Numerical integration involving Cauchy's principal value on x,y data.

For x,y kind of dataset, it is required to perform numerical integration w.r.t. x'.
The italiczied P represents Cauchy's Principal Value. Limits are from 0 to infinity.
It was first tried to approximate the x,y data by combination of mathematical functions but did
not succeed as the data can not be modeled so easily and exactly.
Apologies for not putting very clear image. I do not find any clue how to proceed on it. Thanks.