)^2+0+0$
$\Longrightarrow A=1$
Now equating the coefficient of $t^2$ on both sides of (ii) then
$1=A+B$
$\Longrightarrow 1=1+B$
$\therefore B=0$
Substituting the values of A, B and C in (i) we get
$\displaystyle \frac{t^2-t+1}{t(t+1)^2} = \frac{1}{t}-\frac{3}{(t+1)^2}$
In last substitution $t=x^2$
then             $\displaystyle \frac{x^4-x^2+1}{x^2(x^2+1)^2} =\frac{1}{x^2}-\frac{3}{(x^2+1)^2}$
which are the required partial fractions.
Hence, option 'D' is correct.

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

#### Realted Questions

Q1 Single Correct Hard
$\int 32x^3(log x)^2dx$ is equal to.
• A. $8x^4(log x)^2+c$
• B. $x^4{8(log x)^2-4 log x }+c$
• C. $x^3{(log x)^2+2log x}+c$
• D. $x^4({8(log x)^2-4log x+1})+c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\int x^{\dfrac{13}{2}}.(1+x^{\dfrac{5}{2}})^{\dfrac{1}{2}}dx$
• A. $\dfrac{2}{5}(\dfrac{(1+x^\cfrac32)^3}{2}+\dfrac{(1+x^\cfrac52)}{2}-\dfrac{(1+x^\cfrac52)^2}{2})$
• B. $\dfrac{2}{5}(\dfrac{(1+x^\cfrac72)^3}{2}+\dfrac{(1+x^\cfrac52)}{2}-\dfrac{(1+x^\cfrac52)^2}{2})$
• C. $\dfrac{2}{5}(\dfrac{(1+x^\cfrac92)^3}{2}+\dfrac{(1+x^\cfrac52)}{2}-\dfrac{(1+x^\cfrac52)^2}{2})$
• D. $\dfrac{2}{5}(\dfrac{(1+x^\cfrac52)^3}{2}+\dfrac{(1+x^\cfrac52)}{2}-\dfrac{(1+x^\cfrac52)^2}{2})$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Find the derivative of $x^2$ by first principle

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
$\displaystyle \int_{0}^{1}\log \sin \left ( \frac{\pi }{2}x \right )dx=k \log \frac{1}{2}.$ Find the value of $k$.

Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$