7 TECHNIQUES OF INTEGRATION
TECHNIQUES OF INTEGRATION • 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 ex2 • 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 Example 1 • 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.
TABLES OF INTEGRALS Example 1 • Using the method of cylindrical shells, we see that the volume is:
TABLES OF INTEGRALS Example 1 • In the section of the Table of Integrals titled Inverse Trigonometric Forms, we locate Formula 92:
TABLES OF INTEGRALS Example 1 • So, the volume is:
TABLES OF INTEGRALS Example 2 • 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:
TABLES OF INTEGRALS Example 2 • That is not exactly what we have. • Nevertheless, we will be able to use it if we first make the substitution u = 2x:
TABLES OF INTEGRALS Example 2 • Then, we use Formula 34 with a2 = 5 (so ):
TABLES OF INTEGRALS Example 3 • 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 u3 factor. • However, we can use the reduction formula in entry 84 with n = 3:
TABLES OF INTEGRALS Example 3 • Now, we need to evaluate • We can use the reduction formula in entry 85 with n = 2. • Then, we follow by entry 82:
TABLES OF INTEGRALS Example 3 • Combining these calculations, we get where C = 3K
TABLES OF INTEGRALS Example 4 • Use the Table of Integrals to find • The table gives forms involving , , and , but not . • So, we first complete the square:
TABLES OF INTEGRALS Example 4 • If we make the substitution u = x + 1 (so x = u – 1), the integrand will involve the pattern :
TABLES OF INTEGRALS Example 4 • The first integral is evaluated using the substitution t = u2 + 3:
TABLES OF INTEGRALS Example 4 • For the second integral, we use the formula • with :
TABLES OF INTEGRALS Example 4 • Thus,
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 Example 5 • Use a CAS to find • Maple responds with:
CAS Example 5 • That looks different from the answer in Example 4. • However, it is equivalent because the third term can be rewritten using the identity
CAS Example 5 • Thus, • The resulting extra term can be absorbed into the constant of integration.
CAS Example 5 • Mathematica gives: • It combined the first two terms of Example 4 (and the Maple result) into a single term by factoring.
CAS Example 5 • 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.
CAS Example 6 • Use a CAS to evaluate • Maple and Mathematica give the same answer:
CAS Example 6 • It’s clear that both systems must have expanded (x2 + 5)8by the Binomial Theorem and then integrated each term.
CAS Example 6 • If we integrate by hand instead, using the substitution u = x2 + 5, we get: • For most purposes, this is a more convenient form of the answer.
CAS E. g. 7—Equation 1 • Use a CAS to find • In Example 2 in Section 7.2, we found:
CAS Example 7 • Derive and Maple report: • Mathematica produces:
CAS Example 7 • 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.