$\int\limits_0^2 {\cfrac{{dx}}{{x + 4 - {x^2}}}}$

Let $I = \int\limits_0^2 {\cfrac{{dx}}{{x + 4 - {x^2}}}} = \int\limits_0^2 {\cfrac{{dx}}{{4 - ({x^2} - x)}}}$

$= \int\limits_0^2 {\cfrac{{dx}}{{4 + \cfrac{1}{4} - \left( {{x^2} - x + \cfrac{1}{4}} \right)}}}$

$= \int\limits_0^2 {\cfrac{{dx}}{{\cfrac{{17}}{4} - {{\left( {x - \cfrac{1}{2}} \right)}^2}}} = \int\limits_0^2 {\cfrac{{dx}}{{{{\left( {\cfrac{{\sqrt {17} }}{2}} \right)}^2} - {{\left( {x - \cfrac{1}{2}} \right)}^2}}}} }$

$= \cfrac{1}{{2 \cdot \cfrac{{\sqrt {17} }}{2}}}\left[ {\log \left| {\cfrac{{\cfrac{{\sqrt {17} }}{2} + \left( {x - \cfrac{1}{2}} \right)}}{{\cfrac{{\sqrt {17} }}{2} - \left( {x - \cfrac{1}{2}} \right)}}} \right|} \right]_0^2 = \cfrac{1}{{\sqrt {17} }}\left[ {\log \left( {\cfrac{{\sqrt {17} + 2x - 1}}{{\sqrt {17} - 2x + 1}}} \right)} \right]_0^2$

$= \cfrac{1}{{\sqrt {17} }}\left[ {\log \left( {\cfrac{{\sqrt {17} + 4 - 1}}{{\sqrt {17} - 4 + 1}}} \right) - \log \left( {\cfrac{{\sqrt {17} - 1}}{{\sqrt {17} + 1}}} \right)} \right]$

$= \cfrac{1}{{\sqrt {17} }}\left[ {\log \left( {\cfrac{{\sqrt {17} + 3}}{{\sqrt {17} - 3}}} \right) - \log \left( {\cfrac{{\sqrt {17} - 1}}{{\sqrt {17} + 1}}} \right)} \right]$

$= \cfrac{1}{{\sqrt {17} }}\log \left[ {\left( {\cfrac{{\sqrt {17} + 3}}{{\sqrt {17} - 3}}} \right)\left( {\cfrac{{\sqrt {17} + 1}}{{\sqrt {17} - 1}}} \right)} \right]$

$= \cfrac{1}{{\sqrt {17} }}\log \left[ {\cfrac{{17 + 3 + 3\sqrt {17} + \sqrt {17} }}{{17 + 3 - 3\sqrt {17} - \sqrt {17} }}} \right]$

$= \cfrac{1}{{\sqrt {17} }}\log \left( {\cfrac{{20 + 4\sqrt {17} }}{{20 - 4\sqrt {17} }}} \right) = \cfrac{1}{{\sqrt {17} }}\log \left( {\cfrac{{5 + \sqrt {17} }}{{5 - \sqrt {17} }}} \right)$

$= \cfrac{1}{{\sqrt {17} }}\log \left[ {\cfrac{{5 + \sqrt {17} }}{{5 - \sqrt {17} }} \times \cfrac{{5 + \sqrt {17} }}{{5 + \sqrt {17} }}} \right]$

$= \cfrac{1}{{\sqrt {17} }}\log \left[ {\cfrac{{25 + 17 + 10\sqrt {17} }}{8}} \right] = \cfrac{1}{{\sqrt {17} }}\log \left[ {\cfrac{{42 + 10\sqrt {17} }}{8}} \right]$

$= \cfrac{1}{{\sqrt {17} }}\log \left[ {\cfrac{{21 + 5\sqrt {17} }}{4}} \right]$