Integration points and weights with orthogonal polynomials
Using the above results and the fact that
$$
\begin{equation*}
\int_{-1}^1P_{2N-1}(x)dx=\int_{-1}^1Q_{N-1}(x)dx,
\end{equation*}
$$
we get
$$
\begin{equation*}
\int_{-1}^1P_{2N-1}(x)dx=\int_{-1}^1Q_{N-1}(x)dx=2\alpha_0=
2\sum_{i=0}^{N-1}(L^{-1})_{0i}P_{2N-1}(x_i).
\end{equation*}
$$