Inverse Fourier transform using laplace

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We have to solve the inverse FT of $$\frac{1}{1+4w^2}$$ I tried to do the synthesis but got mediocre results. However this term screams laplace to me. I can see a sine in there. The last lecture they actually showed us a proof how we perform a fourier transform using laplace but not whether there is a similar thing for the inverse. I still have difficulties grasping the whole concept, so I would appreciate any assistance.