I think you mean things like Gaussian quadrature rules? As far as I know, the issue with them is that (1) they work best if the function being integrated has a lot of structure, for example is close to a polynomial; (2) there are no sufficiently accurate variants for higher dimensions. However, if you have a one dimensional integral and you think your function is close to a polynomial, a Gaussian quadrature rule might be the right thing to use.

Sasho

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