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Step-by-step Solution

Integral of 1/(x(x-3))

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Answer

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

Step-by-step explanation

Problem to solve:

$\int\frac{1}{x\cdot \left(x-3\right)}dx$
1

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

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

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

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

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Answer

$-\frac{1}{3}\ln\left|x\right|+\frac{1}{3}\ln\left|x-3\right|+C_0$
$\int\frac{1}{x\cdot \left(x-3\right)}dx$

Main topic:

Integrals by partial fraction expansion

Used formulas:

5. See formulas

Time to solve it:

~ 0.97 seconds

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