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An elementary function is a function that can be represented by a finite number of exponentials, logarithms, nth roots, and constants through composition. Clearly, an non-elementary function that is not elementary.

There are plenty of elementary functions such that integrating said function results in a non-elementary functions. Some quick ones that come to mind are $e^{x^2}$, $x^x$, and $\frac{1}{\ln{x}}$.

My question:

Is it possible to find a non-elementary function $f(x)$ such that $\int f(x)dx$ is an elementary function?

My intuition tells me no, but he's been wrong plenty of times before.

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Think it for a little while. What operation undoes the integration? How does it act on elementary functions? – Mlazhinka Shung Gronzalez LeWy Nov 13 '13 at 2:02
I'm not sure I follow. Differentiating any function finds the infinitesimal change in f(x) with respect to x. On an elementary function, this is no different. – Bonnaduck Nov 13 '13 at 2:42
$(\int f(x)dx)'=f(x)$. How ugly does it look the derivative of an elementary function? – Mlazhinka Shung Gronzalez LeWy Nov 13 '13 at 2:43
Got it. The derivative of an elementary function is always an elementary function. Thus, if $\int f(x) dx$ is elementary, $f(x)$ must be elementary. – Bonnaduck Nov 13 '13 at 2:48
@Bonnaduck it is encouraged to answer your question by yourself once you got it and accept your answer. Once you do it, your question will disappear from the list of unanswered questions, which is designed to contain questions that should be answered, which is not the case of your question anymore. – dbanet Aug 6 '15 at 18:20
up vote 2 down vote accepted

The derivative of an elementary function is always an elementary function. Thus, if $\int f(x)dx$ is elementary, $f(x)$ must be elementary.

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