Based on the method of first integral,the problems of constructing functions,such as the differential mid-value theorem,the solution of integrating factor and thr zero existence of some functions,are explored.

The paper constructs an auxiliary function by using initial integral methodo,ffers a new method to prove Lagrange’s mean-value theorem and Cauchy’s mean-value theoremf,inds several conclusions in the application of differential mean-value theorem.

The paper constructed an auxiliary function by using initial integral method,offered a new method to prove Lagrange s mean-value theorem and found several conclusions in the application of differential calculus.

With the aid of computer symbolic systems Maple,and by the first integral method which is to obtain a series of exact solutions to some nonlinear differential equation,we have solved the Burgers-Huxley equation,and obtained a number of new exact solutions including solutions with arbitrary constant,soliton-like solutions and implicit solutions.

Applying a new method called the first integral method,we obtain some exact solutions of a kind of reaction diffusion equations u_t-u_(xx)+αu+βu~2+γu~3=0.

Given a different method for the proving the Cauchy mean-value theorem by using first integral method to construct assistant functions and in weakened or enhanced condition,get a symmetry result which is a extension of the Cauchy mean-value theorem.