On a calculation of definite integrals by using of the calculation of indefinite integrals

It is known that in the construction of the numerical methods for solving of the initial-value problem of ODE in basically used the methods which have applied to the calculation of the definite integrals. Here for the computing of definite integrals propose to use the methods which have used in solving of the initial-value problem for the ODEs. The definite integrals express by the indefinite integrals which are the solutions of the above-mentioned problems. For the construction of more exact methods for calculation of the definite integrals here propose to use forward-jumping (advanced) methods and the hybrid methods. Here establishes some connection between the Gauss and hybrid methods. And also have determined some necessary conditions for which the coefficients of the proposed methods have to satisfy. Constructed stable methods with the degree p ≤ 8. Shown that, how received here results can be applied to the computing of the double integrals. For this aim, determines some connection between double integrals and single definite integrals. By using this relation have constructed methods which are applied to calculate the double integrals. Advantages of this method illustrated by calculation of model double integral by the constructed here methods.