Partial Fraction Decomposition Calculator

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

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

The form of the partial fraction decomposition is

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

Write the right-hand side as a single fraction:

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

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

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

Expand the right-hand side:

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

Collect up the like terms:

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

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

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

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

Therefore,

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

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