1. 4.9 – Antiderivatives

2. In addition to finding derivatives, there is an important “inverse” problem
Given the derivative, find the function itself.
For example, in physics we may know the velocity 𝑣 𝑡 (the derivative) and wish to compute the position 𝑠 𝑡 of an object.
Since 𝑠 ′ 𝑡 =𝑣 𝑡 , this amounts to finding a function whose derivative is 𝑣 𝑡 .
A function 𝐹 𝑥 whose derivative is 𝑓 𝑥 is called an antiderivative of 𝑓 𝑥 .

3. Definition Antiderivatives
A function 𝐹 𝑥 is an antiderivative of 𝑓 𝑥 on 𝑎, 𝑏 if 𝐹 ′ 𝑥 =𝑓 𝑥 for all 𝑥∈ 𝑎, 𝑏 .

4. Examples
𝐹 𝑥 =− cos 𝑥 is an antiderivative of 𝑓 𝑥 = sin 𝑥 because 𝐹 ′ 𝑥 = 𝑑 𝑑𝑥 − cos 𝑥 = sin 𝑥 =𝑓 𝑥
𝐹 𝑥 = 1 3 𝑥 3 is an antiderivative of 𝑓 𝑥 = 𝑥 2 because 𝐹 ′ 𝑥 = 𝑑 𝑑𝑥 𝑥 3 = 𝑥 2 =𝑓 𝑥

5. One critical observation is that antiderivatives are not unique.
We are free to add a constant C because the derivative of a constant is zero.
If 𝐹 ′ 𝑥 =𝑓 𝑥 , then 𝐹 𝑥 +𝐶 ′ =𝑓 𝑥 .
Each of the following is an antiderivative of 𝑥 2
1 3 𝑥 3
1 3 𝑥 3 +5
1 3 𝑥 3 −4
The process of finding an antiderivative is called integration.

6. Notation Indefinite Integral
The notation 𝑓 𝑥 𝑑𝑥 =𝐹 𝑥 +𝐶 means that 𝐹 ′ 𝑥 =𝑓 𝑥
We say that 𝐹 𝑥 +𝐶 is the general antiderivative or indefinite integral of 𝑓 𝑥

7. Power Rule for Integrals

8. Antiderivative of y = 1/x
The function 𝐹 𝑥 = ln 𝑥 is an antiderivative of 𝑦= 1 𝑥 in the domain 𝑥:𝑥≠0 ; that is
𝑑𝑥 𝑥 = ln 𝑥 +𝐶

9. Linearity of the Indefinite Integral
Sum Rule
𝑓 𝑥 +𝑔 𝑥 𝑑𝑥 = 𝑓 𝑥 𝑑𝑥 + 𝑔 𝑥 𝑑𝑥
Multiples Rule
𝑐𝑓 𝑥 𝑑𝑥 =𝑐 𝑓 𝑥 𝑑𝑥

10. Basic Trigonometric Integrals
sin 𝑥 𝑑𝑥 =− cos 𝑥 +𝐶 cos 𝑥 𝑑𝑥 = sin 𝑥 +𝐶 sec 2 𝑥 𝑑𝑥 = tan 𝑥 +𝐶
csc 2 𝑥 𝑑𝑥 =− cot 𝑥 +𝐶 sec 𝑥 tan 𝑥 𝑑𝑥 = sec 𝑥 +𝐶 csc 𝑥 cot 𝑥 𝑑𝑥 =− csc 𝑥 +𝐶

11. Integrals Involving ex
𝑒 𝑥 𝑑𝑥 = 𝑒 𝑥 +𝐶 𝑒 𝑘𝑥+𝑏 𝑑𝑥 = 1 𝑘 𝑒 𝑘𝑥+𝑏 +𝐶

12. Examples Evaluate cos 𝑥 𝑑𝑥 Evaluate 3 𝑥 4 −5 𝑥 2/3 + 𝑥 −3 𝑑𝑥

13. Examples Evaluate sin 8𝑡−3 +20 cos 9𝑡 𝑑𝑡 Evaluate 3 𝑒 𝑥 −4 𝑑𝑥

14. We can think of an antiderivative as a solution to the differential equation 𝑑𝑦 𝑑𝑥 =𝑓 𝑥
In general, a differential equation is an equation relating an unknown function and its derivatives.
A differential equation with an initial condition is called an initial value problem.

