Evaluate the following integrals:

Let $I=\int x^{3} \cos x^{2} d x$

Put $x^{2}=t$

$2 x d x=d t$

$I=\frac{1}{2} \int t \cos t d t$

Using integration by parts,

$\mathrm{I}=\frac{1}{2}\left(\mathrm{t} \int \cos \mathrm{t} \mathrm{dt}-\int \frac{\mathrm{d}}{\mathrm{dt}} \mathrm{t} \int \cos \mathrm{dt}\right)$

$=\frac{1}{2}\left(\mathrm{t} \times \sin \mathrm{t}-\int \sin \mathrm{t} \mathrm{dt}\right)$

$=\frac{1}{2}(\operatorname{tsin} t+\cos t)+c$

Substitute value for $t$,

$=\frac{1}{2}\left(x^{2} \sin x^{2}+\cos x^{2}\right)+c$