Explanation

(a)
Let $I = \int_{}^{}\frac{dx}{\sqrt{(1 - x)(x - 2)}}$
$= \int_{}^{}\frac{dx}{\sqrt{- x^{2} + 3x - 2)}} = \int_{}^{}\frac{dx}{\sqrt{\frac{1}{4} - \left( x - \frac{3}{2} \right)^{2}}}$
$= \sin^{- 1}{\left( \frac{\left( x - \frac{3}{2} \right)}{\frac{1}{2}} \right) + c = \sin^{- 1}{(2x - 3) + c}}$