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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

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