Module 2 Chapter 8 Indefinite Integrals

1
8
Indefinite Integrals
Case Study
8.1 Concepts of Indefinite Integrals
8.2 Indefinite Integration of Functions
8.3 Integration by Substitution
8.4 Integration by Parts
8.5 Applications of Indefinite Integrals
Chapter Summary
2
Case Study
By measuring the level of radioactivity with a
counter, it is estimated that the number of
radioactive particles, y, in the sample is
decreasing at a rate of 1000e0.046t per hour,
where t is expressed in hours.
According to what we learnt in Section 7.5 (Rates
of Change), we have
In order to express y in terms of t, we need to
find a function y of t such that its derivative
is equal to 1000e0.046t .
The process of finding a function from its
derivative is called integration and will be
discussed in this chapter.
3
8.1 Concepts of Indefinite Integrals
A. Definition of Indefinite Integrals
In previous chapters, we learnt how to find the
derivative of a given function.
Suppose we are given a function x2, by
differentiation, we have
As 2x is the derivative of x2, we call x2 the
primitive function (or antiderivative) of 2x.
Generally,
for any differentiable function F(x), we have the
following definition
Although x2 is a primitive function of 2x, it is
not the unique primitive function.
If we add an arbitrary constant C
to x2 and differentiate it, we have
Thus, x2 C is also a primitive function of 2x
for an arbitrary constant C.
4
8.1 Concepts of Indefinite Integrals
A. Definition of Indefinite Integrals
In order to represent all the primitive functions
of a function f (x), we introduce the concept of
indefinite integral as below

• Note
• In the notation of , f (x) is
  called the integrand, and
• is called the integral sign. The process of
  finding the primitive function is called
  integration.

2. C is called the constant of integration (or
integration constant).
5
8.1 Concepts of Indefinite Integrals
B. Basic Formulas of Indefinite Integrals
As integration is the reverse process of
differentiation, the basic formulas for
integrations can be derived from the
differentiation formulas.
For example Since
,
Note 1. Formula 8.1 is also called the Power
Rule for integration.
6
8.1 Concepts of Indefinite Integrals
B. Basic Formulas of Indefinite Integrals
In addition to the Power Rule, we can use the
similar way to derive the following integration
formulas
7
8.1 Concepts of Indefinite Integrals
C. Basic Properties of Indefinite Integrals
Theorem 8.1 If k is a non-constant, then
.
Proof
By definition, C
, where C is an arbitrary constant.
On the other hand,
Since C and kC are arbitrary constants, the
expressions kg(x) ? C and kg(x) ? kC represent
the same family of primitive functions.
8
8.1 Concepts of Indefinite Integrals
C. Basic Properties of Indefinite Integrals
Proof
Let and
, where C1 and C2 are
arbitrary constants.
By definition,
On the other hand,
Since C1 ? C2 is an arbitrary constant, the
expressions F(x) ? G(x) ? C and F(x) ? G(x) ? C1
? C2 represent the same family of primitive
functions.
9
8.1 Concepts of Indefinite Integrals
C. Basic Properties of Indefinite Integrals
Example 8.1T
Find
Solution
10
8.1 Concepts of Indefinite Integrals
C. Basic Properties of Indefinite Integrals
Example 8.2T
Find
Solution
11
8.1 Concepts of Indefinite Integrals
C. Basic Properties of Indefinite Integrals
Example 8.3T
Find
Solution
12
8.1 Concepts of Indefinite Integrals
C. Basic Properties of Indefinite Integrals
Example 8.4T
Find
Solution
13
8.1 Concepts of Indefinite Integrals
C. Basic Properties of Indefinite Integrals
Example 8.5T
Find
Solution
14
8.1 Concepts of Indefinite Integrals
C. Basic Properties of Indefinite Integrals
Example 8.6T

• Let y ? ln x ln (x ? 1).
• Find .
• Hence find .

