I need to solve the following integral (if it is possible):
$$\int_0^{\infty}dx\,f(x) \left\{ \lim_{t \rightarrow \infty}\int_0^{t}e^{i(x-x_0) \tau}d\, \tau \right \}$$
I found an expression in an article which states that the following holds:
$$ \lim_{t \rightarrow \infty}\int_0^{t}e^{i(x-x_0) \tau}d\, \tau = \pi\delta(x-x_0) + \frac{P}{i(x-x_0)}$$
Where $P$ denotes the Cauchy principal value. But I haven't been able to fully understand how to calculate this principal value, and it is also pretty unclear to me where this term originates.