Mathematica Integrating over the phase in Mathematica


I have a question about integrating over the phase of a function in Mathematica. The origins of this problem is in scale analysis.

[itex] f(x,t)= \sum_n a_n \cos (n(kx-\omega t)) [/itex]

for [itex] k,\omega \in \mathbb{R}[/itex]. I want to integrate an expression dependent on f over the phase

[itex] \theta_n = n(kx-\omega t)[/itex] from [itex] [0,2\pi][/itex]. more specifically, I want to find

[itex] \int_0^{2\pi} g(x,t) \ d \theta[/itex] where, for instance, [itex] g(x,t) = f_t + 1/2f_^2+ f^2 [/itex].

Is there any automated way to do this without having to go back and manually change the phase to be a single symbolic variable?



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