Question

Write the partial fraction decomposition of each rational expression:

$\frac{1}{x(x^2+1)}$.

Short answer

The partial fraction decomposition is,

$\frac{1}{x(x^2+1)}=\frac{1}{x}+\frac{-x}{x^2+1}$.

Step-by-step solution

Step 1. Given Information.

The rational expression is,

$\frac{1}{x(x^2+1)}$.

Step 2. Decompose the partial fractions.

Decompose the rationals expressions to get,

$\frac{1}{x(x^2+1)}=\frac{A}{x}+\frac{Bx+C}{x^2+1}........\left(1\right)$

Multiplying both sides by $x(x^2+1)$,

$$\begin{gathered} 1=A(x^2+1)+(Bx+C)x \\ 1=A{x}^2+A+B{x}^2+Cx \\ 1=(A+B)x^2+Cx+A.........\left(2\right) \end{gathered}$$

Equating the coefficients with like power, to get,

$$\begin{gathered} A=1 \\ B=-1 \\ C=0 \end{gathered}$$

The partial fraction decomposition is,

$\frac{1}{x(x^2+1)}=\frac{1}{x}+\frac{-1x+0}{x^2+1}$

$=\frac{1}{x}+\frac{-x}{x^2+1}$

Step 3. Checking the solution.

Adding the rational expressions,

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

$$\begin{gathered} =\frac{x^2+1-x^2}{x(x^2+1)} \\ =\frac{1}{x(x^2+1)} \end{gathered}$$