Clenshaw–Curtis quadrature

Remarks

The nodes are the extreme points of the Chebyshev polynomial $T_N$ on $[-1, 1]$. Writing $p_N(\cos\theta) = \sum_{j=0}^{N} c_j \cos(j\theta)$ gives $\int_{-1}^{1} p_N = \sum_j c_j \int_0^\pi \cos(j\theta)\sin\theta\,d\theta$, and $\int_0^\pi \cos(j\theta)\sin\theta\,d\theta$ is $2/(1 - j^2)$ for even $j$ and $0$ for odd $j$. The rule is exact for polynomials of degree at most $N$, and of degree at most $N + 1$ when $N$ is even, by symmetry.

Example: $N = 2$. The nodes are $1, 0, -1$, and the interpolatory weights are $\frac13, \frac43, \frac13$. This is Simpson's rule; for $f = x^2$ it gives $\frac13 + 0 + \frac13 = \frac23 = \int_{-1}^{1} x^2\,dx$.

Non-example: the interpolatory rule at the equally spaced points $-1, -\frac13, \frac13, 1$ is a Newton–Cotes rule, not a Clenshaw–Curtis rule, because its nodes are not of the form $\cos(k\pi/3)$, which are $1, \frac12, -\frac12, -1$.