This work proposes a refreshing technique that utilizes the Taylor expansion to improve the computational efficiency of the multi-frequency acoustic scattering problem. The Helmholtz equation in acoustic problems is solved using the boundary element method (BEM). In this work, the Taylor expansion is utilized to separate frequency-dependent terms from the integrand function in the boundary integral equation so that the wave number is independent of the equation system, thereby avoiding the time-consuming frequency sweep analysis. To conquer the non-uniqueness of the solution for the external acoustic field problem, the Burton-Miller method is used to linearly combine the conventional boundary integral equation and the hypersingular boundary integral equation. Moreover, to eliminate the computational difficulties caused by the Burton-Miller method, the Cauchy principal value and the Hadamard finite part integral method are used to solve singular integrals. Two-dimensional numerical examples are exploited to verify the effectiveness and compatibility of the algorithm for the multi-frequency analysis.