Answer
$${\sin ^{ - 1}}\left( {x - 2} \right) + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{dx}}{{\sqrt {\left( {x - 1} \right)\left( {3 - x} \right)} }}} \cr
& {\text{Use FOIL}} \cr
& {\text{ = }}\int {\frac{{dx}}{{\sqrt {3x - {x^2} - 3 + x} }}} \cr
& {\text{ = }}\int {\frac{{dx}}{{\sqrt {4x - {x^2} - 3} }}} \cr
& {\text{Completing the square}} \cr
& 4x - {x^2} - 3 = 1 - {\left( {x - 2} \right)^2} \cr
& {\text{ = }}\int {\frac{{dx}}{{\sqrt {1 - {{\left( {x - 2} \right)}^2}} }}} \cr
& {\text{The integrand contains the form }}{a^2} - {u^2} \cr
& {\text{Use the change of variable }}x - 2 = a\sin \theta \cr
& x - 2 = \sin \theta ,\,\,\,\,\,\,dx = \cos \theta d\theta \cr
& {\text{Use the change of variable}} \cr
& {\text{ = }}\int {\frac{{\cos \theta d\theta }}{{\cos \theta }}} \cr
& = \int {d\theta } \cr
& = \theta + C \cr
& {\text{Write in terms of }}x \cr
& = {\sin ^{ - 1}}\left( {x - 2} \right) + C \cr} $$
