# Symmetry between differentiation and integration [duplicate]

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I want to make clear, that I am interested in the question: Why does integration need a bigger spectrum of functions than differentiation and not why integration is harder!!!

as experience told me, one almost always finds "standard functions" by which I mean common function although this is just another vague description, that are the derivative of a standard function. e.g. it is not a problem at all to do the derivative of $x^x$. on the other hand it is extremely tough or generally more often impossible to find a representation of the integral of some functions like $\int x^x dx$ and $\int e^{-x^2}dx$ in terms of standard functions. I was wondering why this is the case? In my opinion it has to do with the fact that integration turns bad functions, that do not have to be continuous for example into functions with better properties and vice versa for the differentiation. therefore it is harder to find good functions than to find bad ones. but this is not a very "mathematical" approach to this question. so maybe some of you can give me a hint, where this break of symmetry actually comes from?