A　family　of　integral-type　goodness-of-fit　tests　is　investigated.　This　family　includes　some　existing　tests,　such　as　the　Cram,r-von　Mises　test　and　Anderson-Darling　test,　etc.　The　asymptotic　distributions　of　the　tests　in　the　family　under　the　null　and　local　alternative　hypotheses　are　established.　The　almost　sure　convergence　under　a　fixed　underlying　distribution　is　obtained.　Furthermore,　simulations　are　conducted　to　compare　the　powers　of　the　tests　in　the　family.　Simulation　results　show　that　for　different　alternatives,　the　more　powerful　tests　are　different,　and　the　parameter　lambda　has　great　influence　on　the　tests　in　small　sample　cases.