Answer
$$\frac{1}{2}\sinh 2x + C$$
Work Step by Step
$$\eqalign{
& \int {\cosh 2x} dx \cr
& {\text{substitute }}u = 2x,{\text{ }}du = 2dx{\text{ and }}dx = \frac{{du}}{2} \cr
& \int {\cosh 2x} dx = \int {\cosh u} \frac{{du}}{2} \cr
& = \frac{1}{2}\int {\cosh udu} \cr
& {\text{find the antiderivative}} \cr
& = \frac{1}{2}\sinh u + C \cr
& {\text{ with}}\,\,\,u = 2x \cr
& = \frac{1}{2}\sinh 2x + C \cr} $$