I.
$ \eqalign{ & \int {5x^2 e^{x^3 } dx} = \cr & \int {\frac{5} {3}3x^2 e^{x^3 } dx} = \cr & \int {\frac{5} {3}d\left( {e^{x^3 } } \right)} = \cr & \int {\frac{5} {3} \cdot du} = \cr & \frac{5} {3} \cdot u + C \cr & \frac{5} {3}e^{x^3 } + C \cr} $
II.
$
\eqalign{
& \int {x^2 (x^3 - 1)^3 dx} = \cr
& \int {\frac{1}
{3}(x^3 - 1)^3 \cdot 3x^2 } dx = \cr
& \int {\frac{1}
{3}(x^3 - 1)^3 \cdot d(x^3 - 1)} = \cr
& \int {\frac{1}
{3}} \,u^3 du = \cr
& \frac{1}
{3} \cdot \frac{1}
{4}u^4 + C \cr
& \frac{1}
{{12}}u^4 + C \cr
& \frac{1}
{{12}}\left( {x^3 - 1} \right)^4 + C \cr}
$
III.
$
\eqalign{
& \int {\left( {x^3 + x} \right)\sqrt {x^2 + 1} \,\,dx = } \cr
& \int {x\left( {x^2 + 1} \right)\sqrt {x^2 + 1} \,\,dx = } \cr
& \int {x\left( {x^2 + 1} \right)^{1\frac{1}
{2}} \,\,dx = } \cr
& \int {\frac{1}
{2}\left( {x^2 + 1} \right)^{1\frac{1}
{2}} \cdot 2x\,\,dx = } \cr
& \int {\frac{1}
{2}\left( {x^2 + 1} \right)^{1\frac{1}
{2}} \cdot d\left( {x^2 + 1} \right) = } \cr
& \int {\frac{1}
{2}} \,u^{1\frac{1}
{2}} du = \cr
& \frac{1}
{2} \cdot \frac{2}
{5}u^{2\frac{1}
{2}} + C \cr
& \frac{1}
{5}u^{2\frac{1}
{2}} + C \cr
& \frac{1}
{5}u^2 \sqrt u + C \cr
& \frac{1}
{5}\left( {x^2 + 1} \right)^2 \sqrt {x^2 + 1} + C \cr}
$