Generalized hypergeometric functions are used to extend, simplify, and complete the analysis of Stoyanov and Farrel (Math. Comput. 49 (1987) 275-279) and of Wong (Math. Comput. 50 (1998) 229-234), as well as putting their considerations within a wider framework: The integral∫0π/2sin a θcos b θJ ν (λsinθ)J μ (λsinθ)dθ,where J ν is the Bessel function of order ν, is analyzed in terms of generalized hypergeometric functions and a complete asymptotic expansion is given for∫0π/2J ν (λsinθ)J μ (λsinθ)dθ.