Given an ideal I of a local ring (R,m) we are interested in determining when the socle I:m, and in more generality the quasi socle ideals I:m^s where s is a positive integer, is integral over the ideal I. In the first part of the talk we will review what integrality of ideals means and what is known about the problem in the special cases  when I is a complete intersection, height two perfect, Gorenstein ideal.  The focus will be in looking for an explicit way to determine the generators of these quasi-socle ideals in terms of a presentation matrix of the ideal and in a characteristic free fashion. Time permitting, I will describe some of the new result obtained in an ongoing work joint with Shiro Goto, Craig Huneke, Claudia Polini and Bernd Ulrich.