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Unformatted text preview: 538 CHAPTER 6 . . Integration Techniques 6-30 EXPLORATORY EXERCISES 1. In developing the definite integral, we looked at sums such as n i = 1 2 i 2 + i . As with Riemann sums, we are especially inter- ested in the limit as n → ∞ . Write out several terms of the sum and try to guess what the limit is. It turns out that this is one of the few sums for which a precise formula exists, because this is a telescoping sum. To find out what this means, write out the partial fractions decomposition for 2 i 2 + i . Using the partial fractions form, write out several terms of the sum and notice how much cancellation there is. Briefly describe why the term telescoping is appropriate, and determine n i = 1 2 i 2 + i . Then find the limit as n → ∞ . Repeat this process for the telescoping sum n i = 2 4 i 2 − 1 . 2. Use the substitution u = x 1 / 4 to evaluate 1 x 5 / 4 + x dx . Use similar substitutions to evaluate 1 x 1 / 4 + x 1 / 3 dx , 1 x 1 / 5 + x 1 / 7 dx and 1 x 1 / 4 + x 1 / 6 dx . Find the form of the substitution for the general integral 1 x p + x q dx . 6.5 INTEGRATION TABLES AND COMPUTER ALGEBRA SYSTEMS Ask anyone who has ever needed to evaluate a large number of integrals as part of their work (this includes engineers, mathematicians, physicists and others) and they will tell you that they have made extensive use of integral tables and/or a computer algebra system. These are extremely powerful tools for the professional user of mathematics. However, they do not take the place of learning all the basic techniques of integration. To use a table, you often must first rewrite the integral in the form of one of the integrals in the table. This may require you to perform some algebraic manipulation or to make a substitution. While a CAS will report an antiderivative, it will occasionally report it in an inconvenient form. More significantly, a CAS will from time to time report an answer that is (at least technically) incorrect. We will point out some of these shortcomings in the examples that follow. Using Tables of Integrals We include a small table of indefinite integrals at the back of the book. A larger table can be found in the CRC Standard Mathematical Tables . An amazingly extensive table is found in the book Table of Integrals, Series and Products , compiled by Gradshteyn and Ryzhik. EXAMPLE 5.1 Using an Integral Table Use a table to evaluate √ 3 + 4 x 2 x dx . Solution Certainly, you could evaluate this integral using trigonometric substitution. However, if you look in our integral table, you will find √ a 2 + u 2 u du = a 2 + u 2 − a ln a + √ a 2 + u 2 u + c . (5.1) 6-31 SECTION 6.5 . . Integration Tables and Computer Algebra Systems 539 Unfortunately, the integral in question is not quite in the form of (5.1). However, we can fix this with the substitution u = 2 x , so that du = 2 dx . This gives us √ 3 + 4 x 2 x dx = 3 + (2 x ) 2 2 x (2) dx = √ 3 + u 2 u du = 3 + u 2 − √ 3 ln √ 3 + √ 3 + u 2 u + c = 3 + 4 x 2 −...
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This note was uploaded on 10/03/2010 for the course CHE 2C CHE 2C taught by Professor Atsumi during the Spring '10 term at UC Davis.
- Spring '10