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            ELEMENTARY FUNCTIONS, GENERAL METHODS OF INTEGRATION



Elementary functions.

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Consider the following set of functions which we shall call the Simple Elementary Functions. They are assumed to have complex arguments.


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The Simple Elementary Functions: c, x, xa, ex, ax, ln x, loga x, sin x, cos x, tan x, ..., sin-1 x, cos-1 x, tan-1 x, ... , sinh x, cosh x, tanh x, ..., sinh-1 x, cosh-1 x, tanh-1 x, ... where c and a are complex constants and x is a complex variable.

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The simple elementary functions include the familiar exponential, logarithmic, trigonometric and hyperbolic functions plus the variable x and the constant c. Utilizing the term “simple elementary function” we now define the term “elementary function”.


Def. Elementary function. An elementary function is any function that can be built from the simple elementary functions by the four arithmetic operations of addition, subtraction, multiplication and division and the operation of taking a function of a function, each of these operations being performed a finite number of times.



Example. Any complex expression built by these rules from the simple elementary functions is an elementary function. For example


                         ole.gif


is an elementary function.


Note. All of the simple elementary functions themselves also qualify as being elementary functions, from the wording of this definition. However, the concept of an elementary function is that of a complex function built up of simple elementary functions. The simple elementary functions are often referred to as “the elementary functions”, a cause for some confusion.


The derivative of any elementary function is again an elementary function. However, the integral of an elementary function may not be an elementary function. The integral of an elementary function may lead to a function outside the set of elementary functions.

 

General methods of integration. Not every function can be integrated. Even if a function can be integrated the integral may not be an elementary function. Often one finds himself in a position of attempting to integrate some function by one device or another with no certainty that the integral exists. There are, however, certain large and important classes of functions which can be systematically integrated by general methods.


The following three classes of integrals can be systematically integrated in terms of elementary functions by general methods:


ole1.gif

 

                        where R(x) is any rational function of x.


ole2.gif


                        where R(sin x, cos x) is any rational function of sin x and cos x.



ole3.gif


                        where P(x) is any first or second degree polynomial in x and R(x, ole4.gif ) is any rational function of x and ole5.gif .






Integrals of the following class can be integrated in terms of elliptic integrals:


ole6.gif


            where P(x) is any third or fourth degree polynomial in x with no repeated roots and R(x, ole7.gif ) is any rational function of x and ole8.gif .


By a suitable change of variable, such an integral can be reduced to a sum of elementary integrals and integrals of the following types:


             ole9.gif


 


             ole10.gif





             ole11.gif




These are incomplete elliptic integrals of the first, second and third kind, respectively. When expressed in terms of t = sin w, they are Legendre’s normal forms. The constant k (0 < k < 1) is the modulus and a is an arbitrary constant. If f = π/2, the integrals are called complete.


Series evaluation of the elliptic integrals may be made and numerical tables for them are available. They are called elliptic because they were first studied in attempts to determine the circumference of the ellipse.

                                                            The International Dictionary of Applied Mathematics. Elliptic integral







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Integration of integrals of type


          ole12.gif

 

            where R(x) is any rational function of x.



Procedure.


1] If the rational function R(x) is a complex expression, reduce it to a rational fraction i.e. to the form g(x) / G(x) where g(x) and G(x) are polynomials.


2] If the rational fraction g(x) / G(x) is improper reduce it to the sum of a polynomial and a proper fraction h(x) / H(x) ( a proper fraction has a numerator of lower degree than the denominator).


3] Reduce the proper fraction h(x) / H(x) to the sum of partial fractions of the type



             ole13.gif



where n is a positive integer, p2 - 4q < 0, and the denominators of the fractions consist of the linear and irreducible quadratic factors of the polynomial H(x). Any proper fraction can be reduced to the sum of integrals of this type.


4] The evaluation of the integral


                         ole14.gif


is now reduced to the evaluation of the integral of a polynomial plus integrals of the types



ole15.gif



ole16.gif



  ole17.gif  


ole18.gif




I Integration of integrals of types



ole19.gif


and


ole20.gif


These integrals are immediately obtained using the formulas for the integrals of functions 1/u and un.



II Integration of integrals of type



ole21.gif



Case 1. p = 0. If p = 0, then q must be positive since p2 - 4q < 0. Set q = a2 and substitute into 3 giving



  ole22.gif

 



The integration is now immediate using formulas


 

ole23.gif



ole24.gif




Case 2. p ole25.gif 0. We can write


             ole26.gif


                                     ole27.gif



The quantity


                         ole28.gif


is positive, since p2 - 4q < 0.


Set


             ole29.gif



and we have



  ole30.gif



                                                 ole31.gif                                                                                                  

 

                                                 ole32.gif


which can be immediately integrated using formulas (1) and (2) above.




III Integration of integrals of type


  ole33.gif



Again,

                         ole34.gif


                                     ole35.gif



Set


             ole36.gif



and we have



  ole37.gif



                                                 ole38.gif


                                                 ole39.gif




The integral


                         ole40.gif


is immediately integrable using the power formula (i.e. substitute z = t2 + a2) and the integral


             ole41.gif


can be integrated using the reduction formula


             ole42.gif









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Integration of integrals of type


              ole43.gif


                        where R(sin x, cos x) is any rational function of sin x and cos x.



Make the substitution


             ole44.gif


Then


                         ole45.gif



From the following identity


             ole46.gif



we deduce


             ole47.gif




On substituting these values for sin x and cos x into R(sin x, cos x) we see that R(sin x, cos x) is a rational function of t i.e. R(sin x, cos x) = r(t). Thus we can integrate


             ole48.gif


using the method for integrating rational functions.








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Integration of integrals of type


              ole49.gif

 

                        where ole50.gif is any rational function of x and y = ole51.gif .



By “completing the square” we obtain


ole52.gif


where


             ole53.gif


with the lower sign of c(t2 ole54.gif A2) being used when 4ac - b2 is negative.


Thus

             ole55.gif




There are three cases.


Case 1. 4ac- b2 < 0, c < 0. In this case the substitutions


             ole56.gif


will reduce the integral to the form


             ole57.gif


                        where R(sin x, cos x) is any rational function of sin x and cos x.


We then deal with it according to the procedure for that type.





Case 2. 4ac- b2 > 0, c > 0. In this case the substitutions


             ole58.gif


will reduce the integral to the form


             ole59.gif


                        where R(sin x, cos x) is any rational function of sin x and cos x.


We then deal with it according to the procedure for that type.




Case 3. 4ac- b2 < 0, c > 0. In this case the substitutions


             ole60.gif


will reduce the integral to the form


             ole61.gif


                        where R(sin x, cos x) is any rational function of sin x and cos x.


We then deal with it according to the procedure for that type.






References.

Osgood. Advanced Calculus. Chap. 1.


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