I need to inverse a Fourier transfrom F{jw) of some function f(t)

I'm using contour integration to find the function f(t)

f(t) is easy to find when t = 0;

However, when t!=0, there was a problem.

The integral is:

This should equal 0.5 when t=0 and when t is not equal to 0.

Thanks in advance