4
Integration
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Objectives
 Write the general solution of a differential
equation.
 Use indefinite integral notation for
antiderivatives.
 Use basic integration rules to find
antiderivatives.
 Find a particular solution of a differential
equation.
2
Think About It
 Suppose this is
the graph of the
derivative of a function
 What do we know about
the original function?
f '(x)
– Critical numbers
– Where it is increasing, decreasing
 What do we not know?
In this chapter we will
learn how to determine
the original function when
given the derivative.
We will also learn some
applications.
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3
Antiderivatives
Suppose you were asked to find a function F whose
derivative is f(x) = 3x2. From your knowledge of derivatives,
you would probably say that
The function F is an antiderivative of f .
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Anti-Derivatives
 Derivatives give us the rate of change of a
function
 What if we know the rate of change …
– Can we find the original function?
 If F '(x) = f(x)
– Then F(x) is an antiderivative of f(x)
 Example – let F(x) = 12x2
– Then F '(x) = 24x = f(x)
– So F(x) = 12x2 is the antiderivative of f(x) = 24x
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5
Finding An Antiderivative
 Given f(x) = 12x3
– What is the antiderivative, F(x)?
 Use the power rule backwards
– Recall that for f(x) = xn … f '(x) = n • x n – 1
 That is …
– Multiply the expression by the exponent
– Decrease exponent by 1
 Now do opposite
(in opposite order)
12 4
F ( x)  x  3 x 4
4
– Increase exponent by 1
– Divide expression by new exponent
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6
Example 1
 𝑓′ 𝑥 = 𝑥3
Find f(x)
=
=
=
𝑥 3+1
3+1
𝑥4
4
1 4
𝑥
4
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Example 2:
′
 𝑓 𝑥 =𝑥
Find f(x)
=
1
+1
𝑥3
1
+1
3
=
4
𝑥3
4
3
=
3 4
𝑥3
4
1
3
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Example 3
 𝑓′ 𝑥 =
1
𝑥4
Find f(x)
First Rewrite: 𝑥 −4
=
=
=
𝑥 −4+1
−4+1
𝑥 −3
−3
−1
3𝑥 3
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Example 4
 𝑓′ 𝑥 =
1
1
𝑥 3
Find f(x)
Rewrite first: 𝑥
=
𝑥
−1
3
−1 +1
3
−1
3+1
2
=
=
𝑥 3
2
3
3 2
𝑥 3
2
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Example 5
 𝑓 ′ 𝑥 = 2𝑥 3
Find f(x)
=
=
=
2𝑥 3+1
3+1
2𝑥 4
4
1 4
𝑥
2
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Example 6
 𝑓′ 𝑥 = 5
Find f(x)
=
5𝑥 0+1
0+1
= 5𝑥
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Family of Antiderivatives
 Consider a family of parabolas
– f(x) = x2 + n
which differ only by value of n
 Note that f '(x) is the same for
each version of f
 Now go the other way …
– The antiderivative of 2x must be different for each of
the original functions
 So when we take an antiderivative
– We specify F(x) + C
– Where C is an arbitrary constant
This indicates that
multiple
antiderivatives
could exist from
one derivative 13
13
Example
 y’ = 2x
y = (2)(1/2)x1+1 =x2
what are we missing? Could we have started with y
= x2 + 3 or y = x2 + 30??
When we find the antiderivative, we need to remember to
account for the constant that could have been in the
original function.
So we get y = x2 + C
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Indefinite Integral
 The family of antiderivatives of a function f
indicated by
 f ( x)dx
 The symbol is a stylized S to indicate
summation, we will discuss this further in later
sections of this chapter.
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Integration
16
Indefinite Integral
 The indefinite integral is a family of functions
1 4
 x dx  4 x  C
3
2
1
3
x

