Calculadora de descomposición en fracciones parciales

Your input: perform the partial fraction decomposition of $$$\frac{1}{x^{4} + 1}$$$

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

The form of the partial fraction decomposition is

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

Write the right-hand side as a single fraction:

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

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

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

Expand the right-hand side:

$$1=x^{3} A + x^{3} C - \sqrt{2} x^{2} A + x^{2} B + \sqrt{2} x^{2} C + x^{2} D + x A - \sqrt{2} x B + x C + \sqrt{2} x D + B + D$$

Collect up the like terms:

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

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

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

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

Therefore,

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

Answer: $$$\frac{1}{x^{4} + 1}=\frac{\frac{\sqrt{2} x}{4} + \frac{1}{2}}{x^{2} + \sqrt{2} x + 1}+\frac{- \frac{\sqrt{2} x}{4} + \frac{1}{2}}{x^{2} - \sqrt{2} x + 1}$$$