Let $f(x)=\frac{1}{2}-x$ on the interval $[0,1)$, and extend $f$ to be periodic in $\mathbb R$. Show that $\mathcal{F}(0)=0$, and $\mathcal{F}(k)=(2\pi ik)^{-1}$, if $k\ne0$, where my $\mathcal{F}(k) = \int\limits_{-\infty}^{\infty} f(x)\, e^{-2\pi i k x} \, \mathrm{d} x$.
I have proved that $\mathcal{F}(0)=0$, but for the other one, I am not able to as it needs two extra assumptions :
$(i)$ $k \in \mathbf Z$ $\And$
$(ii)$ The integral should be evaluated between just $0$ and $1$.
Can anyone explain me these two steps. Any type of help will be appreciated. Thanks in advance.