Integrals evaluated in terms of Catalan’s constant

Catalan's constant, named after E. C. Catalan (1814-1894) and usually denoted by G, is defined byIt is, of course, a close relative ofThe numerical value is G ≈ 0.9159656. It is not known whether G is irrational: this remains a stubbornly unsolved problem. The best hope for a solution might appear to be the method of Beukers 1 to prove the irrationality of ζ (2) directly from the series, but it is not clear how to adapt this method to G.