The spectral method is considered in solving boundary integral equations. By subtracting known solutions from the Helmholtz integral equation, the singularities of the Helmholtz integral equation are removed. Therefore, any order of basis functions, used in spectral methods, that times nonsingular kernels in the modified Helmholtz integral equation can be integrated easily and efficiently. Instead of solving the variables at collocation points using conventional methods, the generalized Fourier coefficients are solved by the spectral method which reduces the number of unknowns and the storage of matrix elements in computation. Scattering and radiation problems from a sphere are used to illustrate the technique in the present paper.