Advanced Calculus fi..

222 Advanced Calculus, Fifth EditionTABLE 4.1x esLn ];+O ' es'"'dt 1; eslnrdt = F(x)and if these are divided by Ax = 0.1, one should obtain f (x) again approximately.For example,This gives f (x) for x about equal to 0.55. 7The complete indefinite integral (4.22) is now given by F(x) + C, where F(x)is given by Table 4.1, for 0 5 x ( 1. The range of x can clearly be extended as faras desired and the accuracy improved as required, so that the method should meetall practical needs.Elliptic integrals. An integral of the formis an example of an elliptic integral. It can be shown that this integral cannot beevaluated in terms of elementary functions. Accordingly, some numerical method iscalled for. Because of the importance of this integral in applications, elaborate tableshave been computed of the functionfor various values of the constant k. Such tables are given in A Short Table of Integrals,3rd ed., by B. 0. Peirce (Boston: Ginn, 1956) and Tables of Higher Functions6th ed. revised by F. Losch, E. Jahnke and F. Emde (New York: McGraw-Hill,1960)..>I.