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If it is such that the integral cannot equal $0$, it may be that $d0/d\mathbb{I}$ can be any number/ is not defined. In the last example information was lost equation to $2\pi$, i.e the integral of the constant term in te polynomial, this would infer that the handling of a constant in the left hand side was done incorrectly with the rule of "use the regular derivative times x". We must revisit the formal definition of the derivative.  In fact we should probably use our rule that constant terms are made negative.  Infinite limits and functions that terminante at one of the limits seem to work well, this might mean there is a problem with scaling the limits...