Fast computation of Goursat's infinite integral with very high accuracy

We propose an efficient computation method for the infinite integral ∫∞ 0 xdx/(1+ x 6 sin x), whose integrand contains a series of spikes, approximately π apart, growing taller and narrower as x increases. Computing the value of this integral has been a problem since 1984. We herein demonstrate a method using the Hilbert transform for changing this type of singular function into a smooth function and computing the value of the integral to more than one million significant digits using a superconvergent double exponential quadrature method. MSC Classification Codes: 65D32; 65B99; 65Y20; 68W05