Answer
$$2\sin \sqrt x + C $$
Work Step by Step
$$\eqalign{
& \int {\frac{{\cos \sqrt x }}{{\sqrt x }}} dx \cr
& {\text{Integrate by substitution method}} \cr
& {\text{Let }}u = \sqrt x,\,\,\,\,du = \frac{1}{{2\sqrt x }}dx,\,\,\,\,dx = 2\sqrt x du \cr
& {\text{Then}}{\text{,}} \cr
& \int {\frac{{\cos \sqrt x }}{{\sqrt x }}} dx = \int {\frac{{\cos u}}{{\sqrt x }}} \left( {2\sqrt x du} \right) \cr
& {\text{Cancel common factor }}\sqrt x \cr
& = \int {\frac{{\cos u}}{{}}} \left( {2du} \right) \cr
& = 2\int {\cos u} du \cr
& {\text{integrating}} \cr
& = 2\sin u + C \cr
& {\text{replacing }}u = \sqrt x \cr
& = 2\sin \sqrt x + C \cr} $$
