Abstract
We present quadrature rules which integrate exactly not only polynomials up to a certain degree, but also the functions sin kx and cos kx (where k is a free parameter). The formulae we obtain are modified Newton—Cotes formulae. They are derived by replacing the integrand by an interpolation function of the form a cos kx + b sin kx + Σ n-2 j = 0 c j x j based on equally spaced nodes. The total truncation error of the modified quadrature formulae is discussed and a rigorous proof of the error term is given for the modified Simpson's 3 8 rule. Numerical examples show the efficiency of the modified rules and the importance of the error term.
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