I'm attempting to calculate the Fourier Transform of the following function $$ g(t)=\Bigg\{ \begin{array}{ll} \pi, \ |t-3|<1 \\ 0, \ |t-3|>1 \end{array} $$ My approach is using the definition of the Fourier Transform $\mathcal F (\omega) = \frac{1}{2\pi}\int_{-\infty}^{\infty}f(t) e^{-i\omega t} \ dt$ as $$ \frac{1}{2\pi} \int_{2}^{4}\pi e^{-i \omega t}dt = \frac{i}{2 \omega}\left( e^{-i \omega 4} - e^{-i \omega 2}\right)= e^{-3i \omega}\frac{sin(\omega)}{\omega} $$ Observe that I've ommited the integrals where the value of the function is $0$.
I'm seriously doubting of the validity of what I have done and the fact that Wolfram returns a slightly different answer makes me doubt even more. Thank you.