The main objective of the research is to find a new numeric formula to calculate the values of single definite integrals with continuous integrands or these which its integrands are continuous but which is improper at the derivative or whose integrands are improper at one or both boundary of integration by depending on the Mid-point rule which is one of the Newton-Cotes integrative rules, and find the corresponding correction terms when the function of integral is continuous, discontinuous or its derivative is not continuous in the integral region. It s also aim to compare the results of this rule with the results of the Newton-Cotes rules (Trapezoidal, Mid-point and Simpson), and we will show the extent of preference over the above methods, also to explain the importance of the correction limits associated with the new base by using with it Romberg acceleration, where we obtained good results clearly in improving the values of integrations in terms of accuracy and the number of partial periods.