In this paper, the effects of different numerical integration schemes on the distributed Lagrange multiplier/fictitious domain (DLM/FD) method with body-unfitted meshes are studied for solving different types of interface problems: elliptic-, Stokes- and Stokes/elliptic-interface problems. Commonly-used numerical integration schemes, compound type formulas and a specific subgrid integration scheme are presented for the mixed finite element approximation and the comparison between them is illustrated in numerical experiments, showing that different numerical integration schemes have significant effects on approximation errors of the DLM/FD finite element method for different types of interface problems, especially for Stokes- and Stokes/elliptic-interface problems, and that the subgrid integration scheme always results in numerical solutions with the best accuracy.