Solution
(a)
(b) By (a),
15
8.2 Indefinite Integration of Functions
A. Integration of Functions Involving the
Expression (ax b)
The integration formulas mentioned in Section 8.1
enable us to find the indefinite integrals of
simple functions such as ex, sin x and cos x.
But how about e2x, sin 4x and cos (7x 5)?
Using the same method, we can obtain the
following integration formulas
16
8.2 Indefinite Integration of Functions
A. Integration of Functions Involving the
Expression (ax b)
Example 8.7T
Find
Solution
17
8.2 Indefinite Integration of Functions
A. Integration of Functions Involving the
Expression (ax b)
Example 8.8T
Find
Solution
18
8.2 Indefinite Integration of Functions
A. Integration of Functions Involving the
Expression (ax b)
Example 8.9T
Find
Solution
19
8.2 Indefinite Integration of Functions
A. Integration of Functions Involving the
Expression (ax b)
Example 8.10T
Find
Solution
20
8.2 Indefinite Integration of Functions
A. Integration of Functions Involving the
Expression (ax b)
Example 8.11T
Find
Solution
21
8.2 Indefinite Integration of Functions
A. Integration of Functions Involving the
Expression (ax b)
Example 8.12T
Find
Solution
22
8.2 Indefinite Integration of Functions
B. Integration of Trigonometric Functions
To find integrals where the integrand is the
product or power of trigonometric functions, we
can first use double angle formulas and
product-to-sum formulas to express the integrand
in the sum of trigonometric functions.
cos 2A ? 2 cos2 A ? 1 or cos 2A ? 1 ? 2 sin2 A
23
8.2 Indefinite Integration of Functions
B. Integration of Trigonometric Functions
Example 8.13T
Find
Solution
24
8.2 Indefinite Integration of Functions
B. Integration of Trigonometric Functions
Example 8.14T
Find
Solution
25
8.2 Indefinite Integration of Functions
B. Integration of Trigonometric Functions
Example 8.15T
Find
Solution
26
8.2 Indefinite Integration of Functions
B. Integration of Trigonometric Functions
Example 8.16T
Find
Solution
27
8.3 Integration by Substitution
A. Change of Variables
In Section 8.2, we learnt some basic formulas to
find the indefinite integrals of functions.
However, not all functions can be integrated
directly using these formulas.
In this case, we have to use the method of
integration by substitution.
The following shows the basic principle of this
method.
Let and u ? g
(x).
? f (u) g(x)
? f g(x) g(x)
By the definition of integration,
.
28
8.3 Integration by Substitution
A. Change of Variables
For an integral , we
can transform it into a simpler integral
, by the following steps.
Step 1 Separate the integrand into two parts f
g(x) and g(x)dx.
Step 2 Replace every occurrence of g(x) in the
integrand by u.
Step 3 Replace the expression g(x)dx by du.
Let us use this method to find the integral
together.
Note that
,
so we let u ? x2 1, such that ? 2x.
29
8.3 Integration by Substitution
A. Change of Variables
Example 8.17T
Find
Solution
Let u ? 1 x2. Then .
30
8.3 Integration by Substitution
A. Change of Variables
Example 8.18T
Find
Solution
Let u ? x2 7. Then .
31
8.3 Integration by Substitution
A. Change of Variables
With the method of integration by substitution,
we can find the integrals of trigonometric
functions other than sine and cosine, as shown in
the following example.
32
8.3 Integration by Substitution
A. Change of Variables
Example 8.19T
Find
Solution
Alternative Solution
Let u ? csc x cot x.
Let u ? csc x cot x.
33
8.3 Integration by Substitution
A. Change of Variables
It is tedious to write u and du every time when
finding the integrals by substitution, as shown
in the previous examples.
After becoming familiar with the method of
integration by substitution, the working steps
can be simplified by omitting the use of the
variable u.
Let us study the following example.
34
8.3 Integration by Substitution
A. Change of Variables
Example 8.20T
Find
Solution
35
8.3 Integration by Substitution
B. Integrals Involving Powers of Trigonometric
Functions
Sometimes we need to handle indefinite integrals
that involve the products of powers of
trigonometric functions, such as
or , where
m and n are integers.
In the following discussion, we will see how to
apply different strategies according to different
values of m and n.
Strategies for finding integrals in the form
Case 1 m is an odd number. Use sin x dx ?
d(cos x) and express all the other sine terms
as cosine terms.
Case 2 n is an odd number. Use cos x dx ?
