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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}
$

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