Integral exercise using by parts technique

Exercise:

Integrate , $\int{\sin \left( \sqrt{x} \right)dx}$

Solution: Let ${{u}^{2}}=x\Rightarrow 2udu=dx$

So $\int{\sin \left( \sqrt{x} \right)dx}=\int{\sin \left( u \right)\,2u\,du}$ ,

Take $U=2u\,\,\And \,dV=\sin u\,du$ $\Leftrightarrow dU=2du\,\And \,V=-\cos u$

Hence $\int{\sin \left( u \right)2u\,du}=-2u\cos u+2\int{\cos \left( u \right)}du=-2u\cos u+2\sin u+c$

Thus $\int{\sin \left( \sqrt{x} \right)dx}=2\left( -\sqrt{x}\cos \left( \sqrt{x} \right)+\sin \left( \sqrt{x} \right) \right)+c$