I saw in a paper the following integral \begin{align} \int_0^\infty \frac{e^{i\omega x}}{\omega}\,d\omega = -\gamma+\frac{i\pi}{2}-\ln(x+i0)\,, \end{align} where $\gamma$ is the Euler's constant and $x+i0$ implies that this integral only makes sense distributionally. How should one prove this sort of results? For one, the principal value integral does not seem to work because the pole is at the lower limit of the integral.
There is a more complicated identity that is similar to this, and I am guessing the trick is similar but I don't know how to start either: \begin{align} \int_0^\infty \omega^{2\pi i\beta n} \frac{e^{i\omega x}}{\omega}\,d\omega = i e^{i\pi/2}e^{-\pi^2\beta n}\Gamma(2\pi i\beta n)(x+i0)^{-2\pi i\beta n}\,, \end{align} where $\beta>0$ and $n\in \mathbb{Z}$. This one looks very close to the definition of Gamma function except that the exponential has imaginary power.
Is there a systematic way to prove these classes of integrals (or at least similar looking ones)?