Integration points and weights with orthogonal polynomials
We can use Eq.
(14)
to rewrite the above integral as
$$
\begin{equation*}
\int_{-1}^1P_{2N-1}(x)dx=\int_{-1}^1(L_N(x)P_{N-1}(x)+Q_{N-1}(x))dx=\int_{-1}^1Q_{N-1}(x)dx,
\end{equation*}
$$
due to the orthogonality properties of the Legendre polynomials. We see that it suffices
to evaluate the integral over \( \int_{-1}^1Q_{N-1}(x)dx \) in order to evaluate
\( \int_{-1}^1P_{2N-1}(x)dx \). In addition, at the points \( x_k \) where \( L_N \) is zero, we have
$$
\begin{equation*}
P_{2N-1}(x_k)=Q_{N-1}(x_k)\hspace{1cm} k=0,1,\dots, N-1,
\end{equation*}
$$
and we see that through these \( N \) points we can fully define \( Q_{N-1}(x) \) and thereby the
integral. Note that we have chosen to let the numbering of the points run from \( 0 \) to \( N-1 \).
The reason for this choice is that we wish to have the same numbering as the order of a
polynomial of degree \( N-1 \). This numbering will be useful below when we introduce the matrix
elements which define the integration weights \( w_i \).