d(sin x) and express all the other cosine terms
as sine terms.
Case 3 both m and n are even numbers. Use the
double-angle formula to reduce the powers of the
functions.
36
8.3 Integration by Substitution
B. Integrals Involving Powers of Trigonometric
Functions
Example 8.21T
Find
Solution
37
8.3 Integration by Substitution
B. Integrals Involving Powers of Trigonometric
Functions
Example 8.22T
Find
Solution
38
8.3 Integration by Substitution
B. Integrals Involving Powers of Trigonometric
Functions
Example 8.23T
Find
Solution
39
8.3 Integration by Substitution
B. Integrals Involving Powers of Trigonometric
Functions
Similarly, integrals in the form
may be found by using the method of
integration by substitution.
Strategies for finding integrals in the form
Case 1 m is an odd number. Use tan x sec x dx
? d(sec x) and then express all other tangent
terms as secant terms.
Case 2 n is an even number. Use sec2x dx ?
d(tan x) and then express all other secant
terms as tangent terms.
40
8.3 Integration by Substitution
B. Integrals Involving Powers of Trigonometric
Functions
Example 8.24T
Find
Solution
41
8.3 Integration by Substitution
B. Integrals Involving Powers of Trigonometric
Functions
Example 8.25T
Find
Solution
42
8.3 Integration by Substitution
B. Integrals Involving Powers of Trigonometric
Functions
In the above examples, the case that m is even
while n is odd is not considered. This is because
there is no standard technique and the method
varies from case to case.
For example, to find (m ? 0 and n ?
1), we may follow the method in Example 8.19.
In some other cases, such as
(m ? 2 and n ? 1), we may need to use the
technique integration by parts, which will be
discussed later in this chapter.
43
8.3 Integration by Substitution
C. Integration by Trigonometric Substitution
If an indefinite integral involves radicals in
the form , or
, we can use the method of integration
by substitution to eliminate the radicals.
The following three trigonometric identities are
very useful for the elimination
cos2q ? 1 ? sin2q , sec2q ? 1 ? tan2q ,
tan2q ? sec2q ? 1
For example, if we substitute x ? a sin q into
the expression , we have
Then we can express the integrand in terms of q.
After finding the indefinite integral in terms of
q (say, 3q C), the final answer should be
expressed in terms of the original variable, say,
x.
44
8.3 Integration by Substitution
C. Integration by Trigonometric Substitution
In order to express q in terms of x, let us first
introduce the following notations
45
8.3 Integration by Substitution
C. Integration by Trigonometric Substitution
Example 8.26T
Find
Solution
Let x ? sinq. Then dx ? cosq dq.
Since sin q ? x,
46
8.3 Integration by Substitution
C. Integration by Trigonometric Substitution
Example 8.27T
Find
Solution
Let x ? sinq. Then dx ? cosq dq.
Since sin q ? x,
47
8.3 Integration by Substitution
C. Integration by Trigonometric Substitution
Example 8.28T
Find
Solution
Let x ? 3tanq. Then dx ? 3sec2q dq.
Since
48
8.3 Integration by Substitution
C. Integration by Trigonometric Substitution
Example 8.29T
Find
Solution
__________ ?(x ? 2)2 ? 1
Let x 2 ? secq.
Then dx ? secq tanq dq.
Since secq ? x 2,
49
8.4 Integration by Parts
Some indefinite integrals such as ,
and cannot be found
by using the techniques we have learnt so far.
To evaluate them, we need to
introduce another method called integration by
parts.
Proof Suppose u and v are two differentiable
functions.
Since (uv) ? uv vu, by definition,
.
50
8.4 Integration by Parts
From Theorem 8.3, we can see that the problem of
finding can be transformed into the
problem of finding instead.
If the integral is much simpler than
, then the original integral can be found
easily.
If we want to apply the technique of integration
by parts to an integral, such as
, we need to transform the integral into the
form first, such as
or .
51
8.4 Integration by Parts
Example 8.30T
Find
Solution
52
8.4 Integration by Parts
Example 8.31T
Find
Solution
53
8.4 Integration by Parts
In some cases, there may be more than one choice
for u and v.
For example, we can transform
into
However, if we choose the former, then
As a result, we get an integrand x2 cos x which
is more complicated than the original one x sin
x.
Thus we should try sin x dx ? d(cos x) instead.
54
8.4 Integration by Parts
Example 8.32T
Find
Solution
55
8.4 Integration by Parts
Example 8.33T

