What's the integration of $e^{x \sin x}?$ I am a beginner at integration. I sometimes write a random function and try to calculate the integral of that function. So, I tried with $e^{x\sin x}$.
Boy, it's not like others. I googled for it and didn't find anything.
Any hint on this regard? If you just point out, what new method or stuffs, I have to learn about solving it, it will do.
 A: That function does not have a primitive in terms of elementary functions i.e. you are not supposed to be able to integrate that. 
Another example would be
$$\int e^{-x^2} dx$$
Whose primitive (except for a scaling factor) is called the error function, erf.
You can calculate the integrals numerically but you can't write its primitive.
A: There is no solution in terms of standard mathematics so you won't be able to find a "solution". If you try to calculate the definite integral you can use numerics, but no analytical solution can be found.
Hint: Use WolframAlpha to calc such integrals
A: I am not 100% sure, but there's a good chance that this function's integral is not an elementary function, meaning that it cannot be written as a product, sum, and compositum of polynomials, exponential functions, trigonometric functions and the like.
The best known example of this is the function $e^{x^2}$, but in general, if you randomly think of a compositum of elementary functions, there's a high chance it doesn't have an elementary integral.
