CTFS: What happened in this integral?

Question

Specifically, what happened in the last line to obtain the answer? It seems like they ignored the exponential term $e^{-jw_{o}kt}$?

Answer 1

Note that $\delta(t-nT_0)$ is identically zero on the interval $[-T_0/2,T_0/2]$ for all $n \neq 0$, and hence the integral vanishes.

For $n=0$ we have

$$\int_{-T_0/2}^{T_0/2}\delta(t)e^{-j\omega_0 k t}dt=1$$

In summary, the integral is $0$ for all $n\neq 0$ and $1$ for $n=0$, which is exactly the discrete delta function $\delta[n]$.