Evaluate the following integrals:

Let $I=\int \cos \sqrt{x} d x$

$\sqrt{\mathrm{X}}=\mathrm{t} ; \mathrm{x}=\mathrm{t}^{2}$

$d x=2 t d t$

$=\int 2 t \cos t d t$

$I=2 \int t \cos t d t$

Using integration by parts,

$=2\left(\mathrm{t} \times \sin \mathrm{t}-\int \sin \mathrm{t} \mathrm{dt}\right)$

$=2(\mathrm{t} \sin \mathrm{t}+\cos \mathrm{t})+\mathrm{c}$

Replace the value of $t, I=2(\sqrt{x} \sin \sqrt{x}+\cos \sqrt{x})+c$