. $\int\limits_2^3 {\cfrac{{x\,dx}}{{{x^2} + 1}}}$

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

$= \cfrac{1}{2}\left[ {\log 10 - \log 5} \right] = \cfrac{1}{2}\log \cfrac{{10}}{5} = \cfrac{1}{2}\log 2$