Answer
$$2\sqrt x \sin \sqrt x + 2\cos \sqrt x + C $$
Work Step by Step
$$\eqalign{
& \int {\cos \sqrt x } dx \cr
& {\text{Let }}t = \sqrt x,\,\,\,\,\,dt = \frac{1}{{2\sqrt x }}dx,\,\,\,\,\,\,dx = 2\sqrt x dt \cr
& \int {\cos \sqrt x } dx = \int {\cos t} \left( {2\sqrt x } \right)dt \cr
& = \int {2t\cos t} dt \cr
& \cr
& {\text{Using integration by parts method }} \cr
& {\text{Let }}u = 2t,\,\,\,\,du = 2dt,\,\,\,\,dv = \cos tdt,\,\,\,\,v = \sin t \cr
& {\text{Integration by parts then gives}} \cr
& \int {2t\cos t} dt = 2t\sin t - \int {\sin t} \left( {2dt} \right) \cr
& \int {2t\cos t} dt = 2t\sin t - 2\int {\sin t} dt \cr
& \int {2t\cos t} dt = 2t\sin t + 2\cos t + C \cr
& {\text{write in terms of }}x{\text{: substitute }}\sqrt x {\text{ for }}t \cr
& \int {\cos \sqrt x } dx = 2\sqrt x \sin \sqrt x + 2\cos \sqrt x + C \cr} $$
