Many　machine　learning　problems,　such　as　medical　data　summarization　and　social　welfare　maximization,　can　be　modeled　as　the　problems　of　maximizing　monotone　submodular　functions.　Differentially　private　submodular　functions　under　cardinality　constraints　are　first　proposed　and　studied　to　solve　the　Combinatorial　Public　Projects　(CPP)　problem,　in　order　to　protect　personal　data　privacy　while　processing　sensitive　data.　However,　the　research　of　these　functions　for　privacy　protection　has　received　little　attention　so　far.　In　this　paper,　we　propose　to　study　the　differentially　private　submodular　maximization　problem　over　the　integer　lattice.　Our　main　contributions　are　to　present　differentially　private　approximation　algorithms　for　both　DR-submodular　and　integer　submodular　function　maximization　problems　under　cardinality　constraints　and　analyze　the　sensitivity　of　our　algorithms.