Integral of complex periodic signal Integrating   $e^{j(k-n)\omega t} \, dt$ over the interval $0$ to $T$ where $T$ is the fundamental time period of the sinusoids yields zero when $k$ is not equal to $n$.... ? how?
assume ω is the fundamental frequency equal to $2\pi/T$.

It seems to be you mean
$$\int\limits_0^T e^{i(k-n)wt}dt=\frac i{wt(n-k)}e^{i(k-n)wt}=\left.\frac i{wt(n-k)}e^{wti(k-n)}\right|_0^T=$$
$$=\frac i{wt(n-k)}\left(e^{2\pi i(k-n)}-1\right)=0$$
Check that if you had $\;k=n\;$ then you'd have
$$\int\limits_0^T 1\,dt=T\neq 0$$
