Methods Of Integration

During this course we have developed several methods for evaluating indefinite integrals, such as the Sum and Constant Rules, change of variables, integration by parts, and partial fractions. In the integration problems up to this point, the method to be used was usually given. But in a real life integration problem, one will have to decide which method to use on his own.

This section has two purposes. First, to review all the methods of integration. Second, to explain how one might decide which method to use for a given problem.

Almost all the examples and problems in this book involve what are called elementary functions. A real function f(x) is called an elementary function if f(x) is given by a term τ(x) which is built up from constants, sums, differences, products, quotients, powers, roots, exponential functions, logarithmic functions, and trigonometric functions and their inverses. These are the functions for which we have introduced names. Given an elementary function f(x), an indefinite integral ∫ f(x) dx may or may not be an elementary function. For example, it turns out that the integrals

are not elementary functions.

What is meant by the problem "evaluate the indefinite integral ∫f(x)dx"? The problem is really the following.

Given an elementary function f(x), find another elementary function F(x) (if there is one) such that

This is a hard problem. Sometimes the integral is not an elementary function at all. Sometimes the integral is an elementary function but it can be found only by guesswork. There is no routine way to evaluate an indefinite integral. However, one can often find clues which will cut down on the guesswork. We shall point out some of these clues here.

The corresponding problem for differentiation is much easier. Given an elementary function f(x), the derivative f'(x) is always another elementary function. It can be found in a routine way using the rules for differentiation and the Chain Rule.