Existence of Integral for (sin(100*x))^x on [0,2pi]

[missfangula]

Homework Statement

Homework Equations

(sin(100*x))^x , integral from 0 to 2pi

The Attempt at a Solution

I typed it into the wolfram integration calculator and another i found online, and both say that this is probably a nonelementary integral.
Any thoughts?

Thanks,
miss fangula

 

[irycio]

Mathematica also says it doesn't. Even NIntegrate (integrating numerically) returns error messages.

 

[Dickfore]

What is the domain of definition of the integrand

$$\sin^{x}(100 \, x)$$

 

[jackmell]

Any thoughts?

Thanks,
miss fangula

Yes Miss Fangula. You should break it apart and analyze what's going on. Just consider sin(a x). As a increases, it oscillates more. That's a problem for numerical integrators and that is the message Mathematica gives you when a=100. Now what happens when you put a real number as an exponent? For example, what happens when say for example sin(ax)=-0.5 and the exponent is for example 1/2? That's going to be a complex number right? Also, Mathematica will always use the "principal" value for the root and that may not represent the analytic continuation of the function throughout the domain of integration and thus even the numerical answer you get may not be the one you want. So try plotting the function for just some values of a to get some understanding of what it looks like. Also try plotting just the real or imaginary part and keep in mind it's multi-valued in general and also I think it's antiderivative is non-elementary.

 

[missfangula]

Thanks for all the replies. I will try the plotting, jackmell. Dickfore, the domain is [0,100]. any thoughts about that?