1. Show that
2. Hence find .

Solution
(b)

56
8.4 Integration by Parts
Example 8.34T
Find
Solution
57
8.4 Integration by Parts
Example 8.35T
Find
Solution
Therefore,
58
8.5 Applications of Indefinite Integrals
A. Geometrical Applications
In previous chapters, we learnt that of a
curve y ? F(x) is the slope function of the
curve.
Since integration is the reverse process of
differentiation, if we let ? f(x), then by
the definition of integration, we have
where C is an arbitrary constant.
Thus we can see that the equation of a family of
curves y ? F(x) C can be found by
integration, providing that the slope function
of the curve is known.
59
8.5 Applications of Indefinite Integrals
A. Geometrical Applications
Example 8.36T
The equation of the slope of a curve at the point
(x, y) is given by . If
the curve passes through (2, 2), find the
equation of the curve.
Solution
where C is an arbitrary constant
When x ? 2, y ? 2, we have
? The equation of the curve is
60
8.5 Applications of Indefinite Integrals
A. Geometrical Applications
Example 8.37T
It is given that at each point (x, y) on a
certain curve, . If
the curve passes through and
, find the equation of that curve.
Solution
Since and lie on the curve,
we have
? The equation of the curve is
61
8.5 Applications of Indefinite Integrals
B. Applications in Physics
In Section 7.5 of Book 1, we learnt that for a
particle moving along a straight line, its
velocity v and acceleration a at time t are given
by where s is the displacement of the particle
at time t.
Since integration is the reverse process of
differentiation, we have
62
8.5 Applications of Indefinite Integrals
B. Applications in Physics
Example 8.38T
A particle moves along a straight line such that
its velocity v at time t is given by
. Find the displacement s of the particle
at time t, given that s ? 6 when t ? 2.
Solution
When t ? 2, s ? 6, we have
63
8.5 Applications of Indefinite Integrals
B. Applications in Physics
Example 8.39T
A particle moves along a straight line so that
its acceleration a at time t is given by a ?
for t gt 0. When t ? 1, the velocity of the
particle is 8 and its displacement is 24. Find
the displacement of the particle at time t.
Solution
When t ? 1, v ? 8, we have
When t ? 1, s ? 24, we have
64
Chapter Summary
8.1 Concepts of Indefinite Integrals
1. If , then the
indefinite integral of f(x) is defined by
, where C is an
arbitrary constant.
2.
3.
4.
5.
65
Chapter Summary
8.1 Concepts of Indefinite Integrals
6.
7.
8.
9.
10.
11.
12.
13.
66
Chapter Summary
8.2 Indefinite Integration of Functions
Let a and b be real numbers with a ? 0.
1.
2.
3.
4.
5.
67
Chapter Summary
8.3 Integration by Substitution
1. Let u ? g(x) be a differentiable function.
Then,
2. If the integrated involves terms like
, and , we
can simplify the integrand by substituting x ? a
sin q, x ? a tan q or x ? a sec q respectively.
68
Chapter Summary
8.4 Integration by Parts
If u(x) and v(x) are two differentiable
functions, then
69
Chapter Summary
8.5 Applications of Indefinite Integrals

• If the slope of a curve at point (x, y) is f(x),
  then the equation of the family of curves is
  given by
• where F (x) ? f(x).

2. Let s, v and a be the displacement, velocity
and acceleration of a particle moving along a
straight line respectively, then
70
Follow-up 8.1
8.1 Concepts of Indefinite Integrals
C. Basic Properties of Indefinite Integrals
Find
Solution


71
Follow-up 8.2
8.1 Concepts of Indefinite Integrals
C. Basic Properties of Indefinite Integrals
Find
Solution




72
Follow-up 8.3
8.1 Concepts of Indefinite Integrals
C. Basic Properties of Indefinite Integrals
Find
Solution




73
Follow-up 8.4
8.1 Concepts of Indefinite Integrals
C. Basic Properties of Indefinite Integrals
Find
Solution


74
Follow-up 8.5
8.1 Concepts of Indefinite Integrals
C. Basic Properties of Indefinite Integrals
Find
Solution


75
Follow-up 8.6
8.1 Concepts of Indefinite Integrals
C. Basic Properties of Indefinite Integrals

• Let y ? x ln x.
• Find in terms of ln x.
• Hence find .

