Fourier Transform of $ f(t) = e^{-kt}$

Question

I am trying to calculate the fourier transform of the following function: $$ f(t) =\begin{cases} e^{-kt},& t \geq 0 \\ 0,& \text{otherwise}\end{cases} $$

where $k > 0$ is a real number.

I can plot the equation fine but am unsure of where the limits should lie, I think I should be doing something like this:

$$ f(\omega) = \int_0^T e^{-t(k+j\omega)}dt\ $$ Would this be correct or am I going the wrong way about completing this?

Answer 1

Rather than the limits being T and 0 they had to be infinity and zero as the equation continued onto infinity asymptotic to the x axis.