How Should Exponential Terms Be Integrated in Fourier Transforms?

[Martin89]

Hi All! I've been looking at this Fourier Transform integral and I've realized that I'm not sure how to integrate the exponential term to infinity. I would expect the result to be infinity but that wouldn't give me a very useful function. So I've taken it to be zero but I have no idea if you can do this...
Thanks!

 

[Cryo]

With your current limits (with $-\infty$) what you have done is wrong. The exponential will blow up at negative times. The integral will not converege.

 

[Martin89]

With your current limits (with $-\infty$) what you have done is wrong. The exponential will blow up at negative times. The integral will not converege.

Thanks, I realize my mistake now. The limits should be zero to infinity as negative time is not possible