Quadrature rule for spline interpolation Consider an integrable function $f$ on $[-1,1]$. We denote $\left(x_j\right)_{-N}^{N}$ the equally spaced grid on $[-1,1]$, and wish to compute the integral $I = \int\limits_{-1}^{1} f(x) \, dx$ using only $f(x_j)$. It is well known that if we interpolate $f$ using a piecewise linear function, and integrate it, the resulting integration rule is the $\frac{1}{3}$-Simpson (composite) rule. Furthermore, if we interpolate $f$ using a piecewise quadratic function, the resulting integration is the $\frac{3}{8}$-Simpson (composite) rule. **My question:** Is there a well known integration rule that results from integrating the cubic-spline interpolant? Could you refer to a textbook/paper that discusses it?

The resulting integration rule depends on the choice of end point conditions. If clamped conditions are used you get what is variably called the 'corrected trapezoidal rule' or the 'composite Hermite rule', $$ \int_a^b f(x) dx = h \left( \frac{1}{2}f_0 + f_2 + \dots + f_{n-1} + \frac{1}{2}f_{n} \right) + \frac{1}{12} h^2 \left( f_0' - f_{n}' \right) + \mathcal{O}(h^4). $$ This rule is usually derived directly from the Euler-Maclaurin formula and is indeed the simplest (apart from the trapezoidal rule) among the Euler-Maclaurin integration rules.

Different end point conditions result in different formulas of a similar type. There is a short paper by Holmes (2014), _Connections Between Cubic Splines and Quadrature Rules_ , detailing the derivation of the above rule and some related results.