Find the Fourier Transform of piecewise finction

Question

$$f(x) = \begin{cases} 0 & |x|> a \\ 1 & |x|< a \end{cases}$$

I have most of the solution, I'm just faltering on obtaining the sin(ax) part of the solution, I'm missing an exponential somewhere.

Answer 1

$$\begin{align} \int_{-\infty}^{\infty} f(x) e^{ikx}dx&=\int_{-a}^{a} 1\, e^{ikx}\,dx+\int_{|x|>a} 0\,e^{ikx}\, dx\\\\ &=\frac{e^{ika}-e^{-ika}}{ik}\\\\ &=2\,\frac{\sin ka}{k} \end{align}$$

This last step follows from Euler's identity $e^{iz}=\cos z+i\sin z$, from which we see that $\sin z=\frac{e^{iz}-e^{-iz}}{2i}$.