I've taken courses in Numerical Analysis, but all the proofs we've examined for the error of the Trapezoidal method hinge on a uniform grid. However, I'm working on a quadrature method which depends on the difference of the function values, not the input values. Specifically, it looks something like this: $$I(f,a,b) \approx bf(b) - af(a) - \frac12 \sum_{i = 1}^{N}[x_{i-1} + x_i]\Delta f(x_i)$$
Now, I don't need the proof for this exact statement derived, though I'm not opposed to hints. But if anyone can point out a good resource for the error analysis of the Trapezoidal rule with a non-uniform grid, which looks like this: $$I(f,a,b) \approx \frac12 \sum_{i = 1}^{N}[f(x_{i-1}) + f(x_i)]\Delta x_i$$
or a derivation of it, then that would be extremely helpful.