部分分数分解計算機

Your input: perform the partial fraction decomposition of $$$\frac{1}{\left(u^{2} - 1\right) \left(u^{2} + 1\right)}$$$

Simplify the expression: $$$\frac{1}{\left(u^{2} - 1\right) \left(u^{2} + 1\right)}=\frac{1}{u^{4} - 1}$$$

Factor the denominator: $$$\frac{1}{u^{4} - 1}=\frac{1}{\left(u - 1\right) \left(u + 1\right) \left(u^{2} + 1\right)}$$$

The form of the partial fraction decomposition is

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

Write the right-hand side as a single fraction:

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

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

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

Expand the right-hand side:

$$1=u^{3} A + u^{3} B + u^{3} D - u^{2} A + u^{2} C + u^{2} D + u A - u B + u D - A - C + D$$

Collect up the like terms:

$$1=u^{3} \left(A + B + D\right) + u^{2} \left(- A + C + D\right) + u \left(A - B + D\right) - A - C + D$$

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

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

Solving it (for steps, see system of equations calculator), we get that $$$A=- \frac{1}{4}$$$, $$$B=0$$$, $$$C=- \frac{1}{2}$$$, $$$D=\frac{1}{4}$$$

Therefore,

$$\frac{1}{\left(u - 1\right) \left(u + 1\right) \left(u^{2} + 1\right)}=\frac{- \frac{1}{4}}{u + 1}+\frac{- \frac{1}{2}}{u^{2} + 1}+\frac{\frac{1}{4}}{u - 1}$$

Answer: $$$\frac{1}{\left(u^{2} - 1\right) \left(u^{2} + 1\right)}=\frac{- \frac{1}{4}}{u + 1}+\frac{- \frac{1}{2}}{u^{2} + 1}+\frac{\frac{1}{4}}{u - 1}$$$