Calculatrice de décomposition de fractions partielles

Trouver des fractions partielles étape par étape

Cette calculatrice en ligne permet de trouver la décomposition en fractions partielles de la fonction rationnelle, avec les étapes indiquées.

Your input: perform the partial fraction decomposition of $\frac{1}{1 - x^{2}}$

Simplify the expression: $\frac{1}{1 - x^{2}}=\frac{-1}{x^{2} - 1}$

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

The form of the partial fraction decomposition is

$\frac{-1}{\left(x - 1\right) \left(x + 1\right)}=\frac{A}{x + 1}+\frac{B}{x - 1}$

Write the right-hand side as a single fraction:

$\frac{-1}{\left(x - 1\right) \left(x + 1\right)}=\frac{\left(x - 1\right) A + \left(x + 1\right) B}{\left(x - 1\right) \left(x + 1\right)}$

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

$-1=\left(x - 1\right) A + \left(x + 1\right) B$

Expand the right-hand side:

$-1=x A + x B - A + B$

Collect up the like terms:

$-1=x \left(A + B\right) - A + B$

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

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

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

Therefore,

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

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