Using Fourier transform to filter frequencies

Let $ Vleft(tright)=V_{0}taufrac{t}{t^{2}+tau^{2}} $. and assume I want to filter out all the frequencies of $ omega>frac{1}{tau} $.

Now I have the result:

$ widetilde{Vleft(wright)}=int_{-infty}^{infty}V_{o}taufrac{t}{t^{2}+tau^{2}}e^{-iomega t}dt=ipi V_{0}tau e^{-|omega|tau}sgnleft(wright) $

So in order to get all the frequencies $ omega<frac{1}{tau} $

What would be the right way to calculate :

$ int_{-infty}^{frac{1}{tau}}widehat{Vleft(tright)}e^{iomega t}domega $

or

$ int_{-frac{1}{tau}}^{frac{1}{tau}}widehat{Vleft(tright)}e^{iomega t}domega $

(Im just not sure what wold be the right way to use the formula).

Thanks in advance.