Reading a book on fractional calculus reminded me that I'd like to know more on the following method/idea.
Given an integral: $\displaystyle \int_{-\pi}^\pi{\frac{\displaystyle\sum_{j=a}^b{e^{i\cdot c_j \cdot t}}}{\displaystyle\sum_{k=a}^b{e^{i \cdot d_k \cdot t}}} dt}$
I can use recurrences and generating functions to find all of the derivatives. Using all of the derivatives (evaluated at a point), I can essentially create another function (for example, the function at point $n$ is the $n$th derivative). The function brings up a curiousity: how accurate could it be at predicting the integral itself?
EDIT An additional thought: If this method fails, perhaps the derivatives could still be used to aid in a very good spline extrapolation using only a few points.
So I'm wondering what problems arise when trying to extrapolate/interpolate an integral of the form above using derivatives. I'd like to know any studies of this problem if they exist, and what I can expect. I realize I'm being a bit vague, but I'm not exactly sure what the good questions are regarding this issue. I would like to explore this possibility as much as possible, and hope that answers arrive with hope and how to evaluate the integral this way.