Numerical integration of functions with a very small significant support

In some applications, one has to deal with the problem of integrating, over a bounded interval, a smooth function taking significant values, with respect to the machine precision or to the accuracy one wants to achieve, only in a very small part of the domain of integration. In this paper, we propose a simple and efficient numerical approach to compute or discretize integrals of this type. We also consider a class of second kind integral equations whose integral operator has the above behavior. Some numerical testing is presented.