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:

\frac{1}{2\pi}\lim_{R\to+\infty} \int_0^{\pi} e^{j{Re^{j\theta}}}.d\theta

This should equal 0.5 when t=0 and $\displaystyle e^t$ when t is not equal to 0.

Thanks in advance