Any techniques to solve analytically or estimate $\int_2^N \cos ^N\left(\frac{2 \pi N}{m}\right) \, dm $?

Question

I was wondering if there was a way to solve,

$$\int_2^N \cos ^N\left(\frac{2 \pi a}{m}\right) \, dm \,\,(1)$$ or if more tractable$$\int_2^N \cos ^N\left(\frac{2 \pi N}{m}\right) \, dm \,\,(2)$$

using non numeric integration techniques?

I've just started this very nice open access epub book Copley, Leslie Mathematics for the Physical Sciences,

now in Chapter 6 - Asymptotic Expansions, there are methods for estimating difficult integrals.

Would this technique or some other estimation technique work on (1) or (2)?

Perhaps increasing the upper limit well beyond N as suggested in that chapter?

Answer 1

I'm assuming $N$ is a positive integer. You can expand $\cos^N(x)$ as a linear combination of $\cos(kx)$ for $k$ from $0$ to $N$, and use $$ \int \cos(2\pi a/m)\; dm = m \cos(2\pi a/m) + 2 \pi a\; \text{Si}(2 \pi a/m) + c$$ where Si is the sine-integral function.