$\cfrac{x}{{\left( {x - 1} \right)\left( {x - 2} \right)\left( {x - 3} \right)}}$

.: Let $I = \int {\cfrac{{x\,dx}}{{\left( {x - 1} \right)\left( {x - 2} \right)\left( {x - 3} \right)}}}$

We write,
$\cfrac{x}{{\left( {x - 1} \right)\left( {x - 2} \right)\left( {x - 3} \right)}} = \cfrac{A}{{x - 1}} + \cfrac{B}{{x - 2}} + \cfrac{C}{{x - 3}}$

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

..(i)
Putting $x = 1$ in (i),

we get
$1 = A\left( {1 - 2} \right)\left( {1 - 3} \right)$ $\Rightarrow$ $A = \cfrac{1}{2}$
Putting $x = 2$ in (i),

we get
$2 = B\left( {2 - 1} \right)\left( {2 - 3} \right)$ $\Rightarrow$ $B = - 2$
Putting $x = 3$ in (i),

we get
$3 = C\left( {3 - 1} \right)\left( {3 - 2} \right)$ $\Rightarrow$ $C = \cfrac{3}{2}$

$\therefore$ $\cfrac{x}{{\left( {x - 1} \right)\left( {x - 2} \right)\left( {x - 3} \right)}} = \cfrac{1}{{2\left( {x - 1} \right)}} - \cfrac{2}{{x - 2}} + \cfrac{3}{{2\left( {x - 3} \right)}}$

$\therefore$ $\int {\cfrac{x}{{\left( {x - 1} \right)\left( {x - 2} \right)\left( {x - 3} \right)}}dx} = \cfrac{1}{2}\int {\cfrac{{dx}}{{x - 1}}} - 2\int {\cfrac{{dx}}{{x - 2}}} + \cfrac{3}{2}\int {\cfrac{{dx}}{{x - 3}}}$

$= \cfrac{1}{2}\log \left| {x - 1} \right| - 2\log \left| {x - 2} \right| + \cfrac{3}{2}\log \left| {x - 3} \right| + C$