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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 $$$\frac{1}{v \left(v - 1\right)}$$$
The form of the partial fraction decomposition is
$$\frac{1}{v \left(v - 1\right)}=\frac{A}{v}+\frac{B}{v - 1}$$
Write the right-hand side as a single fraction:
$$\frac{1}{v \left(v - 1\right)}=\frac{v B + \left(v - 1\right) A}{v \left(v - 1\right)}$$
The denominators are equal, so we require the equality of the numerators:
$$1=v B + \left(v - 1\right) A$$
Expand the right-hand side:
$$1=v A + v B - A$$
Collect up the like terms:
$$1=v \left(A + B\right) - A$$
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,
$$\frac{1}{v \left(v - 1\right)}=\frac{-1}{v}+\frac{1}{v - 1}$$
Answer: $$$\frac{1}{v \left(v - 1\right)}=\frac{-1}{v}+\frac{1}{v - 1}$$$