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Partial Fraction Decomposition Calculator

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1

Solved example of partial fraction decomposition

$fracciones\:parciales\:\frac{X-1}{X^2-4}$
2

Simplify $\sqrt{x^2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $\frac{1}{2}$

$\frac{x-1}{\left(x+\sqrt{4}\right)\left(\sqrt{x^2}-\sqrt{4}\right)}$
3

Calculate the power $\sqrt{4}$

$\frac{x-1}{\left(x+2\right)\left(\sqrt{x^2}-\sqrt{4}\right)}$

4

Simplify $\sqrt{x^2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $\frac{1}{2}$

$\frac{x-1}{\left(x+2\right)\left(x-\sqrt{4}\right)}$
5

Calculate the power $\sqrt{4}$

$\frac{x-1}{\left(x+2\right)\left(x- 2\right)}$
6

Multiply $-1$ times $2$

$\frac{x-1}{\left(x+2\right)\left(x-2\right)}$
7

Rewrite the fraction $\frac{x-1}{\left(x+2\right)\left(x-2\right)}$ in $2$ simpler fractions using partial fraction decomposition

$\frac{x-1}{\left(x+2\right)\left(x-2\right)}=\frac{A}{x+2}+\frac{B}{x-2}$
8

Find the values for the unknown coefficients: $A, B$. The first step is to multiply both sides of the equation from the previous step by $\left(x+2\right)\left(x-2\right)$

$x-1=\left(x+2\right)\left(x-2\right)\left(\frac{A}{x+2}+\frac{B}{x-2}\right)$
9

Multiplying polynomials

$x-1=\frac{\left(x+2\right)\left(x-2\right)A}{x+2}+\frac{\left(x+2\right)\left(x-2\right)B}{x-2}$
10

Simplifying

$x-1=\left(x-2\right)A+\left(x+2\right)B$
11

Expand the polynomial

$x-1=\left(x-2\right)A+\left(x+2\right)B$
12

Assigning values to $x$ we obtain the following system of equations

$\begin{matrix}-3=-4A&\:\:\:\:\:\:\:(x=-2) \\ 1=4B&\:\:\:\:\:\:\:(x=2)\end{matrix}$
13

Proceed to solve the system of linear equations

$\begin{matrix} -4A & + & 0B & =-3 \\ 0A & + & 4B & =1\end{matrix}$
14

Rewrite as a coefficient matrix

$\left(\begin{matrix}-4 & 0 & -3 \\ 0 & 4 & 1\end{matrix}\right)$
15

Reducing the original matrix to a identity matrix using Gaussian Elimination

$\left(\begin{matrix}1 & 0 & \frac{3}{4} \\ 0 & 1 & \frac{1}{4}\end{matrix}\right)$
16

The integral of $\frac{x-1}{\left(x+2\right)\left(x-2\right)}$ in decomposed fraction equals

$\frac{3}{4\left(x+2\right)}+\frac{1}{4\left(x-2\right)}$

Final Answer

$\frac{3}{4\left(x+2\right)}+\frac{1}{4\left(x-2\right)}$

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