I am trying to compute the Fourier transform of $\operatorname{rect}$, where
$$\operatorname{rect}(t) = \begin{cases}1 &, 0 \leqslant t \leqslant 1\\ 0 &, \text{otherwise.} \end{cases}$$
I used
$$\int_0^1 e^{-i\omega t}\,dt$$
and after inserting limits and substituting Euler's formula got back
$$\frac{i\cos \omega + \sin \omega}{\omega} - \frac{i}{\omega}.$$
But the answer is supposed to be
$$\frac{\sin \omega}{\omega}.$$
Where is the mistake?