Thank you very much for playing with the integral, yes this one is divergent !! But this is at one node of the problem, I am also dealing with integrals that are not divergent like this one:

Integrate[(Sqrt[(0. + 0.0078125*(1 - xi) + 
        0.0078125*xi)^2]*(-0.0078125 + 0.0078125*(1 - xi)*(1 + xi) + 
      0.0078125*xi*(1 + xi))*
        BesselK[0, 
     6283.185307179586*
      Sqrt[0. + (-0.0078125 + 0.0078125*(1 - xi)*(1 + xi) + 
           0.0078125*xi*(1 + xi))^2]])/
     (0. + (-0.0078125 + 0.0078125*(1 - xi)*(1 + xi) + 
       0.0078125*xi*(1 + xi))^2), {xi, -1, 1}]

If the correct numerical integration is used, this integral is given = 0. So if the constant is used 6283.185307179586, one need to solve as a cauchy principal value integral, if the constant is small one could solve as a weakly singular integral. I would love to remove that constant from the Bessel Term and then add it somehow after the integration. But Thank you very much for your answer !

This is a singular integral, maybe mathematica is having some problems integrating it ?

Yeah, I think Mathematica has some problems with the integration of Bessel functions. I am trying to somehow avoid the evaluation of the integral in the Cauchy principal value sense by removing the constant and thus making the integral weakly singular.

The Large value expression for the Bessel K0 has a series of finite part integrals that Mathematica should have problems solving it. But the small argument expression has only weakly singular integrals. thats why I am trying to somehow remove that constant.

Thank you for proving my point !

For large arguments, you might consider looking into the asymptotic expansion of the modified Bessel function (e.g. this one).