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# Step-by-step Solution

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## Answer

$\ln\left|x+1\right|-\ln\left|x+2\right|+C_0$

## Step-by-step explanation

Problem to solve:

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

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

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

Find the values of the unknown coefficients. The first step is to multiply both sides of the equation by $\left(x+1\right)\left(x+2\right)$

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

## Answer

$\ln\left|x+1\right|-\ln\left|x+2\right|+C_0$

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

### Main topic:

Integrals by partial fraction expansion

~ 0.77 seconds