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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*}