4
dx


3
x
 4x  C


 The + C represents an arbitrary constant
– The constant of integration
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Integration Example 1
 Solve: ∫ 3𝑥𝑑𝑥
=
3𝑥 2
2
+ 𝑐 𝑜𝑟
3 2
𝑥
2
+𝑐
Check:
𝑑 3 2
𝑥 + 𝑐 = 3𝑥
𝑑𝑥 2
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Properties of Indefinite Integrals
 The power rule
1 n 1
 x dx  n  1 x  C , n  1
n
 The integral of a sum (difference) is the
sum (difference) of the integrals
  f ( x)  g ( x) dx   f ( x)dx   g ( x)dx
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Properties of Indefinite Integrals
 The derivative of the indefinite integral is
the original function
d
f ( x)dx  f ( x)

dx
 A constant can be factored out of the
integral
k

f
(
x
)
dx

k

f
(
x
)
d
x


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Try It Out
 Determine the indefinite integrals as specified
below
6
dx

t

1/ 4
dt
3
x

5
dx



12
y


3
 6 y  8 y  5  dy
2
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Differential Equations
 We will study this topic in
chapter 6.
 For now, here’s a taste of it.
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Differential Equations
The family of functions represented by G is the
general antiderivative of f, and G(x) = x2 + C is
the general solution of the differential equation
G'(x) = 2x.
Differential equation
A differential equation in x and y is an equation
that involves x, y, and derivatives of y.
For instance, y' = 3x and y' = x2 + 1 are examples
of differential equations.
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Example 1 – Solving a Differential Equation
Find the general solution of the differential
equation y' = 2.
Solution:
To begin, you need to find a function whose
derivative is 2.
One such function is
y = 2x.
2x is an antiderivative of 2.
Now, you can use Theorem 4.1 to conclude that
the general solution of the differential equation is
y = 2x + C.
General solution
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Example 1 – Solution
cont’d
The graphs of several functions of the form y = 2x + C
are shown in Figure 4.1.
Figure 4.1
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Notation for Antiderivatives
When solving a differential equation of the form
it is convenient to write it in the equivalent
differential form
The operation of finding all solutions of this
equation is called antidifferentiation (or
indefinite integration) and is
denoted by an integral sign ∫.
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Notation for Antiderivatives
The general solution is denoted by
The expression ∫f(x)dx is read as the antiderivative of f with
respect to x. So, the differential dx serves to identify x as
the variable of integration. The term indefinite integral is a
synonym for antiderivative.
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Basic Integration Rules (p.250)
The inverse nature of integration and differentiation can be
verified by substituting F'(x) for f(x) in the indefinite
integration definition to obtain
Moreover, if ∫f(x)dx = F(x) + C, then
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Basic Integration Rules
These two equations allow you to obtain integration
formulas directly from differentiation formulas, as shown in
the following summary.
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Basic Integration Rules
cont’d
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Basic Integration Rules
Note that the general pattern of integration is similar to that
of differentiation.
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Example 2 – Applying the Basic Integration Rules
Describe the antiderivatives of 3x.
Solution:
So, the antiderivatives of 3x are of the form
C is any constant.
where
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Initial Conditions and Particular Solutions
You have already seen that the equation y = ∫f(x)dx has
many solutions (each differing from the others by a
constant).
This means that the graphs of any two antiderivatives of f
are vertical translations of each other.
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Initial Conditions and Particular Solutions
For example, Figure 4.2 shows the
graphs of several antiderivatives
of the form
for various integer values of C.
Each of these antiderivatives is a solution
of the differential equation
Figure 4.2
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Initial Conditions and Particular Solutions
In many applications of integration, you are given enough
information to determine a particular solution. To do this,
you need only know the value of y = F(x) for one value of x.
This information is called an initial condition.
For example, in Figure 4.2, only one curve passes through
the point (2, 4).
To find this curve, you can use the following information.
F(x) = x3 – x + C
General solution
F(2) = 4
Initial condition
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Initial Conditions and Particular Solutions
By using the initial condition in the general solution, you
can determine that F(2) = 8 – 2 + C = 4, which implies that
C = –2.
So, you obtain
F(x) = x3 – x – 2.
Particular solution
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Example 7 – Finding a Particular Solution
Find the general solution of
and find the particular solution that satisfies the initial
condition F(1) = 0.
Solution:
To find the general solution, integrate to obtain
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Example 7 – Solution
cont’d
Using the initial condition F(1) = 0, you
can solve for C as follows.
So, the particular solution, as shown
in Figure 4.3, is
Figure 4.3
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Homework:
 Pg 255-256 #2-44E
 Great to try:
p.255#1,4,9,11,13,14,22,31,34,35,38,40,43,47,49,50,59,
62,63,65,66,68,71,72,87,90,92,94,95
 This is the bare minimum. It is recommended that you do
all of the problems in each section to ensure mastery of
the concepts. (this includes the word problems!)
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