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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, x^{a}, e^{x}, a^{x}, ln x, log_{a }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

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:

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

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

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

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

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

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

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

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

where n is a positive integer, p^{2} - 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

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

I Integration of integrals of types

and

These integrals are immediately obtained using the formulas for the integrals of functions 1/u and
u^{n}.

II Integration of integrals of type

Case 1. p = 0. If p = 0, then q must be positive since p^{2} - 4q < 0. Set q = a^{2} and substitute
into 3 giving

The integration is now immediate using formulas

Case 2. p 0. We can write

The quantity

is positive, since p^{2} - 4q < 0.

Set

and we have

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

III Integration of integrals of type

Again,

Set

and we have

The integral

is immediately integrable using the power formula (i.e. substitute z = t^{2} + a^{2}) and the integral

can be integrated using the reduction formula

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Integration of integrals of type

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

Make the substitution

Then

From the following identity

we deduce

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

using the method for integrating rational functions.

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Integration of integrals of type

where is any rational function of x and y = .

By “completing the square” we obtain

where

with the lower sign of c(t^{2}
A^{2}) being used when 4ac - b^{2} is negative.

Thus

There are three cases.

Case 1. 4ac- b^{2 }< 0, c < 0. In this case the substitutions

will reduce the integral to the form

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- b^{2 }> 0, c > 0. In this case the substitutions

will reduce the integral to the form

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- b^{2 }< 0, c > 0. In this case the substitutions

will reduce the integral to the form

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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