Question

Integrate $\frac{1}{x(x-3)}$ with respect to $x.$

Answer 1

$I=\int \frac{dx}{x(x-3)}$
$=\int \frac{dx}{x^2-3x+\frac{9}{4}-\frac{9}{4}}$
$=\int \frac{dx}{{(x-\frac{3}{2})}^2-{\left(\frac{3}{2}\right)}^2}$
$=\frac{1}{2}\times \frac{2}{3}\log \left|\frac{x-3}{x}\right|+C$
$=\frac{1}{3}\log \left|\frac{x-3}{x}\right|+C$