1. Copyright © Cengage Learning. All rights reserved.4
Integration
Copyright © Cengage Learning. All rights reserved.

2. 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.

3. We will also learn some applications.
Think About It
Suppose this is the graph of the derivative of a function
What do we know about the original function?
Critical numbers
Where it is increasing, decreasing
What do we not know?
f '(x)
In this chapter we will learn how to determine the original function when given the derivative.
We will also learn some applications. 3

4. 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 .

5. 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 5

6. 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)
Increase exponent by 1
Divide expression by new exponent 6

7. Example 1
𝑓 ′ 𝑥 = 𝑥 3
Find f(x)
= 𝑥
= 𝑥 4 4
= 𝑥 4

8. Example 2: 𝑓 ′ 𝑥 = 𝑥 1 3 Find f(x) = 𝑥 1 3 +1 1 3 +1 = 𝑥 4 3 4 3
𝑓 ′ 𝑥 = 𝑥 1 3
Find f(x)
= 𝑥
= 𝑥
= 𝑥 4 3

9. Example 3 𝑓 ′ 𝑥 = 1 𝑥 4 Find f(x) First Rewrite: 𝑥 −4 = 𝑥 −4+1 −4+1
𝑓 ′ 𝑥 = 1 𝑥 4
Find f(x)
First Rewrite: 𝑥 −4
= 𝑥 −4+1 −4+1
= 𝑥 −3 −3
= −1 3 𝑥 3

10. Example 4 𝑓 ′ 𝑥 = 1 𝑥 1 3 Find f(x) Rewrite first: 𝑥 −1 3
𝑓 ′ 𝑥 = 1 𝑥 1 3
Find f(x)
Rewrite first: 𝑥 −1 3
= 𝑥 − −1 3 +1
= 𝑥
= 𝑥 2 3

11. Example 5
𝑓 ′ 𝑥 =2 𝑥 3
Find f(x)
= 2 𝑥
= 2 𝑥 4 4
= 𝑥 4

12. Example 6
𝑓 ′ 𝑥 =5
Find f(x)
= 5 𝑥
= 5𝑥

13. 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

14. Example y’ = 2x y = (2)(1/2)x1+1 =x2
what are we missing? Could we have started with y = x or y = x ??
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

15. Indefinite Integral
The family of antiderivatives of a function f indicated by
The symbol is a stylized S to indicate summation, we will discuss this further in later sections of this chapter.

16. Integration

17. Indefinite Integral The constant of integration
The indefinite integral is a family of functions
The + C represents an arbitrary constant
The constant of integration

18. Integration Example 1 Solve: ∫3𝑥𝑑𝑥 = 3 𝑥 2 2 +𝑐 𝑜𝑟 3 2 𝑥 2 +𝑐 Check:
= 3 𝑥 𝑐 𝑜𝑟 𝑥 2 +𝑐
Check:
𝑑 𝑑𝑥 𝑥 2 +𝑐 =3𝑥

19. Properties of Indefinite Integrals
The power rule
The integral of a sum (difference) is the sum (difference) of the integrals 19

20. Properties of Indefinite Integrals
The derivative of the indefinite integral is the original function
A constant can be factored out of the integral

21. Try It Out
Determine the indefinite integrals as specified below

22. Differential Equations
We will study this topic in chapter 6.
For now, here’s a taste of it.

23. 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.

24. 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 x 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

25. 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

26. 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 ∫.

27. 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.

28. 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

29. Basic Integration Rules
These two equations allow you to obtain integration formulas directly from differentiation formulas, as shown in the following summary.

30. Basic Integration Rules
cont’d

31. Basic Integration Rules
Note that the general pattern of integration is similar to that of differentiation.

32. Example 2 – Applying the Basic Integration Rules
Describe the antiderivatives of 3x.
Solution:
So, the antiderivatives of 3x are of the form where C is any constant.

33. 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.

34. 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

35. 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) = Initial condition

36. 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 – Particular solution

37. 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

38. Example 7 – Solution Using the initial condition F(1) = 0, you
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

39. Homework:
Pg #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!)