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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

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.

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

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.

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.

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

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.

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.

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

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

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

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

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

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

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:

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

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

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

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

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

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

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

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.

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.

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.

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)

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

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.

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.

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.

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

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)

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.

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.

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.

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

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.

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.

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

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

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.

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.