Why is it that differentiation of a function that is a composition of elementary functions (such as $\sin \:2^x$ or $\ln(\mathrm{arcsec}\: x^3)$ or $x^{1/x}$) always produces a composition of elementary functions, while integration sometimes does not? For example, the derivative of $\sin \:x^2$ is $2x\cos \:x^2$, but the integral of $\sin \:x^2$ produces the Fresnel sine function, the definition of which is simply the integral? To put it loosely, why does differentiation make functions simpler and integration make functions more complex? I guess what I'm trying to get at is what fundamental aspect of integration sometimes produces a non-elementary function from an elementary function?
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J. M. ain't a mathematician
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Yunfei Ma
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The derivative of $\sin x^2$ is $2x\cos x^2$. – choco_addicted May 14 '16 at 01:35
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Oops, I've corrected it. – Yunfei Ma May 14 '16 at 01:36
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But there also some functions that differentiating them is harder than integrating. For example $2x\cos x^2$, its anti-derivative is $\sin x^2$ and its derivative is $2\cos x^2-4x^2 \sin x^2 $ – Ruzayqat May 14 '16 at 01:44
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1One actually does not need to go further than the natural logarithm. Who'd expect that $\int \frac1{u}\mathrm du$ results in a transcendental function? – J. M. ain't a mathematician May 14 '16 at 01:59
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The phrase you are looking for is "integration in finite terms".
Here is one of the key papers in the field:
http://www.ams.org/journals/tran/1969-139-00/S0002-9947-1969-0237477-8/S0002-9947-1969-0237477-8.pdf
A useful book which I read many years ago is J. F. Ritt, Integration in finite terms, Liouville's theory of elementary methods, Columbia Univ. Press, New York, 1948.
marty cohen
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