|
|
|
|
|
|
|
|
|
|
|
|
Partial Fraction Decomposition Calculator
Find partial fractions step by step
This online calculator will find the partial fraction decomposition of the rational function, with steps shown.
Solution
Your input: perform the partial fraction decomposition of $$$\frac{1}{\left(x^{2} - 1\right)^{2}}$$$
Factor the denominator: $$$\frac{1}{\left(x^{2} - 1\right)^{2}}=\frac{1}{\left(x - 1\right)^{2} \left(x + 1\right)^{2}}$$$
The form of the partial fraction decomposition is
$$\frac{1}{\left(x - 1\right)^{2} \left(x + 1\right)^{2}}=\frac{A}{x + 1}+\frac{B}{\left(x + 1\right)^{2}}+\frac{C}{x - 1}+\frac{D}{\left(x - 1\right)^{2}}$$
Write the right-hand side as a single fraction:
$$\frac{1}{\left(x - 1\right)^{2} \left(x + 1\right)^{2}}=\frac{\left(x - 1\right)^{2} \left(x + 1\right) A + \left(x - 1\right)^{2} B + \left(x - 1\right) \left(x + 1\right)^{2} C + \left(x + 1\right)^{2} D}{\left(x - 1\right)^{2} \left(x + 1\right)^{2}}$$
The denominators are equal, so we require the equality of the numerators:
$$1=\left(x - 1\right)^{2} \left(x + 1\right) A + \left(x - 1\right)^{2} B + \left(x - 1\right) \left(x + 1\right)^{2} C + \left(x + 1\right)^{2} D$$
Expand the right-hand side:
$$1=x^{3} A + x^{3} C - x^{2} A + x^{2} B + x^{2} C + x^{2} D - x A - 2 x B - x C + 2 x D + A + B - C + D$$
Collect up the like terms:
$$1=x^{3} \left(A + C\right) + x^{2} \left(- A + B + C + D\right) + x \left(- A - 2 B - C + 2 D\right) + A + B - C + D$$
The coefficients near the like terms should be equal, so the following system is obtained:
$$\begin{cases} A + C = 0\\- A + B + C + D = 0\\- A - 2 B - C + 2 D = 0\\A + B - C + D = 1 \end{cases}$$
Solving it (for steps, see system of equations calculator), we get that $$$A=\frac{1}{4}$$$, $$$B=\frac{1}{4}$$$, $$$C=- \frac{1}{4}$$$, $$$D=\frac{1}{4}$$$
Therefore,
$$\frac{1}{\left(x - 1\right)^{2} \left(x + 1\right)^{2}}=\frac{\frac{1}{4}}{x + 1}+\frac{\frac{1}{4}}{\left(x + 1\right)^{2}}+\frac{- \frac{1}{4}}{x - 1}+\frac{\frac{1}{4}}{\left(x - 1\right)^{2}}$$
Answer: $$$\frac{1}{\left(x^{2} - 1\right)^{2}}=\frac{\frac{1}{4}}{x + 1}+\frac{\frac{1}{4}}{\left(x + 1\right)^{2}}+\frac{- \frac{1}{4}}{x - 1}+\frac{\frac{1}{4}}{\left(x - 1\right)^{2}}$$$