Answer
$[\frac{1}{2}][\frac{-1}{x+1} + \frac{1}{x-1}]$
Work Step by Step
We must first find the Partial Fraction Decomposition:
$\frac{A}{x+1} + \frac{B}{x-1} = \frac{1}{(x+1)(x-1)}$
We must then solve for the constants:
$A(x-1) + B(x+1) = 1$
$Ax - A + Bx + B = 1$
$Ax + Bx = 0$
$-A + B = 1$
This can be represented with the following matrix:
$\begin{bmatrix}
1 & 1 & |0\\
-1 & 1 & |1\\
\end{bmatrix}$ ~ $\begin{bmatrix}
1 & 1 & |0\\
0 & 2 & |1\\
\end{bmatrix}$
$2B = 1$
$B = \frac{1}{2}$
$A + \frac{1}{2} = 0$
$A = \frac{-1}{2}$
The Partial Fraction is:
$[\frac{1}{2}][\frac{-1}{x+1} + \frac{1}{x-1}]$
Checking the Result:
$[\frac{1}{2}]\frac{-x+1+x+1}{(x+1)(x-1)} = [\frac{1}{2}]\frac{2}{(x+1)(x-1)} = \frac{1}{(x+1)(x-1)}$
Therefore, the answer is correct.
