Integrate the unknown function please

Or use Boole's Rule :

$2\left(7f(0)+32f(1)+12f(2)+32f(3)+7f(4)\right)=2(-7-32+48+608+336)=\boxed{1906}$

Using the Poyla Form,

${ n }^{ 3 }=6\left( \begin{matrix} n \\ 3 \end{matrix} \right) +6\left( \begin{matrix} n \\ 2 \end{matrix} \right) +\left( \begin{matrix} n \\ 1 \end{matrix} \right)$

and ${ 2 }^{ n }=\left( \begin{matrix} n \\ 4 \end{matrix} \right) +\left( \begin{matrix} n \\ 3 \end{matrix} \right) +\left( \begin{matrix} n \\ 2 \end{matrix} \right) +\left( \begin{matrix} n \\ 1 \end{matrix} \right) +\left( \begin{matrix} n \\ 0 \end{matrix} \right) \quad for\quad 0\le x\le 4\quad and\quad x\quad is\quad an\quad integer$

So, ${ n }^{ 3 }-{ 2 }^{ n }=-\left( \begin{matrix} n \\ 4 \end{matrix} \right) +5\left( \begin{matrix} n \\ 3 \end{matrix} \right) +5\left( \begin{matrix} n \\ 2 \end{matrix} \right) -\left( \begin{matrix} n \\ 0 \end{matrix} \right) \quad for\quad 0\le x\le 4\quad and\quad x\quad is\quad an\quad integer$

$\int _{ 0 }^{ 4 }{ \left( -\frac { { x }^{ 4 } }{ 24 } +\frac { 13{ x }^{ 3 } }{ 12 } -\frac { 11{ x }^{ 2 } }{ 24 } -\frac { { 7x } }{ 12 } -1 \right) } dx={ \left[ -\frac { { x }^{ 5 } }{ 120 } +\frac { 13{ x }^{ 4 } }{ 48 } -\frac { 11{ x }^{ 3 } }{ 72 } -\frac { 7{ x }^{ 2 } }{ 24 } -x \right] }_{ 0 }^{ 4 }\\ =\frac { 1906 }{ 45 }$

$\\ \frac { 1906 }{ 45 } \times 45=1906$