Asymptotic Representation of Certain Real Integrals with Parameter-dependent Limits of Integration

In this paper we consider a function f(x) defined byAll quantities are taken to be real, it is assumed that R is a function of the variable x, b is a constant, N and G are functions of the variable t and all the functions are such that the integral (1) exists when x is large enough. We wish to find an asymptotic representation of f(x) as x → + ∞, assuming that we are given certain information about the limiting behaviours of the functions R, N and G.