I need to perform this integral:
$$ \int_0^{\omega_{\large s}/\left(4\pi\right)} \frac{\mathrm{i}\omega\,\mathrm{e}^{\mathrm{i}\omega t}} {\left(\mathrm{i}\omega - \omega_{0}\right) \left(\mathrm{i}\omega - \omega_{1}\right)}\,\mathrm{d}\omega $$
I was thinking of trying contour integration in some way to use the Residue Theorem but I really cannot see a proper contour to use. Is it possible to use a quarter of circle centered at $0$ and then subtract the result from the same contour centered at $\omega_{s}/\left(4\pi\right)$ ?. Or is there an easier way that I am not seeing ?. Thank you in advance.
I believe that you can write: $$\frac{1}{(i\omega-\omega_0)(i\omega-\omega_1)}=\frac{1}{(\omega_0-\omega_1)(i\omega-\omega_0)}-\frac{1}{(\omega_0-\omega_1)(i\omega-\omega_1)}$$ And so: $$\int_0^{\omega_s/4\pi}\frac{i\omega e^{i\omega t}}{(i\omega-\omega_0)(i\omega-\omega_1)}d\omega=\int_0^{\omega_s/4\pi}\frac{i\omega e^{i\omega t}}{(\omega_0-\omega_1)(i\omega-\omega_0)}d\omega-\int_0^{\omega_s/4\pi}\frac{i\omega e^{i\omega t}}{(\omega_0-\omega_1)(i\omega-\omega_1)}d\omega$$ $$=\int_0^{\omega_s/4\pi}\frac{\omega_0e^{i\omega t}}{(\omega_0-\omega_1)(i\omega-\omega_0)}d\omega-\int_0^{\omega_s/4\pi}\frac{\omega_1e^{i\omega t}}{(\omega_0-\omega_1)(i\omega-\omega_1)}d\omega$$ $$=\frac{\omega_0}{\omega_0-\omega_1}\int_0^{\omega_s/4\pi}\frac{e^{i\omega t}}{i\omega-\omega_0}d\omega-\frac{\omega_1}{\omega_0-\omega_1}\int_0^{\omega_s/4\pi}\frac{e^{i\omega t}}{i\omega-\omega_1}d\omega$$