$\begin{align}  & \int\limits_{2}^{2\sqrt{3}}{\left( \frac{dx}{{{x}^{2}}+4} \right)}=\int\limits_{2}^{2\sqrt{3}}{\frac{dx}{{{x}^{2}}+{{2}^{2}}}}=\left[ \frac{1}{2}{{\tan }^{-1}}\frac{x}{2} \right]_{2}^{2\sqrt{3}} \\ & \int\limits_{2}^{2\sqrt{3}}{\left( \frac{dx}{{{x}^{2}}+4} \right)}=\frac{1}{2}\left[ {{\tan }^{-1}}\frac{2\sqrt{3}}{2}-{{\tan }^{-1}}\frac{2}{2} \right] \\ & \int\limits_{2}^{2\sqrt{3}}{\left( \frac{dx}{{{x}^{2}}+4} \right)}=\frac{1}{2}\left[ {{\tan }^{-1}}\sqrt{3}-{{\tan }^{-1}}1 \right] \\ & \int\limits_{2}^{2\sqrt{3}}{\left( \frac{dx}{{{x}^{2}}+4} \right)}=\frac{1}{2}\left[ {{60}^{\circ }}-{{45}^{\circ }} \right]=\frac{{{15}^{\circ }}}{2} \\\end{align}$