部分分式分解计算器

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

Simplify the expression: $$$\frac{1}{u^{2} - 2 u}=\frac{1}{u \left(u - 2\right)}$$$

The form of the partial fraction decomposition is

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

Write the right-hand side as a single fraction:

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

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

$$1=u B + \left(u - 2\right) A$$

Expand the right-hand side:

$$1=u A + u B - 2 A$$

Collect up the like terms:

$$1=u \left(A + B\right) - 2 A$$

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

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

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

Therefore,

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

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