If i have some function expressed as points on a non-uniform grid (I'm specifically interested in logarithmic grids, but general results are also interesting), and I want to integrate it, I believe there are two general possibilities:
Interpolate, and use some method like Gauss Quadrature, or Clenshaw-Curtis Quadrature. The downside is that you now have stacked errors from the interpolation and the integration method. The upside is that these quadrature methods are known to be very accurate.
Use some integration method like the trapezoidal rule, or Simpson's rule, which use the points that you have available to you.
Is there a general rule of thumb for which method is better? Does it depend on some basic property of the function (for example: periodic)?