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 $$$\frac{1}{x^{2} + x}$$$
Simplify the expression: $$$\frac{1}{x^{2} + x}=\frac{1}{x \left(x + 1\right)}$$$
The form of the partial fraction decomposition is
$$\frac{1}{x \left(x + 1\right)}=\frac{A}{x}+\frac{B}{x + 1}$$
Write the right-hand side as a single fraction:
$$\frac{1}{x \left(x + 1\right)}=\frac{x B + \left(x + 1\right) A}{x \left(x + 1\right)}$$
The denominators are equal, so we require the equality of the numerators:
$$1=x B + \left(x + 1\right) A$$
Expand the right-hand side:
$$1=x A + x B + A$$
Collect up the like terms:
$$1=x \left(A + B\right) + A$$
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,
$$\frac{1}{x \left(x + 1\right)}=\frac{1}{x}+\frac{-1}{x + 1}$$
Answer: $$$\frac{1}{x^{2} + x}=\frac{1}{x}+\frac{-1}{x + 1}$$$