Partial Fraction Decomposition Calculator

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

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

The form of the partial fraction decomposition is

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

Write the right-hand side as a single fraction:

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

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

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

Expand the right-hand side:

$$1=u A + u B - A + B$$

Collect up the like terms:

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

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,

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

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