1. 4.1 Antiderivatives and Indefinite Integration

2. Definition of an Antiderivative
In many cases, we would like to know what is the function F(x) whose derivative is a given function f(x). Or,
F’(x) = f(x)
For instance, if a given function f(x) = –sinx, then the derivative of F(x) = cosx + 4 is the given function f(x).

3. Definition of an Antiderivative
Notice that even we could find the antiderivative of a given function, the answer is not unique!
If f(x) = sinx + cosx
then F1(x) = sinx – cosx – 1
F2(x) = sinx – cosx + 4
F3(x) = sinx – cosx – 9
This is why F(x) is called an antiderivative of f(x), rather then the antiderivative of f(x). In fact, all antideritives of f(x) differs at a constant.

4. Antiderivatives and Indefinite Integration
Differentiable Functions
ALL functions D
derivative operator
antiderivative operator/ indefinite integral operator
This expression denotes (all) the antiderivatives of f(x).
is read as “the antiderivative of f with respect to x” or “the indefinite integral of f with respect to x”.

5. Theorem 4.1 Representation of Antiderivatives
This Theorem tells us that we can represent the entire family of antideritives of a given function by adding a constant to a known antiderivative.
If f(x) = sinx + cosx
then F(x) = sinx – cosx + C

6. Theorem 4.1 Representation of Antiderivatives
The constant C is called the constant of integration. The family of functions represented by G is the general antiderivative of f.
If f(x) = sinx + cosx
then F(x) = sinx – cosx + C is the general solution of the differential equation F ’(x) = sinx + cosx

7. Notation of Antiderivatives
Let F(x) be an antiderivative of f(x), and y be all antiderivatives f(x), then
(*)
We differentiate (*), then
The operation of finding all solutions of this equation is called antidifferentiation (or indefinite integration) denoted by the sign

8. Notation of Antiderivatives
Variable of Integration
Constant of Integration
Integrand
single function
bag of functions

9. Indefinite Integration
Example Find the antiderivatives of 5x
Solution
single function
bag of functions
iff F’(x) = f(x)

10. Example
Example Find the antiderivatives of 3
Solution
constant

11. Example 3 Find the antiderivatives of the following
and
Solution
m is not necessary to be an integer. mψR

12. Example

13. Basic Integration Rules

14. Example Example 4 Find the antiderivatives of the following and
Solution

15. Example Example 5 Find the antiderivatives of the following Solution
Also refer to middle of P. 251 for another example.

16. Example Example 6 Find the antiderivatives of the following Solut1ion
Tip: Sometimes, simplifying or rewriting before integration
Example
Example Find the antiderivatives of the following
Solut1ion

17. Note
(1) Please be aware that the indefinite integral
is NOT equal to

18. Note The indefinite integration with respect to a polynomial
can NOT be written as

19. Practice

20. Practice
Check:

21. Practice
check:

22. Initial Conditions and Particular Solutions
In the beginning of this section, we have already known that all antideritives of f(x) differs at a constant. This means that the graphs of any two antiderivatives of f are VERTICAL translation of each other.
C=2
C=1
C=0
C=–1
C=–2

23. Initial Conditions and Particular Solutions
Graph must pass through point (0, –2).
C=2
So, x = 0, y = –2 is the solution of the equation
C=1
C=0
C=–1
Or,
C=–2

24. (To find a particular solution, we need an initial condition)
Example 7 Find f(x) if f ’(x) = 6x – 4 and f(1) = 2.
differential equation
initial condition
use this to get a bag of functions containing f(x)
use this to reach into bag and pull out a particular f(x)

25. Example 7 Find f(x) if f’ (x) = 6x – 4 and f(1) = 2.
Solut1ion
check:

26. Example Example 8 Find f(x) if Solut1ion Answer: differential equation
initial condition
Solut1ion
Answer:

27. USED MOST OFTEN

28. USED MOST OFTEN

29. Example
Example 9
Solut1ion
Check:

30. Example Example 10 Find the antiderivatives of the following Solut1ion
check:

31. Motion Along a Line s(t) position function v(t) = s’ (t) v(t) velocity
a(t) = v’ (t)
a(t)
acceleration
v(t) = s’ (t)
a(t) = v’ (t)
a(t) = s’’ (t)

32. Free Falling Objects
NOTE Velocity is negative when falling (or positive when thrown up)
acceleration due to gravity

33. Example
Example The Grand Canyon is 1800 meters deep at its deepest point. A rock is dropped from the rim above this point. Write the height of the rock as function of the time t in the seconds. How long will it take the rock to hit the canyon floor?
Solut1ion
Let h(t), v(t) and a(t) be the height, the velocity, and the acceleration of the rock at time t. Then

34. Example
Solut1ion
Let h(t), v(t) and a(t) be the height, the velocity, and the acceleration of the rock at time t. Then So

35. Example
Solut1ion
When h(t) = 0, the rock will hit the canyon floor. So
(The negative root will be discard)

36. Homework
Pg odd, odd not 27, 47, 48, 49, 51, 67, 69, 75, 81