Evaluate the following integrals:

Let $\mathrm{I}=\int 2 \mathrm{x}^{3} \mathrm{e}^{\mathrm{x}^{2}} \mathrm{dx}$

Put $x^{2}=t$

$2 x d x=d t$

$I=\int t e^{t} d t$

Using integration by parts,

$=\mathrm{t} \int \mathrm{e}^{\mathrm{t}} \mathrm{dt}-\int \frac{\mathrm{d}}{\mathrm{dt}} \mathrm{t} \int \mathrm{e}^{\mathrm{t}} \mathrm{dt}$

We have, $\int e^{x} d x=e^{x}$

$=t e^{t}-e^{t}+c$

$=e^{t}(t-1)+c$

Substitute value for $t$,

$I=e^{x^{2}}\left(x^{2}-1\right)+c$