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Step-by-step Solution

Integral of $\frac{x}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}$ with respect to x

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Answer

$-\frac{1}{2}\ln\left|x+1\right|+2\ln\left|x+2\right|-\frac{3}{2}\ln\left|x+3\right|+C_0$

Step-by-step explanation

Problem to solve:

$\int \frac { x d x } { ( x + 1 ) ( x + 2 ) ( x + 3 ) }$
1

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

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

Find the values of the unknown coefficients. The first step is to multiply both sides of the equation by $\left(x+1\right)\left(x+2\right)\left(x+3\right)$

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

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Answer

$-\frac{1}{2}\ln\left|x+1\right|+2\ln\left|x+2\right|-\frac{3}{2}\ln\left|x+3\right|+C_0$