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In this section, we will learn: How to use tables and computer algebra systems in integrating functions that have elementary antiderivatives.
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
The antiderivatives of functions like ex2
The other functions at the end of Section 7.5
4.
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.
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.
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.
TABLES OF INTEGRALS Example 1
Using the method of cylindrical shells,
we see that the volume is:
1
V 2 x arctan x dx
0
10.
TABLES OF INTEGRALS Example 1
In the section of the Table of Integrals
titled Inverse Trigonometric Forms,
we locate Formula 92:
2
1 u 1 1 u
u tan u du tan u C
u 2
11.
TABLES OF INTEGRALS Example 1
1
So, the volume is: V 2 x tan 1 x dx
0
2 1
x 1 1 x
2 tan x
2 20
2 1 1
( x 1) tan x x
0
1
(2 tan 1 1)
[2( / 4) 1]
12 2
12.
TABLES OF INTEGRALS Example 2
Use the Table of Integrals to find
2
x
5 4x dx
2
If we look at the section of the table titled
‘Forms involving a 2 u 2 ,’ we see that
the closest entry is number 34:
u2 u 2 2 a 2
1 u
du a u sin C
a2 u2 2 2 a
13.
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:
x2 (u / 2)2 du
dx
2 2 2
5 4x 5 u
1 u2
du
8 5 u2
14.
TABLES OF INTEGRALS Example 2
Then, we use Formula 34 with a2 = 5
(so a 5 ):
x2 1 u2
dx du
5 4x2 8 5 u2
1 u 2 5 1 u
5 u sin C
8 2 2 5
x 2 5 1 2x
5 4 x sin C
8 16 5
15.
TABLES OF INTEGRALS Example 3
Use the Table of Integrals to find
3
x sin x dx
If we look in the section Trigonometric Forms, we see
that none of the entries explicitly includes a u3 factor.
3 3 2
x sin x dx x cos x 3 cos x dx
x
However, we can use the reduction formula in entry 84
with n = 3:
16.
TABLES OF INTEGRALS Example 3
2
Now, we need to evaluate x cos x dx
We can use the reduction formula in entry 85
n n n 1
u cos u du u sin u n sin u du
u
with n = 2.
Then, we follow by entry 82:
2 2
x cos x dx x sin x 2 x sin x dx
x 2 sin x 2(sin x x cos x) K
17.
TABLES OF INTEGRALS Example 3
Combining these calculations, we get
3 3 2
x sin x dx x cos x 3x sin x
6 x cos x 6sin x C
where C = 3K
18.
TABLES OF INTEGRALS Example 4
Use the Table of Integrals to find
2
x x 2 x 4 dx
The table gives forms involving a 2 x 2 , a 2 x 2 ,
and x 2 a 2 , but not ax 2 bx c .
So, we first complete the square:
x 2 2 x 4 ( x 1) 2 3
19.
TABLES OF INTEGRALS Example 4
If we make the substitution u = x + 1
(so x = u – 1), the integrand will involve
2 2
the pattern a u :
2 2
x x 2 x 4 dx (u 1) u 3 du
2 2
u u 3 du u 3 du
20.
TABLES OF INTEGRALS Example 4
The first integral is evaluated using
the substitution t = u2 + 3:
2
u u 3 du 1
2 t dt
2 3/ 2
t
1
2 3
2 3/ 2
(u 3)
1
3
21.
TABLES OF INTEGRALS Example 4
For the second integral, we use the formula
2
u a
a 2 u 2 du a 2 u 2 ln(u a 2 u 2 ) C
2 2
with a 3 :
2 u 2 2
u 3 du u 3 2 ln(u u 3)
3
2
22.
TABLES OF INTEGRALS Example 4
2
x x 2 x 4 dx
2 32 x 1 2
( x 2 x 4)
1
3 x 2x 4
2
2
ln( x 1 x 2 x 4) C
3
2
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
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.
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:
1
3x 2 3 ln 3x 2 C
dx 1
However, Derive, Mathematica, and Maple
return: 1
ln(3 x 2)
3
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 23 .
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.
This time, though, we ask a machine for
the answer.
31.
CAS Example 5
2
Use a CAS to find x x 2 x 4 dx
Maple responds with:
2 32 2
1
3 ( x 2 x 4) (2 x 2) x 2 x 4
1
4
3 3
arc sinh (1 x)
2 3
32.
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
arc sinh x ln( x x 2 1)
33.
CAS Example 5
3 3 2
arc sinh (1 x) ln (1 x) 1
3 (1 x) 1
3 3
1
ln 1 x (1 x) 2 3
3
1
ln
3
2
ln x 1 x 2 x 4
The resulting extra term 32 ln 1/ 3 can be absorbed
into the constant of integration.
34.
CAS Example 5
Mathematica gives:
5 x x2 2 3 1 x
x 2 x 4 arc sinh
6 6 3 2 3
It combined the first two terms of Example 4
(and the Maple result) into a single term by factoring.
35.
CAS Example 5
Derive gives:
2 2
6 x 2 x 4 (2 x x 5)
ln
3
2 2
x 2 x 4 x 1
The first term is like the first term in the Mathematica
answer.
The second is identical to the last term in Example 4.
36.
CAS Example 6
2 8
Use a CAS to evaluate x( x 5) dx
Maple and Mathematica give the same answer:
1
18 x18 52 x16 50 x14 1750
3 x12
4375 x10
21875 x8 218750
3 x 6
156250 x 4
390625 2
2 x
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:
2 8 2 9
x( x 5) dx ( x 5) C
1
18
For most purposes, this is a more convenient
form of the answer.
39.
CAS E. g. 7—Equation 1
5 2
Use a CAS to find sin x cos x dx
In Example 2 in Section 7.2, we found:
5 2
sin x cos x dx
13 cos3 x 52 cos5 x 71 cos7 x C
40.
CAS Example 7
Derive and Maple report:
17 sin 4 x cos3 x 4
35 sin 2 x cos3 x 8
105 cos3 x
Mathematica produces:
5
64 cos x 1
192 cos 3x 320
3
cos 5 x 1
448 cos 7 x
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.
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