Rationalizing Substitutions

Abstract

In this chapter we look at a few more substitutions that can be used effectively to transform some types of integrals to those involving rational functions. In this way we may be able to integrate the original functions by referring to the method of Partial Fractions from Chapter 8. Before we start though we need to remind the reader of a notion called the least common multiple of two given positive integers. As the phrase suggests, the least common multiple (abbr. lcm) of two numbers x, y (assumed integers) is the smallest number that is a multiple of each one of x and y. For example, the lcm{2, 4} = 4, since 4 is the smallest number that is a multiple of both 2 and itself. Other examples include, the lcm{2, 3, 4} = 12, lcm{2, 3} = 6, lcm{2, 5} = 10, lcm{2, 4, 6} = 12 etc. Thus, given two fractions, say 1/2 and 1/3, the least common multiple of their denominators is 6.