Answer
$\frac{x+2}{x^2+1}-\frac{1}{x}$
Work Step by Step
Step 1. Factorize the denominator as: $x(x^2+1)$
Step 2. Assume the partial fraction decomposition is:
$\frac{A}{x}+\frac{Bx+C}{x^2+1}$
Step 3. Combine the rational functions above:
$\frac{A(x^2+1)+(Bx+C)(x)}{x(x^2+1)}=\frac{(A+B)x^2+(C)x+(A)}{x(x^2+1)}$
Step 4. Compare the above function with the original to setup the following system of equations:
$\begin{cases} A+B=0\\C=2\\A=-1 \end{cases}$
Step 5. Solve the above equations: $A=-1, B=1, C=2$
Step 6. The final result is:
$\frac{-1}{x}+\frac{x+2}{x^2+1}$ or $\frac{x+2}{x^2+1}-\frac{1}{x}$
