Answer
$\frac{2x+1}{2x^2+1}-\frac{x}{x^2+1}$
Work Step by Step
Step 1. Factor the denominator as: $(2x^2+1)(x^2+1)$
Step 2. Assume the end results as: $\frac{Ax+B}{2x^2+1}+\frac{Cx+D}{x^2+1}$
Step 3. Combine the above functions as:
$\frac{(Ax+B)(x^2+1)+(Cx+D)(2x^2+1)}{(2x^2+1)(x^2+1)}=\frac{(A+2C)x^3+(B+2D)x^2+(A+C)x+(B+D)}{(2x^2+1)(x^2+1)}$
Step 4. Compare the above result with the original expression to set up the following system of equations:
\begin{cases} A+2C=0 \\ B+2D=1\\A+C=1\\B+D=1 \end{cases}
Step 5. Use substitution to solve the above equations and get $A=2, B=1,C=-1,D=0$
Step 6. Write the final results as:
$\frac{2x+1}{2x^2+1}-\frac{x}{x^2+1}$
