# Step-by-step Solution

## Integral of 1/(x(x+1)*(2x+3))

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### Videos

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

## Step-by-step explanation

Problem to solve:

$\int\left(\frac{1}{x\left(x+1\right)\left(2x+3\right)}\right)dx$
1

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

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

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

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

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

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$\int\left(\frac{1}{x\left(x+1\right)\left(2x+3\right)}\right)dx$

### Main topic:

Integrals by partial fraction expansion

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