GAUSS, I believe, gave a very large number of forms for approximate evaluation of definite integrals between finite limits. His formulæ are all based, like Tehebycheff's rules, on the assumption that the integrand is expressible approximately by a finite number of terms, of the series a + bx + cx2 + dx3 + … His plan was to use a minimum number of suitably weighted ordinates to give him the exact value of the integral for a specified number of terms.
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BAKER, T. A New Method for Approximate Evaluation of Definite Integrals between Finite Limits. Nature 105, 486 (1920). https://doi.org/10.1038/105486a0