Partial Fraction Decomposition Calculator

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

Simplify the expression: $\frac{1}{1 - u^{2}}=\frac{-1}{u^{2} - 1}$

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