Solution
(a)
(b) By (a),
76
Follow-up 8.7
8.2 Indefinite Integration of Functions
A. Integration of Functions Involving the
Expression (ax b)
Find
Solution
77
Follow-up 8.8
8.2 Indefinite Integration of Functions
A. Integration of Functions Involving the
Expression (ax b)
Find
Solution
78
Follow-up 8.9
8.2 Indefinite Integration of Functions
A. Integration of Functions Involving the
Expression (ax b)
Find
Solution
79
Follow-up 8.10
8.2 Indefinite Integration of Functions
A. Integration of Functions Involving the
Expression (ax b)
Find
Solution
80
Follow-up 8.11
8.2 Indefinite Integration of Functions
A. Integration of Functions Involving the
Expression (ax b)
Find
Solution
81
Follow-up 8.12
8.2 Indefinite Integration of Functions
A. Integration of Functions Involving the
Expression (ax b)
Find
Solution
82
Follow-up 8.13
8.2 Indefinite Integration of Functions
B. Integration of Trigonometric Functions
Find
Solution
83
Follow-up 8.14
8.2 Indefinite Integration of Functions
B. Integration of Trigonometric Functions
Find
Solution
84
Follow-up 8.15
8.2 Indefinite Integration of Functions
B. Integration of Trigonometric Functions
Find
Solution
85
Follow-up 8.16
8.2 Indefinite Integration of Functions
B. Integration of Trigonometric Functions
Find
Solution
86
Follow-up 8.17
8.3 Integration by Substitution
A. Change of Variables
Find
Solution
Let u ? 1 x3. Then .
87
Follow-up 8.18
8.3 Integration by Substitution
A. Change of Variables
Find
Solution
Let u ? x 4. Then .
88
Follow-up 8.19
8.3 Integration by Substitution
A. Change of Variables
Find
Solution
Note that .
Let u ? sin x. Then .
89
Follow-up 8.20
8.3 Integration by Substitution
A. Change of Variables
Find the following integrals. (a) (b)
Solution
(a)
(b)
90
Follow-up 8.21
8.3 Integration by Substitution
B. Integrals Involving Powers of Trigonometric
Functions
Find
Solution
91
Follow-up 8.22
8.3 Integration by Substitution
B. Integrals Involving Powers of Trigonometric
Functions
Find
Solution
92
Follow-up 8.23
8.3 Integration by Substitution
B. Integrals Involving Powers of Trigonometric
Functions
Find
Solution
93
Follow-up 8.24
8.3 Integration by Substitution
B. Integrals Involving Powers of Trigonometric
Functions
Find
Solution
94
Follow-up 8.25
8.3 Integration by Substitution
B. Integrals Involving Powers of Trigonometric
Functions
Find
Solution
95
Follow-up 8.26
8.3 Integration by Substitution
C. Integration by Trigonometric Substitution
Find
Solution
Let x ? 3 sinq. Then dx ? 3 cosq dq.
96
Follow-up 8.27
8.3 Integration by Substitution
C. Integration by Trigonometric Substitution
Find
Solution
Since sin q ? 2x,
Let . Then
97
Follow-up 8.28
8.3 Integration by Substitution
C. Integration by Trigonometric Substitution
Find
Solution
Let Then
Since
98
Follow-up 8.29
8.3 Integration by Substitution
C. Integration by Trigonometric Substitution
Find
Solution

Let x ? 8 secq .
Then dx ? 8 tanq secq dq.
_______ ? x2 ? 64
99
Follow-up 8.30
8.4 Integration by Parts
Find
Solution
100
Follow-up 8.31
8.4 Integration by Parts
Find
Solution
101
Follow-up 8.32
8.4 Integration by Parts
Find
Solution
102
Follow-up 8.33
8.4 Integration by Parts

1. Find
2. Hence find

Solution
(a) Let y ? 2x.
(b)
ln y ? x ln 2
Differentiate both sides with respect to x,
103
Follow-up 8.34
Find
Solution
104
Follow-up 8.35
8.4 Integration by Parts
Find
Solution
Therefore,
105
Follow-up 8.36
8.5 Applications of Indefinite Integrals
A. Geometrical Applications
The equation of the slope of a curve at the point
(x, y) is given by ? 16 sin x cos x. If
the curve passes through , find its
equation.
Solution
where C is an arbitrary constant.
When y ? ?7, we have
? The equation of the curve is y ? 4 cos 2x
5.
106
Follow-up 8.37
8.5 Applications of Indefinite Integrals
A. Geometrical Applications
It is given that at each point (x, y) on a
certain curve, ? 18e3x. If ? 1
and y ? 1 when x ? 0, find the equation of that
curve.
Solution
When x ? 0, y ? 1, we have
? The equation of the curve is y ? 2e3x
7x 1.
When x 0, ? 1, we have
107
Follow-up 8.38
8.5 Applications of Indefinite Integrals
B. Applications in Physics
A particle moves along a straight line such that
its acceleration a at time t is given by
. Find the velocity v of the particle at time
t if v ? 7 when t ? 4.
Solution
When t ? 4, v ? 7, we have
108
Follow-up 8.39
8.5 Applications of Indefinite Integrals
B. Applications in Physics
A particle is moving along a straight line such
that its acceleration a cm s2 after t seconds
is given by a ? 6(t 1). The displacement of the
particle is 13 cm after 2 seconds and 45 cm after
4 seconds. Find the displacement of the particle
when its velocity is at its minimum.
Solution
(2) (1),
Substituting C1 ? 6 into (1),
When t ? 2, s ? 13, we have
When t ? 4, s ? 45, we have
? The velocity is minimum when t 1.
When t ? 1,
? The displacement is 9 cm.