The　flexoelectricity,　which　is　a　new　electromechanical　coupling　phenomenon　between　strain　gradients　and　electric　polarization,　has　a　great　influence　on　the　fracture　analysis　of　flexoelectric　solids　due　to　the　large　gradients　near　the　cracks.　On　the　other　hand,　although　the　flexoelectricity　has　been　extensively　investigated　in　recent　decades,　the　study　on　flexoelectricity　in　nonhomogeneous　materials　is　still　rare,　especially　the　fracture　problems.　Therefore,　in　this　manuscript,　the　conservation　integrals　for　nonhomogeneous　flexoelectric　materials　are　obtained　to　solve　the　fracture　problem.　Application　of　operators　such　as　grad,　div,　and　curl　to　electric　Gibbs　free　energy　and　internal　energy,　the　energy-momentum　tensor,　angular　momentum　tensor,　and　dilatation　flux　can　also　be　derived.　We　examine　the　correctness　of　the　conservation　integrals　by　comparing　with　the　previous　work　and　discuss　the　operator　method　here　and　Noether　theorem　in　the　previous　work.　Finally,　considering　the　flexoelectric　effect,　a　nonhomogeneous　beam　problem　with　crack　is　solved　to　show　the　application　of　the　conservation　integrals.