Chapter 11, Problem 63RE

To determine

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

Answer to Problem 63RE

Solution: $\frac{1}{10}(\frac{x+9}{x^2+9}-\frac{1}{x+1})$

Explanation of Solution

  $\frac{x}{(x^2+9)(x+1)}$

  $\begin{array}{l}=\frac{Ax+B}{x^2+9}+\frac{C}{x+1}\\ \Rightarrow x=A(x+B)(x+1)+C(x^2+9)\end{array}$

  $\begin{array}{l}a.A+C=0\\ b.B+9C=0\\ c.A+B=1\\ d.C-A\Rightarrow B-C=1\end{array}$

Subtract $d$ from $b$

  $\begin{array}{l}10C=-1\\ C=\frac{-1}{10}\\ B=\frac{9}{10}\\ A=\frac{1}{10}\end{array}$

  $\frac{1}{10}(\frac{x+9}{x^2+9}-\frac{1}{x+1})$