We have investigated the internal relation among the first integral, Hojman's method and Birkhoff's equations. There is a special equivalent relationship between Birkhoffian and Birkhoffian functions constructed by Hojman's method, and from this we can derive a more general form of Hojman's method. Then, combining the special equivalent relationship and Birkhoff's equations, we can derive the cyclic integral in Birkhoffian sense. An example is given to illustrate the application of the results.