This article presents approximations of the hypersingular integrals ∫ba g(x)(x-t)αdx and ∫ba g(x)｜x-t|αdx with arbitrary singular point t ∈ (a,b) and negative fraction number α ＜-1.These general expansions are applicable to a large range of hypersingular integrals,including both popular hypersingular integrals discussed in the literature and other important ones which have not been addressed yet.The corresponding mid-rectangular formulas and extrapolations,which can be calculated in fairly straightforward ways,are investigated.Numerical examples are provided to illustrate the features of the numerical methods and verify the theoretical conclusions.