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5

With V10 one can see that the special case p = -1 is explicitly excluded: FunctionDomain[Integrate[x^p, x], p, Reals] Reason: The general formula Integrate[x^p, x] would result in a division by zero error with p = -1: Limit[Integrate[x^p, x], p -> -1] Integrating over a certain interval one gets the expected results: Integrate[x^#, x] ...


4

The original limit is definitely computed correctly: Limit[n*Sin[2*Pi*Exp[1]*n!], n -> Infinity] (* Out: Interval[{-Infinity, Infinity}] *) Keep in mind that Limit assumes the variable (n, in this case) is continuous. Of course, there is a mechanism for specifying the domain so I'd consider the following to be a bug (though, hardly, egregious): ...


2

The reason is because Mathematica does not know anything about f[x]. What if f[x] was 1? then the integrand will be zero. One way is to make your own rule for this case. ClearAll[x, y, h, f]; rul0[x_] := (x /. Integrate[1 - f[y_], {y_, a_, b_}] :> ((b - a) - Integrate[f[y], {y, a, b}])) Simplify[Integrate[1 - f[x], {x, 0, h}], TransformationFunctions ...



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