Partial Fraction Decomposition Calculator

Your input: perform the partial fraction decomposition of $\frac{1}{x^{2} - 1}$

Factor the denominator: $\frac{1}{x^{2} - 1}=\frac{1}{\left(x - 1\right) \left(x + 1\right)}$

The form of the partial fraction decomposition is

$$\frac{1}{\left(x - 1\right) \left(x + 1\right)}=\frac{A}{x + 1}+\frac{B}{x - 1}$$

Write the right-hand side as a single fraction:

$$\frac{1}{\left(x - 1\right) \left(x + 1\right)}=\frac{\left(x - 1\right) A + \left(x + 1\right) B}{\left(x - 1\right) \left(x + 1\right)}$$

The denominators are equal, so we require the equality of the numerators:

$$1=\left(x - 1\right) A + \left(x + 1\right) B$$

Expand the right-hand side:

$$1=x A + x B - A + B$$

Collect up the like terms:

$$1=x \left(A + B\right) - A + B$$

The coefficients near the like terms should be equal, so the following system is obtained:

$$\begin{cases} A + B = 0\\- A + B = 1 \end{cases}$$

Solving it (for steps, see system of equations calculator), we get that $A=- \frac{1}{2}$, $B=\frac{1}{2}$

Therefore,

$$\frac{1}{\left(x - 1\right) \left(x + 1\right)}=\frac{- \frac{1}{2}}{x + 1}+\frac{\frac{1}{2}}{x - 1}$$

Answer: $\frac{1}{x^{2} - 1}=\frac{- \frac{1}{2}}{x + 1}+\frac{\frac{1}{2}}{x - 1}$