Chapter 11, Problem 67RE

To determine

To calculate: the partial function decomposition of each rational expression.

Answer to Problem 67RE

Solution: $\frac{1}{4(x-1)}-\frac{1}{4(x+1)}+\frac{1}{2(x^2+1)}$

Explanation of Solution

  $\frac{x^2}{(x^2+1)(x^2-1)}$

  $\begin{array}{l}=\frac{1}{2}\left[\frac{x^2+1+x^2-1}{(x^2+1)(x^2-1)}\right]\\ =\frac{1}{2}\left[\frac{1}{x^2+1}+\frac{1}{x^2-1}\right]\\ =\frac{1}{2}\left[\frac{1}{(x+1)(x-1)}+\frac{1}{x^2-1}\right]\\ =\frac{1}{2}\left[\frac{(x+1)-(x-1)}{2(x+1)(x-1)}+\frac{1}{x^2-1}\right]\end{array}$

  $=\frac{1}{4(x-1)}-\frac{1}{4(x+1)}+\frac{1}{2(x^2+1)}$