Answer
$$2\cos \sqrt x + 2\sqrt x \sin \sqrt x + C$$
Work Step by Step
$$\eqalign{
& \int {\cos \sqrt x } dx \cr
& {\text{Make an appropiate }}u{\text{ - substitution }} \cr
& u = \sqrt x ,\,\,\,\,\,\,\,du = \frac{1}{{2\sqrt x }}dx,\,\,\,\,\,\,dx = 2\sqrt x du,\,\,\,\,\,\,dx = 2udu\,\,\, \cr
& {\text{write in terms of }}u \cr
& \int {\cos \sqrt x } dx = \int {\cos u} \left( {2udu} \right) \cr
& = 2\int {u\cos u} du \cr
& {\text{Use the Endpaper Integral Table to evaluate the integral}} \cr
& {\text{By formula 45}} \cr
& \left( {45} \right):\,\,\,\,\,\,\int {u\cos udu = } \cos u + u\sin u + C \cr
& 2\int {u\cos u} du = 2\cos u + 2u\sin u + C \cr
& {\text{write in terms of }}x{\text{; replace }}\sqrt x {\text{ for }}u \cr
& = 2\cos \sqrt x + 2\sqrt x \sin \sqrt x + C \cr} $$
