Aim: Find Fourier coefficients of $\phi(x)=3\cos(2x)$, $x\in[-\pi,\pi]$.
I have written $3\cos(2x)=\frac{3}{2}(e^{2ix}+e^{-2ix})$ using the exponential form of cosine.
Then used the formula for Fourier coefficients $$\hat{\phi}(k)=\frac{1}{2\pi}\int^{\pi}_{-\pi}\phi(x)e^{-ikx}dx=\frac{3}{4\pi}\int^{\pi}_{-\pi}(e^{2ix}+e^{-2ix})e^{-ikx}dx$$ Now I am unsure how to solve this integral. The example in my notes goes straight from this to the answer without steps. Any help or points on where to look would be appreciated. Thank you.
By using Euler's formula, you already have its Fourier series, and thus its coefficients.. They're both $\frac{3}{2}$. These integrals can be done by distributing the exponential, then doing them one by one. You'll need to consider the cases $k\neq \pm 2$ separate from the cases where $k = \pm 2$ with the integrals (otherwise you get divide by $0$ issues).