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comment Why is differentiating mechanics and integration art?
A map is a rule of preparing for each object from A one object from B. Let me illustrate. Suppose you prepare from each decimal number the sum of its digits: from 231 you get 6, and so on. This is absolutely routine. But now you ask, which numbers give you 6, the preimages of 6? Looks more challenging, isn't it? Finding preimages is not a map, because for one object you may get several. And what if you are asked to find minimal of those preimages? This weird task is in the spirit of integration problems of traditional calculus: you are asked to find a very special form of answer.
May
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comment Why is differentiating mechanics and integration art?
Exactly for the same reason as for polynomials above: differentiating is a map from functions to functions and integration is trying to revert it - to find a preimage. Nobody ever promised that the inverse action should be as easy as the original one.
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awarded  Teacher
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answered Why is differentiating mechanics and integration art?