# Step-by-step Solution

## Integral of $\frac{1}{x^2\left(x+2\right)}$ with respect to x

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

## Step-by-step explanation

Problem to solve:

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

Rewrite the fraction $\frac{1}{x^2\left(x+2\right)}$ in $3$ simpler fractions using partial fraction decomposition

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

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

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

$\frac{-\frac{1}{2}}{x}+\frac{1}{4}\ln\left|x+2\right|-\frac{1}{4}\ln\left|x\right|+C_0$
$\int\frac{1}{x^2\left(x+2\right)}dx$