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.