techniques of integration 7. 7.6 integration using tables and computer algebra systems in this...
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TECHNIQUES OF INTEGRATION
7
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
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:
1
02 arctanV x x dx
Example 1
TABLES OF INTEGRALS
In the section of the Table of Integrals
titled Inverse Trigonometric Forms,
we locate Formula 92:
21 11
tan tan2
u uu u du u C
u
Example 1
TABLES OF INTEGRALS
So, the volume is:1 1
0
121
0
12 1
0
1
212
2 tan
12 tan
2 2
( 1) tan
(2 tan 1 1)
[2( / 4) 1]
V x x dx
x xx
x x x
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:
2
25 4
xdx
x
Example 2
2 2a u
2 22 2 1
2 2sin
2 2
u u a udu a u C
aa u
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:
2 2
2 2
2
2
( / 2)
25 4 5
1
8 5
x u dudx
x u
udu
u
Example 2
TABLES OF INTEGRALS
Then, we use Formula 34 with a2 = 5 (so ):5a
2 2
2 2
2 1
2 1
1
85 4 5
1 55 sin
8 2 2 5
5 25 4 sin
8 16 5
x udx du
x u
u uu C
x xx C
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 u3 factor.
However, we can use the reduction formula in entry 84 with n = 3:
3 sinx x dx
Example 3
3 3 2sin cos 3 cosx x dx x x x x dx
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:
2 2
2
cos sin 2 sin
sin 2(sin cos )
x x dx x x x x dx
x x x x x K
2 cosx x dx
1cos sin sinn n nu u du u u n u u du
Example 3
TABLES OF INTEGRALS
Combining these calculations, we get
where C = 3K
3 3 2sin cos 3 sin
6 cos 6sin
x x dx x x x x
x x x C
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:
2 2 4x x x dx
Example 4
2 2a x 2 2a x2 2x a 2ax bx c
2 22 4 ( 1) 3x x x
TABLES OF INTEGRALS
If we make the substitution u = x + 1
(so x = u – 1), the integrand will involve
the pattern :2 2a u
2 2
2 2
2 4 ( 1) 3
3 3
x x x dx u u du
u u du u du
Example 4
TABLES OF INTEGRALS
The first integral is evaluated using
the substitution t = u2 + 3:
2 12
3/ 21 22 3
2 3/ 213
3
( 3)
u u du t dt
t
u
Example 4
TABLES OF INTEGRALS
For the second integral, we use the formula
with :
22 2 2 2 2 2ln( )
2 2
u aa u du a u u a u C
3a
2 2 2323 3 ln( 3)
2
uu du u u u
Example 4
TABLES OF INTEGRALS
Thus,
2
2 3 2 213
232
2 4
1( 2 4) 2 4
2
ln( 1 2 4)
x x x dx
xx x x x
x x x C
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:
13
1ln 3 2
3 2dx x C
x
13 ln(3 2)x
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.
23
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:
2 2 4x x x dx
2 3 2 21 13 4( 2 4) (2 2) 2 4
3 3arcsinh (1 )2 3
x x x x x
x
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
2arcsinh ln( 1) x x x
Example 5
CAS
Thus,
The resulting extra term can be absorbed into the constant of integration.
213
2
2
3 3arcsinh (1 ) ln (1 ) (1 ) 1
3 3
1ln 1 (1 ) 3
31
ln ln 1 2 43
x x x
x x
x x x
Example 5
32 ln 1/ 3
CAS
Mathematica gives:
It combined the first two terms of Example 4 (and the Maple result) into a single term by factoring.
225 3 12 4 arcsinh
6 6 3 2 3
x x xx x
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.
2 21
6
232
2 4 (2 5)
ln 2 4 1
x x x x
x x x
Example 5
CAS
Use a CAS to evaluate
Maple and Mathematica give the same answer:
2 8( 5)x x dx
18 16 14 12 105 1750118 2 3
8 6 4 2218750 3906253 2
50 4375
21875 156250
x x x x x
x x x x
Example 6
CAS
It’s clear that both systems
must have expanded (x2 + 5)8
by the Binomial Theorem and then
integrated each term.
Example 6
CAS
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.
2 8 2 9118( 5) ( 5)x x dx x C
Example 6
CAS
Use a CAS to find
In Example 2 in Section 7.2, we found:
5 2sin cosx x dx
5 2
3 5 71 2 13 5 7
sin cos
cos cos cos
x x dx
x x x C
E. g. 7—Equation 1
CAS
Derive and Maple report:
Mathematica produces:
4 3 2 3 381 47 35 105sin cos sin cos cos x x x x x
5 31 164 192 320 448cos cos3 cos5 cos7x x x x
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