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Are there potential methods to solve this non-elementary integration problem? At the beginning of break my Calc BC teacher has given us this problem to use "all necessary resources" to see if we can indeed figure out a way to, even theoretically, crack it:

$$ \int e^{\sin x}\ dx\ =\ ? $$

I've worked towards transforming it a little using some differential equations, such as the following: $$ y = e^{\sin x},\quad y'=\cos x\left(e^{\sin x}\right)\\ \int y'\sec x\ dx = \int y\ dx \\ y\sec x\ -\ \int y'\sec x\tan x\ dx = \int y\ dx $$ However, I'm having trouble getting anywhere with it. Just looking at the graph and its (obviously) cyclical qualities begs me to find the function that defines its antiderivative. It just looks like it has one! Surely, of course, I am mistaken, but I want to have it proven to me that, having exhausted all of the possibilities, there is no antiderivative.

Please help me in uncovering the mystery of this integral!(that is, if the possibility of surprising my teacher does not entertain you...) Of course, I appologize for the naivety of my endeavor, but I believe it is my duty as a learner of calculus to be exploratory if not scrutinous of the many claims my teacher makes to prove to myself that it is in fact true.

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There is nothing naive in your endeavour: the reason why you are struggling with this integral is the fact that its result cannot be expressed as a composition of elementary functions (I don't know of any reference to give you, this is "standard mathematical wisdom"). I believe that your teacher gave it to you precisely in order for you to cut your teeth on it and understand through bitter personal experience that no all "easy" functions admit "easy" antiderivatives.

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  • $\begingroup$ Thanks for this clarification, but what defines an "elementary function?" If the antiderivative isn't defined by these elementary functions, what is it defined by? How do you, then, define a non-elementary function? $\endgroup$ – dsillman2000 Nov 24 '16 at 21:08
  • $\begingroup$ @dsillman2000: You may want to check Liouville's theorem on this subject, and then the concept of Differential Galois theory and finally Risch's algorithm. I warn you, though, that the subject is probably more than your current mathematical knowledge allows you to handle! $\endgroup$ – Alex M. Nov 24 '16 at 21:11
  • $\begingroup$ I like a challenge ;) Thanks for taking the time to help me out here! $\endgroup$ – dsillman2000 Nov 24 '16 at 21:13

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