I'm just curious as to the theory of integration, and why, given continuity and differentiability, all functions have somewhat easy and very calculatable dirivatives, while there are many functions which exist that either do not have a algebraic integral. Why is it that our math works perfectly and via set rules one way, but not the other. Is it something similar to all numbers having a square, but not all numbers having an integer square root. Obviously, that's not a mathematically sound way of saying it, nor are the two concepts really related, but I hope you understand where i'm going with that. Have there been any proofs as to why integration fails in certain circumstances. I understand that there are some functions, such as(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\int_a^b e^{x^2} dx[/tex]

that have no integral, but as they do have some calculatable area under them, shouldn't they have some integral of some form, Is it just that we lack the math to express the form? I know there is a theorem that there can be no integral for the above function, but is that just for our current concept of mathematics, otherwise how can a function without an integral really exist?

Edit: Thanks graphic7

~Lyuokdea

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# Integration Anomalies

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