Integrating Trigonometric Integral: Simplifying e^sinx Expression

[psholtz]

Homework Statement

I'm trying to integrate the following form:

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

The Attempt at a Solution

I thought about trying to write something like:

$$y = \int e^{\frac{i}{2}e^{-ix} - \frac{i}{2}e^{ix}} dx$$

But this seems to lead down the road of trying to integrate the form

$$\int e^{e^x} dx$$

which seems similarly intractable.

Is there a way to reduce the expression to something simpler, or are you just left w/ leaving the expression in a form like:

$$y(x) = \int_{x_0}^x e^{\sin t} dt$$

 

[jgens]

Wolframalpha couldn't find a result in terms of standard mathematical functions, so I would guess that evaluating that indefinite integral is pretty much hopeless.

http://www.wolframalpha.com/input/?i=integrate+e^(sin(x))

 

[psholtz]

Yes, I tried Wolfram before posting as well and it came up empty for me too..

My guess is that the integral:

$$y = \int e^{\sin x + 2x} dx$$

is just as intractable as the first one, yes?

Thanks for your help..

 

[jgens]

Again, since wolframalpha can't find a solution in terms of standard mathematical functions, I doubt that you'll be able to evaluate the indefinite integral.