A comparative study on different integration schemes is presented in this thesis. The aim is to develop an integration method for discontinuous problems. The motivation comes from the finite cell method with the three types of cells: the cells that are completely outside the physical domain, the cells that are completely inside the physical domain, and the cells that are cut by the physical domain boundary. This method is based on higher order finite elements for which the mesh does not necessarily conform to the physical boundaries. Therefore, for some elements, the material properties are not continuous and there are jumps from the material to the void in one element. In order to calculate stiffness matrix and load vector of the third types of cells, we encounter with the numerical integration of discontinuous functions. It is shown that the mapping method can well do this service and the error is in the acceptable range of computational mechanics. The effort reveals that the problem of meshing of the domain for FEM purposes can be replaced by less expensive effort to perform accurate integration over the domain discretized approximately. In this thesis, we focus on the problem of the numerical integration of discontinuous functions in order to increase the accuracy of solution for the third type of cells. The thesis has been arranged as follows: At first, we express the history of the finite elements method. In the second chapter, the finite elements method is studied. The finite cell method is investigated in chapter three. In chapter four, introduced which are useful for second type of cells. Some numerical integration methods for the discontinuous functions and some examples are presented in chapter five. Finally, these methods are compared and then some conclusion and suggestions are stated in the last chapter of thesis. It is worthy of mention that, although the formulation used in this thesis is applied to linear elasticity problems and examined for 2D cases, the concepts are generally valid.