Partial Fraction Decomposition Calculator

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

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

The form of the partial fraction decomposition is

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

Write the right-hand side as a single fraction:

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

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

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

Expand the right-hand side:

$$\frac{1}{2}=u A + u B - \frac{\sqrt{2} A}{2} + \frac{\sqrt{2} B}{2}$$

Collect up the like terms:

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

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

$$\begin{cases} A + B = 0\\- \frac{\sqrt{2} A}{2} + \frac{\sqrt{2} B}{2} = \frac{1}{2} \end{cases}$$

Solving it (for steps, see system of equations calculator), we get that $$$A=- \frac{\sqrt{2}}{4}$$$, $$$B=\frac{\sqrt{2}}{4}$$$

Therefore,

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

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