I have made some progress, but the problem is not yet solved. Doing some further numerical studies, I found the value of the integral for a set of . For the value seems to be (dashed curve in the figure).
Edit: Never mind about the fit (dashed curve) =P. For even smaller the integral curve has the same curvature in the log-log plot, so that the proportionality constant (0.85) and the exponent (-0.32) continuously increases.