I'm working on the Newton cote formula, and I would like to be sure if the only method to find the degree of precision is to compute each integral for each monomial.
Example :
$\displaystyle \int_{-1}^1f(x)\cdot dx = f\left(-\dfrac{\sqrt{3}}{3}\right)+f\left(\dfrac{\sqrt{3}}{3}\right)$
Must I compute each $(I_i)$ to verify if the formula is exact for each one.
let $p_n(x)=x^n$
$(I_i):\displaystyle \int_{-1}^1p_i(x)\cdot dx =p_i\left(-\dfrac{\sqrt{3}}{3}\right)+p_i\left(\dfrac{\sqrt{3}}{3}\right)$
Suppose one formula is exact until $i=10$, and not exact for $i=11$, must I compute ten integrals? Is there an another method more efficient?
The demands of the brute force approach can be reduced in your special case. In particular, $$\int_{-1}^1 x^{2k-1} dx = 0, \quad k \in \mathbb{N},$$ because the interval $[-1,1]$ is symmetric around $0$ and the functions $x \rightarrow x^{2k-1}$ are odd.
The alternative to brute force is theoretical analysis.