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  "body": "\n## Summary\n\nClenshaw–Curtis quadrature integrates a function on $[-1, 1]$ by interpolating it at the points $\\cos(k\\pi/N)$ and integrating the interpolating polynomial exactly.\nThe substitution $x = \\cos\\theta$ turns the problem into a cosine series, so the weights can be computed fast with the discrete cosine transform.\nIts nodes for $N$ are contained in its nodes for $2N$, so doubling $N$ reuses all earlier function values.\nFor many smooth functions it is about as accurate as [[Gauss–Legendre quadrature|RGGXDRCJVE]] with the same number of points.\n\n## Explanations\n\n### L1 human\n\nSample the function at points that are the shadows on the $x$-axis of equally spaced points on a half circle.\nThese points crowd towards the ends of the interval, which avoids the wild oscillations of interpolation at equally spaced points.\nFit the polynomial through the samples and integrate that polynomial exactly.\nFor example, with $N = 2$ the points are $-1, 0, 1$, and the result is [[Simpson's rule|MNXC26G3BQ]].\n",
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