Answer
$$ - 2\sqrt x \cos \sqrt x + 2\sin \sqrt x + C$$
Work Step by Step
$$\eqalign{
& \int {\sin \sqrt x dx} \cr
& {\text{Let }}w = \sqrt x ,{\text{ }}dw = \frac{1}{{2\sqrt x }}dx,{\text{ }}dx = 2wdw \cr
& \int {\sin \sqrt x } dx = 2\int {w\sin w} dw \cr
& {\text{Integrate by parts}} \cr
& {\text{Let }}u = 2w \Rightarrow du = 2dw; \cr
& dv = \sin wdw \Rightarrow v = - \cos w \cr
& {\text{Integration by parts formula}} \cr
& 2\int {w\sin w} dw = \underbrace {2w}_u\underbrace {\left( { - \cos w} \right)}_v - 2\int {\underbrace {\left( { - \cos w} \right)}_v} \underbrace {dw}_{du} \cr
& = - 2w\cos w + 2\int {\cos w} dw \cr
& = - 2w\cos w + 2\sin w + C \cr
& {\text{Write in terms of }}x,{\text{ substitute }}\sqrt x {\text{ for }}w \cr
& = - 2\sqrt x \cos \sqrt x + 2\sin \sqrt x + C \cr} $$
