TECHNIQUES OF INTEGRATION 7

7.6 Integration Using Tables and Computer Algebra Systems In this section, we will learn: How to use tables and computer algebra systems in integrating functions that have elementary antiderivatives.

TABLES & COMPUTER ALGEBRA SYSTEMS However, you should bear in mind that even the most powerful computer algebra systems (CAS) can’t find explicit formulas for:  The antiderivatives of functions like e x 2  The other functions at the end of Section 7.5

TABLES OF INTEGRALS Tables of indefinite integrals are very useful when:  We are confronted by an integral that is difficult to evaluate by hand.  We don’t have access to a CAS.

TABLES OF INTEGRALS A relatively brief table of 120 integrals, categorized by form, is provided on the Reference Pages.

TABLES OF INTEGRALS More extensive tables are available in:  CRC Standard Mathematical Tables and Formulae, 31st ed. by Daniel Zwillinger (Boca Raton, FL: CRC Press, 2002), which has 709 entries  Gradshteyn and Ryzhik’s Table of Integrals, Series, and Products, 6e (San Diego: Academic Press, 2000), which contains hundreds of pages of integrals

TABLES OF INTEGRALS Remember, integrals do not often occur in exactly the form listed in a table.  Usually, we need to use substitution or algebraic manipulation to transform a given integral into one of the forms in the table.

TABLES OF INTEGRALS The region bounded by the curves y = arctan x, y = 0, and x = 1 is rotated about the y-axis. Find the volume of the resulting solid. Example 1

TABLES OF INTEGRALS Using the method of cylindrical shells, we see that the volume is: Example 1

TABLES OF INTEGRALS In the section of the Table of Integrals titled Inverse Trigonometric Forms, we locate Formula 92: Example 1

TABLES OF INTEGRALS So, the volume is: Example 1

TABLES OF INTEGRALS Use the Table of Integrals to find  If we look at the section of the table titled ‘Forms involving,’ we see that the closest entry is number 34: Example 2

TABLES OF INTEGRALS  That is not exactly what we have.  Nevertheless, we will be able to use it if we first make the substitution u = 2x: Example 2

TABLES OF INTEGRALS  Then, we use Formula 34 with a 2 = 5 (so ): Example 2

TABLES OF INTEGRALS Use the Table of Integrals to find  If we look in the section Trigonometric Forms, we see that none of the entries explicitly includes a u 3 factor.  However, we can use the reduction formula in entry 84 with n = 3: Example 3

TABLES OF INTEGRALS Now, we need to evaluate  We can use the reduction formula in entry 85 with n = 2.  Then, we follow by entry 82: Example 3

TABLES OF INTEGRALS Combining these calculations, we get where C = 3K Example 3

TABLES OF INTEGRALS Use the Table of Integrals to find  The table gives forms involving,, and, but not.  So, we first complete the square: Example 4

TABLES OF INTEGRALS If we make the substitution u = x + 1 (so x = u – 1), the integrand will involve the pattern : Example 4

TABLES OF INTEGRALS The first integral is evaluated using the substitution t = u 2 + 3: Example 4

TABLES OF INTEGRALS For the second integral, we use the formula with : Example 4

TABLES OF INTEGRALS Thus, Example 4

COMPUTER ALGEBRA SYSTEMS We have seen that the use of tables involves matching the form of the given integrand with the forms of the integrands in the tables.

CAS Computers are particularly good at matching patterns. Also, just as we used substitutions in conjunction with tables, a CAS can perform substitutions that transform a given integral into one that occurs in its stored formulas.  So, it isn’t surprising that CAS excel at integration.

CAS That doesn’t mean that integration by hand is an obsolete skill.  We will see that, sometimes, a hand computation produces an indefinite integral in a form that is more convenient than a machine answer.

CAS VS. MANUAL COMPUTATION To begin, let’s see what happens when we ask a machine to integrate the relatively simple function y = 1/(3x – 2)

CAS VS. MANUAL COMPUTATION Using the substitution u = 3x – 2, an easy calculation by hand gives: However, Derive, Mathematica, and Maple return:

CAS VS. MANUAL COMPUTATION The first thing to notice is that CAS omit the constant of integration.  That is, they produce a particular antiderivative, not the most general one.  Thus, when making use of a machine integration, we might have to add a constant.

CAS VS. MANUAL COMPUTATION Second, the absolute value signs are omitted in the machine answer.  That is fine if our problem is concerned only with values of x greater than.  However, if we are interested in other values of x, then we need to insert the absolute value symbol.

CAS In the next example, we reconsider the integral of Example 4. This time, though, we ask a machine for the answer.

CAS Use a CAS to find  Maple responds with: Example 5

CAS That looks different from the answer in Example 4. However, it is equivalent because the third term can be rewritten using the identity Example 5

CAS Thus,  The resulting extra term can be absorbed into the constant of integration. Example 5

CAS Mathematica gives:  It combined the first two terms of Example 4 (and the Maple result) into a single term by factoring. Example 5

CAS Derive gives:  The first term is like the first term in the Mathematica answer.  The second is identical to the last term in Example 4. Example 5

CAS Use a CAS to evaluate  Maple and Mathematica give the same answer: Example 6

CAS It’s clear that both systems must have expanded (x 2 + 5) 8 by the Binomial Theorem and then integrated each term. Example 6

CAS If we integrate by hand instead, using the substitution u = x 2 + 5, we get:  For most purposes, this is a more convenient form of the answer. Example 6

CAS Use a CAS to find  In Example 2 in Section 7.2, we found: E. g. 7—Equation 1

CAS  Derive and Maple report:  Mathematica produces: Example 7

CAS We suspect there are trigonometric identities that show these three answers are equivalent.  Indeed, if we ask Derive, Maple, and Mathematica to simplify their expressions using trigonometric identities, they ultimately produce the same form of the answer as in Equation 1. Example 7