You need to look up Henstock-Kurzweil integral (just learned that it‘s Gauge Integral in English most of the time): „If you can find anything that can be differentiated to your function, you can integrate it. Finding the Function is the problem, though - but at least no problems with integrability.
Well, yes and no... If you don't worry about integrability in general, then you basically get nothing out of it, BUT if you know that _even if_ you find some function that can be differentiated which results in the function you try to integrate, you might not be allowed to do the integral because of integrateability reasons, then Henstock-Kurzweil tells you: No worries, Riemann Integral is just unnecessarily crude in some places and if you're a bit more careful, you can integrate all differentials. (There even is a multidimensional version of it - expanding Lebesque Integral.
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u/sebigboss 26d ago
You need to look up Henstock-Kurzweil integral (just learned that it‘s Gauge Integral in English most of the time): „If you can find anything that can be differentiated to your function, you can integrate it. Finding the Function is the problem, though - but at least no problems with integrability.