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Integrals by partial fraction expansion Calculator

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1

Example

$\int\frac{1}{x^2\left(x+2\right)}dx$
2

Rewrite the fraction $\frac{1}{x^2\left(2+x\right)}$ using partial fraction decomposition as

$\frac{1}{x^2\left(2+x\right)}=\frac{A}{2+x}+\frac{B}{x^2}+\frac{C}{x}$
3

Find the values of the unknown coefficients. The first step is to multiply both sides of the equation by $x^2\left(2+x\right)$

$1=x^2\left(\frac{A}{2+x}+\frac{B}{x^2}+\frac{C}{x}\right)\left(2+x\right)$
4

Multiplying polynomials

$1=\frac{Ax^2\left(2+x\right)}{2+x}+\frac{Bx^2\left(2+x\right)}{x^2}+\frac{Cx^2\left(2+x\right)}{x}$
5

Simplifying

$1=Ax^2+B\left(2+x\right)+Cx\left(2+x\right)$
6

Expand the polynomial

$1=Ax^2+2B+Bx+2Cx+Cx^2$
7

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

$\begin{matrix}1=2B&\:\:\:\:\:\:\:(x=0) \\ 1=4A+2B-2B-4C+4C&\:\:\:\:\:\:\:(x=-2) \\ 1=4A+4B+20C&\:\:\:\:\:\:\:(x=2)\end{matrix}$
8

Proceed to solve the system of linear equations

$\begin{matrix}0A & + & 2B & + & 0C & =1 \\ 4A & + & 0B & + & 0C & =1 \\ 4A & + & 4B & + & 20C & =1\end{matrix}$
9

Rewrite as a coefficient matrix

$\left(\begin{matrix}0 & 2 & 0 & 1 \\ 4 & 0 & 0 & 1 \\ 4 & 4 & 20 & 1\end{matrix}\right)$
10

Reducing the original matrix to a identity matrix using Gaussian Elimination

$\left(\begin{matrix}1 & 0 & 0 & \frac{1}{4} \\ 0 & 1 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & -\frac{1}{10}\end{matrix}\right)$
11

The decomposed integral equivalent is

$\int\left(\frac{\frac{1}{4}}{2+x}+\frac{\frac{1}{2}}{x^2}+\frac{-\frac{1}{10}}{x}\right)dx$
12

The integral of a sum of two or more functions is equal to the sum of their integrals

$\int\frac{\frac{1}{4}}{2+x}dx+\int\frac{\frac{1}{2}}{x^2}dx+\int\frac{-\frac{1}{10}}{x}dx$
13

The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$

$\int\frac{\frac{1}{4}}{2+x}dx+\int\frac{\frac{1}{2}}{x^2}dx-\frac{1}{10}\ln\left|x\right|$
14

Apply the formula: $\int\frac{n}{b+x}dx$$=n\ln\left|b+x\right|$, where $b=2$ and $n=\frac{1}{4}$

$\frac{1}{4}\ln\left|2+x\right|+\int\frac{\frac{1}{2}}{x^2}dx-\frac{1}{10}\ln\left|x\right|$
15

Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$

$\frac{1}{4}\ln\left|2+x\right|+\int\frac{1}{2}x^{-2}dx-\frac{1}{10}\ln\left|x\right|$
16

Take the constant out of the integral

$\frac{1}{4}\ln\left|2+x\right|+\frac{1}{2}\int x^{-2}dx-\frac{1}{10}\ln\left|x\right|$
17

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$\frac{1}{4}\ln\left|2+x\right|-\frac{1}{2}x^{-1}-\frac{1}{10}\ln\left|x\right|$
18

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$\frac{1}{4}\ln\left|2+x\right|-\frac{1}{2}\cdot\frac{1}{x}-\frac{1}{10}\ln\left|x\right|$
19

Apply the formula: $a\frac{1}{x}$$=\frac{a}{x}$, where $a=-\frac{1}{2}$

$\frac{1}{4}\ln\left|2+x\right|+\frac{-\frac{1}{2}}{x}-\frac{1}{10}\ln\left|x\right|$
20

Add the constant of integration

$\frac{1}{4}\ln\left|2+x\right|+\frac{-\frac{1}{2}}{x}-\frac{1}{10}\ln\left|x\right|+C_0$