Integrating square roots

Even though the integral exists, direct numerical qua& rature is not particularly satisfactory because of the singularity of the integrand at y = y . . A technique is described here for removing the singularity and evaluating (1) through the use of s tandard numerical methods for second order ordinary differential equations. I t will be necessary to compute if(y), but no other special properties of f wiI1 be used. In the application [1] which led to this note, f(y) is a certain rational function involving piecewise cubic splines. I t is easy to evaluate f '(y), but difficult to transform the integral to any form where special weighted quadrature formulas are applicable. Our technique has been quite successful in this particular application. By letting