Answer
$${\text{False}}$$
Work Step by Step
$$\eqalign{
& \int_1^2 {\frac{1}{{x\left( {x - 3} \right)}}} dx \cr
& {\text{The integrand is given by the expression }}\frac{1}{{x\left( {x - 3} \right)}},{\text{ which is}} \cr
& {\text{not defined for }}x = 0{\text{ or }}x = 3,{\text{ these values are not in the}} \cr
& {\text{interval integration }}\left[ {1,2} \right],{\text{ therefore the integral is not improper}} \cr
& {\text{The statement is False}}{\text{.}} \cr} $$
