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## TECHNIQUES OF INTEGRATION

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**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.