1. 7
TECHNIQUES OF INTEGRATION

2. 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.

3. 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

4. Tables of indefinite integrals are very useful when:
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.

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

6. More extensive tables are available in:
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

7. 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.

8. Find the volume of the resulting solid.
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.

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

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

11. TABLES OF INTEGRALS
Example 1
So, the volume is:

12. Use the Table of Integrals to find
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:

13. That is not exactly what we have.
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:

14. Then, we use Formula 34 with a2 = 5 (so ):
TABLES OF INTEGRALS
Example 2
Then, we use Formula 34 with a2 = 5 (so ):

15. Use the Table of Integrals to find
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:

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

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

18. Use the Table of Integrals to find
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:

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

20. The first integral is evaluated using the substitution t = u2 + 3:
TABLES OF INTEGRALS
Example 4
The first integral is evaluated using the substitution t = u2 + 3:

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

22. TABLES OF INTEGRALS
Example 4
Thus,

23. 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.

24. Computers are particularly good at matching patterns.
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.

25. That doesn’t mean that integration by hand is an obsolete skill.
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.

26. 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)

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

28. 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.

29. 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.

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

31. CAS
Example 5
Use a CAS to find
Maple responds with:

32. That looks different from the answer in Example 4.
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

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

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

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

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

37. CAS
Example 6
It’s clear that both systems must have expanded (x2 + 5)8 by the Binomial Theorem and then integrated each term.

38. 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.

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

40. Derive and Maple report:
CAS
Example 7
Derive and Maple report:
Mathematica produces:

41. 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.