  Filon-Simpson quadrature rules are derived for integrals of the type \int_a^b
dx f(x) sin(xy)/(xy) and \int_a^b dx f(x) 4 sin^2(xy/2)/(xy)^2 which are needed
in applications of the worldline variational approach to Quantum Field Theory.
These new integration rules reduce to the standard Simpson rule for y = 0 and
are exact for y \to \infty when a = 0 and f(0) \ne 0.The subleading term in the
asymptotic expansion is also reproduced more and more precisely when the number
of integration points is increased. Tests show that the numerical results are
indeed stable over a wide range of y-values whereas usual Gauss-Legendre
quadrature rules are more precise at low y but fail completely for large values
of y. The associated Filon-Simpson weights are given in terms of sine and
cosine integrals and have to be evaluated for each value of y. A Fortran
program to calculate them in a fast and accurate manner is available. A
detailed comparison with the double exponential method of Ooura and Mori is
made.
