Answer
$$2{\sec ^{ - 1}}2x + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{2}{{x\sqrt {4{x^2} - 1} }}dx} \cr
& {\text{substitute }}u = 2x,{\text{ }}du = 2dx{\text{ and }}dx = \frac{{du}}{2} \cr
& \int {\frac{2}{{x\sqrt {4{x^2} - 1} }}dx} = \int {\frac{2}{{\left( {u/2} \right)\sqrt {{u^2} - 1} }}} \frac{{du}}{2} \cr
& = 2\int {\frac{1}{{u\sqrt {{u^2} - 1} }}} du \cr
& {\text{find the antiderivative}} \cr
& = 2{\sec ^{ - 1}}u + C \cr
& {\text{ with}}\,\,\,u = 2x \cr
& = 2{\sec ^{ - 1}}2x + C \cr} $$