In this paper, the acoustic scattering by an obstacle across a wide frequency range of sound waves is investigated on the basis of the Helmholtz integral formulation. To overcome the nonuniqueness difficulties, the methods proposed by Burton [NPL Report NAC 30 (Jan 1973)] and by Burton and Miller [Proc. R. Soc. London, Ser. A323, 201–210 (1971)] are adopted for the Dirichlet and Neumann problems, respectively. The aim of this paper is twofold. The first is to bring together completely regular formulations of the Helmholtz integral equation and its normal derivative. The second is to extend these formulations to treat the higher-frequency problems. The weakly singular integrals are regularized by subtracting out one term and adding it back. Depending on the problem concerned, the additional integral can finally be expressed in an explicit form or results in solving a surface source distribution of the equipotential body. The hypersingular kernels are regularized by the method of using some properties of the associated Laplace equation, originally proposed by Chien et al. [J. Acoust. Soc. Am.88, 918–937 (1990)]. The completely regularized integral equations are amenable to computation by direct use of the standard quadrature methods. To study the acoustic scattering due to higher-frequency waves, Filon’s quadrature method [Proc. R. Soc. Edinburgh49, 38–47 (1928)] is extended to treat the rapidly oscillatory integrands. Numerical examples consist of acoustic scattering from a hard or soft sphere of radiusaacross a wide spectrum of wave numberska=π–20π.Comparisons of the numerical results with the exact solutions demonstrate the validity and efficiency of the implementation.