Explanation

(b)
Let $I = \int_{2}^{3}{\frac{x + 1}{x^{2}(x - 1)}dx}$
$= \int_{2}^{3}{\left( \frac{- 2}{x} - \frac{1}{x^{2}} + \frac{2}{x - 1} \right)dx}$
$= \left\lbrack - 2\log{x +}\frac{1}{x} + 2\log(x - 1) \right\rbrack_{2}^{3}$
$= \left\lbrack 2\log{\left( \frac{x - 1}{x} \right) + \frac{1}{x}} \right\rbrack_{2}^{3} = \left\lbrack 2\left( \log{\frac{2}{3} - \log\frac{1}{2}} \right) + \frac{1}{3} - \frac{1}{2} \right\rbrack$
$= 2\log{\frac{4}{3} - \frac{1}{6} = \log{\frac{16}{9} - \frac{1}{6}}}$