How to integrate a periodic exponent function?

This is really interesting.  I do not know the answer.   

Here is the graph

https://www.desmos.com/calculator/s4cswnczix

and here is what Wolfram|alpha has to say.

http://www.wolframalpha.com/input/?i=y%3Dk%5Esinx%20%20find%20integral

Thanks for your reply Melody, and for your effort with Wolfram which I have heard of but never used. That Taylor series is not something I would have come up with but I'm glad it exists.

I hope you realized the question wasn't an urgent one. In fact, it dates from about 1962. I was leaving a lively party with a couple of guys from school; all three of us were maths nerds and all rather sozzled. Having failed to get anywhere with the girls we amused ourselves by tossing culculus questions at one another as we walked home. After a few solvable problems, I came up a challenge to integrate e^sin(x). No one could think of an answer. "It's not integrable," said Tony rather haughtily (and he went on to study Maths at Cambridge the next year). On reflection I couldn't accept that -- I'm a good visualizer and I could imagine what the curve was like -- but let the matter lie.

It's just that I recently came across Tony's name on the internet (he's now a famous classical musician) and it made me think back to that boozy walk home and the simple but "unintegrable" formula. I'm turning 70 next week so I think I can permit myself a bit of nostalgia.

Nice to see such a lively forum.

all the best, Vic