$\begin{align}  & \int\limits_{0}^{\tfrac{3}{2}}{\frac{dx}{{{x}^{2}}+9}}=\int\limits_{0}^{\tfrac{3}{2}}{\frac{dx}{{{x}^{2}}+9}}=\frac{1}{3}\left[ \arctan \frac{x}{3} \right]_{0}^{\tfrac{3}{2}} \\ & \int\limits_{0}^{\tfrac{3}{2}}{\frac{dx}{{{x}^{2}}+9}}=\frac{1}{3}\left[ \arctan \frac{\tfrac{3}{2}}{3}-\arctan 0 \right]=\frac{1}{3}\arctan \frac{1}{2} \\\end{align}$