Polynomial is sum of terms, which is built with constant, variables, and exponent (Pierce, 2021), where the exponent is greater and equal to zero (Gruber & user103828, 2015). Then integral of any polynomial is simply sum of the integrals of its tems, whose formula of its indefinite integral is well known (Nave, 2017) and its definite integral can be calculated with the help of the formula, e.g. using Python (Mehta, 2021). Simpson’s rule can calculate integral of polynomial function by approxymating it with set of quadratic functions (Kong et al., 2020), but not recommended since it is not very efficient compared to the use of its indefinite integral formula.
integral of a term Link to heading
Suppose that there is a tems
$$\tag{1} u_n(x) = a_n x^n $$
with $a_n \in \mathbb{R}$, $x \in \mathbb{R}$, $n \in \mathbb{W}$, where $\mathbb{r}$ stands for real number and $\mathbb{W}$ stands for whole number, natural number and zero.
Integral of Equation (1) is simply
$$\tag{2} \begin{array}{rcl} v_{n+1}(x) & = & \displaystyle \int u_n(x) \ dx \newline & = & \displaystyle \int a_n x^n \ dx \newline & = & \displaystyle a_n \int x^n \ dx \newline & = & \displaystyle a_n \left( \frac{1}{n+1} x^{n+1} \right) + c \newline & = & \displaystyle \left( \frac{a_n}{n+1} \right) x^{n+1} + c \newline & = & \displaystyle b_{n+1} x^{n+1} + c. \end{array} $$