Evaluate $\displaystyle \int \dfrac{dx}{x^2 + 3 x + 3}$
$\begin{aligned} \text{Let } I & = \int \dfrac{dx}{x^2 + 3 x + 3} \\\\ & = \int \dfrac{dx}{\left(x^2 + 3 x + \dfrac{9}{4}\right) + 3 - \dfrac{9}{4}} \\\\ & \left[\text{Note: Completion of square in the denominator}\right] \\\\ & = \int \dfrac{dx}{\left(x + \dfrac{3}{2}\right)^2 + \left(\dfrac{\sqrt{3}}{2}\right)^2} \\\\ & = \dfrac{2}{\sqrt{3}} \tan^{-1} \left(\dfrac{x + \dfrac{3}{2}}{\dfrac{\sqrt{3}}{2}}\right) + c \;\;\; \left[\text{Note: } \int \dfrac{dx}{x^2 + a^2} = \dfrac{1}{a} \tan^{-1} \left(\dfrac{x}{a}\right)\right] \\\\ & = \dfrac{2}{\sqrt{3}} \tan^{-1} \left(\dfrac{2 x + 3}{\sqrt{3}}\right) + c \end{aligned}$