X-Nico

Clenshaw–Curtis quadrature

Unlike computation of arbitrary integrals, however, Fourier-series integrations for periodic functions (like $f(\backslash cos\backslash theta)$, by construction), up to the Nyquist frequency $k=N$, are accurately computed by the $N+1$ equally spaced and equally weighted points $\backslash theta\; n\; =\; n\; \backslash pi\; /\; N$ for $n\; =\; 0,\backslash ldots,N$ (except the endpoints are weighted by 1/2, to avoid double-counting, equivalent to the trapezoidal rule or the Euler–Maclaurin formula).

see also