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Calculadora de Decomposição em Frações Parciais
Encontre a decomposição em frações parciais passo a passo
Esta calculadora online encontrará a decomposição em frações parciais da função racional, com as etapas exibidas.
Solution
Your input: perform the partial fraction decomposition of $$$\frac{1}{u \left(u - 1\right)}$$$
The form of the partial fraction decomposition is
$$\frac{1}{u \left(u - 1\right)}=\frac{A}{u}+\frac{B}{u - 1}$$
Write the right-hand side as a single fraction:
$$\frac{1}{u \left(u - 1\right)}=\frac{u B + \left(u - 1\right) A}{u \left(u - 1\right)}$$
The denominators are equal, so we require the equality of the numerators:
$$1=u B + \left(u - 1\right) A$$
Expand the right-hand side:
$$1=u A + u B - A$$
Collect up the like terms:
$$1=u \left(A + B\right) - A$$
The coefficients near the like terms should be equal, so the following system is obtained:
$$\begin{cases} A + B = 0\\- A = 1 \end{cases}$$
Solving it (for steps, see system of equations calculator), we get that $$$A=-1$$$, $$$B=1$$$
Therefore,
$$\frac{1}{u \left(u - 1\right)}=\frac{-1}{u}+\frac{1}{u - 1}$$
Answer: $$$\frac{1}{u \left(u - 1\right)}=\frac{-1}{u}+\frac{1}{u - 1}$$$