I want to compute
$\displaystyle(\mathcal{F}f)(k)=\int_\mathbb{R} \chi_{[-0.5,0.5]} e^{-ikx}\,\mathrm{d}x$
I proceeded as follows:
$\displaystyle(\mathcal{F}f)(k) =\int_{-0.5}^{0.5}\cos(kx)+i\sin(kx)\,\mathrm{d}x$
Then I argumented that, since $\sin$ is odd, the integral must be zero, so I am left with
$...= \int_{-0.5}^{0.5}\cos(kx)\,\mathrm{d}x=\frac{2}{k}\sin(k/2)$
Is this correct?