Fourier series of $cos(\omega_0 t)$ in continuous time Can any one please help me with understanding how we can calculate the Fourier series of Cos(w0t) using the formula:


I saw that they did the following calculus, but I Don't really understand how we calculate the integral of Xs[k] in this case:

  

It'll be so helpful if someone can solve it step by step.
They did not actually solve the integral, they just used a version of Euler's formula:

$$\cos(\omega_0t)=\frac12\left(e^{j\omega_0t}+e^{-j\omega_0t}\right)$$

which already is the Fourier series of $\cos(\omega_0t)$. Comparing $(1)$ to the formula for the Fourier series, you see that all coefficients are zero except for the ones with indices $k=1$ and $k=-1$, and they equal $\frac12$.