today I tried to integrate $x^x$ by applying a reverse chain rule which turned out to be false. I was told $\int e^{f(x)}\,dx$ can be done when $f(x)$ is linear. This made me wonder what conditions we can find so that $\int e^{f(x)}\,dx$ can be expressed in terms of elementary functions, but I'm not sure what to do.
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asked May 28, 2013 at 2:15
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1$\begingroup$ Integrability and "having a simple expression" are two different things. Just because you cannot find a simple expression does not mean that the function is not integrable. $\endgroup$– parsiadMay 28, 2013 at 2:24
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$\begingroup$ The question is unclear. Note for example that every continuous function has an antiderivative, whether it has the form $e^f$ or not. Do you mean you want to know for which elementary functions $f$ does $e^f$ have an elementary antiderivative, in the sense of en.wikipedia.org/wiki/Elementary_function? $\endgroup$– Jonas MeyerMay 28, 2013 at 2:26
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$\begingroup$ Possible duplicate of math.stackexchange.com/questions/155/…. $\endgroup$– lhfMay 28, 2013 at 3:00
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$\begingroup$ I've clarified the question. $\endgroup$– Little miss sunshineMay 28, 2013 at 3:05
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1 Answer
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When $f$ is a polynomial, it is a consequence of a celebrated theorem by Liouville that $e^{f(x)}$ has an elementary antiderivative iff there is a polynomial $h$ such that $1=h'+hf$. This implies that $f$ has degree at most 1. In particular, the function $e^{x^2}$ does not have an elementary antiderivative.
See How can you prove that a function has no closed form integral?.
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1$\begingroup$ Thanks, can anything be said when $f$ is neither a polynomial nor a log? $\endgroup$ Jun 1, 2013 at 14:58
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$\begingroup$ @Littlemisssunshine, yes, Liouville's theorem gives you the precise conditions. $\endgroup$– lhfJun 1, 2013 at 15:23
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