$\int\limits_2^3 {\cfrac{{dx}}{{{x^2} - 1}}}$

$\int\limits_2^3 {\cfrac{{dx}}{{{x^2} - 1}}} = \left[ {\cfrac{1}{2}\log \left( {\cfrac{{x - 1}}{{x + 1}}} \right)} \right]_2^3 = \cfrac{1}{2}\left[ {\log \left( {\cfrac{{3 - 1}}{{3 + 1}}} \right) - \log \left( {\cfrac{{2 - 1}}{{2 + 1}}} \right)} \right]$

$= \cfrac{1}{2}\left[ {\log \left( {\cfrac{2}{4}} \right) - \log \left( {\cfrac{1}{3}} \right)} \right] = \cfrac{1}{2}\log \left( {\cfrac{{2/4}}{{1/3}}} \right) = \cfrac{1}{2}\log \cfrac{3}{2}$