Integration points and weights with orthogonal polynomials
We develope then
Q_{N-1}(x) in terms of Legendre polynomials,
as done in Eq.
(13),
\begin{equation}
Q_{N-1}(x)=\sum_{i=0}^{N-1}\alpha_iL_i(x).
\tag{15}
\end{equation}
Using the orthogonality property of the Legendre polynomials we have
\begin{equation*}
\int_{-1}^1Q_{N-1}(x)dx=\sum_{i=0}^{N-1}\alpha_i\int_{-1}^1L_0(x)L_i(x)dx=2\alpha_0,
\end{equation*}
where we have just inserted
L_0(x)=1 !