Integrating some infinite oscillating tails (Q1063377)

In numerical integration of a slowly decaying function f(x) with alternative sign, the tail of f(x) is often expressed as \(f(x)=g(x)j(x)\), where j(x) is a Bessel function \(J_ 0(x)\) or \(J_ 1(x)\) and g(x) is a ultimately positive function. For making it easier to calculate, this proposal approximates the consecutive zeros of the Bessel function by \(J_ 1(x)\approx \sqrt{2/\pi x}(\cos (x-(3/4)\pi)).\)