Answer
$$x\sqrt {{x^2} + 9} + 36\ln \left( {x + \sqrt {{x^2} + 9} } \right) + C$$
Work Step by Step
$$\eqalign{
& \int {\sqrt {4{x^2} + 36} dx} \cr
& {\text{Rewritting}} \cr
& {\text{ = }}\int {\sqrt {4\left( {{x^2} + 9} \right)} dx} \cr
& = 2\int {\sqrt {{x^2} + 9} dx} \cr
& {\text{Integrate by tables using the formula }} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\int {\sqrt {{x^2} + {a^2}} } dx = \frac{x}{2}\sqrt {{x^2} + {a^2}} + \frac{{{a^2}}}{2}\ln \left( {x + \sqrt {{x^2} + {a^2}} } \right) + C \cr
& {\text{Let }}a = 6,\,\,\,{\text{then}} \cr
& 2\int {\sqrt {{x^2} + 9} dx} = 2\left[ {\frac{x}{2}\sqrt {{x^2} + 9} + \frac{{36}}{2}\ln \left( {x + \sqrt {{x^2} + 9} } \right)} \right] + C \cr
& {\text{Simplifying}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = x\sqrt {{x^2} + 9} + 36\ln \left( {x + \sqrt {{x^2} + 9} } \right) + C \cr} $$
