If I have a general integral equation with a semi-bounded limits from zero to infinity and I want to apply this equation on some deterministic values. The integration in this case must be converted to a deterministic integral, but how?
for example, as shown in equation attached here, I want to find the integral for a specific range of frequencies.
Your question is half-complete. This is one part of the Kramer's Kronig relationships. See,
https://en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relations
And as the link says,
http://lampx.tugraz.at/~hadley/ss2/linearresponse/causality.php
Note that the singularity is stronger in this form making it less suitable for a numerical evaluation.
Also see, https://www.phy.duke.edu/~rgb/Class/phy319/phy319/node56.html
I suggest try to obtain an analytical expression for a given problem. Or, if you intend to do it numerically, then chose the right method depending on what you have in the numerator.
Also, take care in ithe way you evaluate the integral: there is a simple pole out there. The integral in the KK formula is usually so specified that you must take the principal part. So you should somehow first appropriately divide the integral in a part close to the singularity, and another part more distant from it, treating the first one, perhaps, semi-analytically (developing, say, the function in a power series around the position of the pole) and the second numerically.
The equation I mentioned is on of KK relations and I desire to apply the integrand of both real and imaginary parts of KK relations in a limited frequency range [w1,w2] , where w1 >>0 and w2