Answer
$$2\sinh\sqrt x + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{\cosh \sqrt x }}{{\sqrt x }}} dx \cr
& {\text{substitute }}u = \sqrt x \cr
& du = \frac{1}{{2\sqrt x }}dx \cr
& 2du = \frac{1}{{\sqrt x }}dx \cr
& = \int {\cosh \sqrt x \frac{1}{{\sqrt x }}} dx \cr
& \int {\cosh u\left( 2 \right)} du \cr
& = 2\int {\cosh udu} \cr
& {\text{find the antiderivative }} \cr
& = 2\sinh u + C \cr
& {\text{write in terms of }}x,{\text{ replace }}u = \sqrt x \cr
& = 2\sinh\sqrt x + C \cr} $$
