Do numerical integration methods ever utilize evaluations of the derivative of the function?

Question

Suppose one is performing numerical integration of some function $f(x)$, but in addition to being able to evaluate its value at points $f(x_1), \dots, f(x_n)$, one also can additionally obtain its derivative at those points for free $f'(x_1), \dots, f'(x_n)$. Are there numerical integration methods which are able to utilize the derivative values? If so, do such methods lead to improved order of accuracy?

Answer 1

I take it that you mean to address simple quadrature, not ODE integration.

One step in the derivation of Gauß quadrature is the discussing of the general quadrature based on Hermite interpolation. In Gauß quadrature, the nodes are selected so that the quadrature weights or coefficients of the derivative values are zero.

So if the nodes are in some other configuration, the quadrature method will have non-trivial contributions from the derivative values and the same order $2s$ for $s$ nodes with function and derivative values.

Answer 2

I think derivatives don't necessarily help. Take the single point Gauss-Laguerre rule $\int_0^\infty\,e^{-x}f(x)\, dx \approx w_1f(x_1)$, where $w_1 = 1$, $x_1 = 1$. This is exact for $f(x) = 1, x$. A different rule, in which the derivative is included, \begin{equation} \int_0^\infty\,e^{-x}f(x)\, dx \approx Af(\xi) + Bf'(\xi) \end{equation} would presumably be more accurate, say exact also for $x^2$, if $A$, $B$, and $\xi$ satisfied \begin{align*} 1 &= A,\\ 1 &= A\xi + B , \\ 2 &= A\xi^2 + 2B\xi, \end{align*} corresponding to exactness for $f(x) = 1, x, x^2$. A bit of algebra shows this would require $\xi^2 - 2\xi + 2 = 0$, which doesn't have real roots, so this simple extension doesn't exist, at least in the sense described.