Let us consider following table which I want to calculate myself
$$ x(t)=\frac{\sin(\omega_bt)}{\pi t}\quad\iff\quad X(j\omega)= \begin{cases} 1 & \text{if $|\omega|<\omega_b$}, \\[8pt] 0 & \text{if $|\omega|>\omega_b$}. \end{cases} $$ I want to prove it myself, so I have tried
$$\int_{-\infty}^{\infty} \frac{(e^{j\omega_bt}-e^{-j\omega_bt}) e^{j\omega t}}{2j\pi t}\ dt$$
now separately
$$e^{j \omega_b t} e^{j\omega t}=e^{j t(\omega_b+\omega)}$$ and $$e^{-j \omega_b t} e^{j\omega t}=e^{-j t(\omega_b+\omega)}$$
How can I continue from this? Thanks.