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Sep 14, 2015 at 13:27 comment added Wolfgang Bangerth Your integrand is piecewise smooth. Subdivide the circle into a set of triangles or quadrilaterals and apply Gaussian quadrature there -- you'll get the (almost) exact answers with basically no effort at all.
Sep 11, 2015 at 15:10 comment added Eliad I updated the question.
Sep 11, 2015 at 14:12 history answered Wolfgang Bangerth CC BY-SA 3.0