15. Examples
Solve 𝑑𝑦 𝑑𝑥 =4 𝑥 7 with an initial condition 𝑦 0 =4.

16. Examples
Solve the differential equation 𝑑𝑦 𝑑𝑡 = sin 𝜋𝑡 with the initial condition 𝑦 2 =2

17. Homework
Text page 281 #s even, even, all

18. Speed Quiz

19. 5.3 – The Fundamental Theorem of Calculus, Part I

20. Fundamental Theorem of Calculus Part I
Assume that 𝑓 𝑥 is continuous on 𝑎, 𝑏 . If 𝐹 𝑥 is an antiderivative of 𝑓 𝑥 on 𝑎, 𝑏 , then
𝑎 𝑏 𝑓 𝑥 𝑑𝑥 =𝐹 𝑏 −𝐹 𝑎 OR
𝑎 𝑏 𝑓 ′ 𝑥 𝑑𝑥 =𝑓 𝑏 −𝑓 𝑎

21. Other Important Integral Theorems
𝑎 𝑏 𝑓 𝑥 𝑑𝑥 = signed area of region between the graph and x-axis over [a, b]
For a < b, 𝑏 𝑎 𝑓 𝑥 𝑑𝑥 =− 𝑎 𝑏 𝑓 𝑥 𝑑𝑥
𝑎 𝑎 𝑓 𝑥 𝑑𝑥 =0
Let 𝑎≤𝑏≤𝑐, and assume that 𝑓 𝑥 is integrable. 𝑎 𝑐 𝑓 𝑥 𝑑𝑥= 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 + 𝑏 𝑐 𝑓 𝑥 𝑑𝑥
𝑎 𝑏 𝑑𝑥 𝑥 = ln 𝑏 − ln 𝑎 = ln 𝑏 𝑎
For an object in linear motion with velocity 𝑣 𝑡 , then
Displacement during 𝑡 1 , 𝑡 2 = 𝑡1 𝑡2 𝑣 𝑡 𝑑𝑡
Distance traveled during 𝑡 1 , 𝑡 2 = 𝑡1 𝑡2 𝑣 𝑡 𝑑𝑡

22. Example
Calculate the area under the graph of 𝑓 𝑥 = 𝑥 3 over 2, 4

23. Example
Calculate −𝜋/4 𝜋/4 sec 2 𝑥 𝑑𝑥 and sketch the corresponding region.

24. Example
Calculate 0 𝜋 sin 𝑥 𝑑𝑥 and sketch the corresponding region.

25. Example
Calculate 0 2𝜋 sin 𝑥 𝑑𝑥 and sketch the corresponding region.

26. Example
Calculate 0 2 𝑓 𝑥 𝑑𝑥 given 𝑓 𝑥 = 3 𝑥 for 𝑥<1 2𝑥+1 for 𝑥≥1

27. Example
Calculate −2 5 𝑓 𝑥 𝑑𝑥 given 𝑓 𝑥 = 2𝑥− 𝑥 2 for 𝑥< for 𝑥≥1

28. Example
Calculate −3 2 𝑥 𝑑𝑥 by first rewriting the integral without absolute value.

29. Example
Calculate −1 4 𝑥−2 𝑑𝑥 by first rewriting the integral without absolute value.

30. Homework
Text pages #s 6-14 even, 28, 30, 38, 43, 44, 55, 56

31. 5.4 – The Fundamental Theorem of Calculus, Part II

32. Example
Find a formula for the area function 𝐴 𝑥 = 3 𝑥 𝑡 2 𝑑𝑡 .

33. Example
Find the formula for the area function 𝐴 𝑥 = 2 𝑥 2 2𝑥+5 𝑑𝑥

34. Fundamental Theorem of Calculus Part II
Assume that 𝑓 𝑥 is continuous on an open interval I and let 𝑎∈𝐼. Then the area function
𝐴 𝑥 = 𝑎 𝑥 𝑓 𝑡 𝑑𝑡
is an antiderivative of 𝑓 𝑥 on I; that is, 𝐴 ′ 𝑥 =𝑓 𝑥 . Equivalently,
𝑑 𝑑𝑥 𝑎 𝑥 𝑓 𝑡 𝑑𝑡 =𝑓 𝑥
Furthermore, 𝐴 𝑥 satisfies the initial condition 𝐴 𝑎 =0
If the upper bound is a function of x, apply the chain rule and multiply by the derivative of the upper bound.

35. Example
Find the derivative of 𝐴 𝑥 = 2 𝑥 1+ 𝑡 3 𝑑𝑡 . Evalute 𝐴 ′ 2 , 𝐴 ′ 3 , and 𝐴 2 .

36. Example
Find the derivative of 𝐺 𝑥 = −2 𝑥 2 sin 𝑡 𝑑𝑡

37. Homework
Text page 320 #s 4-10 even, 28

38. Quiz

39. 5.6 – Substitution Method

40. Integration (antidifferentiation) is generally more difficult than differentiation.
There are no sure-fire methods, and many antiderivatives cannot be expressed in terms of elementary functions.
However, there are a few important general techniques. One such technique is the Substitution Method, which uses the Chain Rule “in reverse.”
Consider the integral 2𝑥 cos 𝑥 2 𝑑𝑥 . We can evaluate it if we remember the Chain Rule
𝑑 𝑑𝑥 sin 𝑥 2 =2𝑥 cos 𝑥 2
This tells us that sin 𝑥 2 is an antiderivative of 2𝑥 cos 𝑥 2 , and therefore
2𝑥 Derivative of the inside function cos 𝑥 2 Inside function 𝑑𝑥=sin⁡ 𝑥 2 +𝐶

41. The integrand is the product of a composite function and the derivative of the inside function.
The Chain Rule does not help if the derivative of the inside function is missing.
For instance, we cannot use the Chain Rule to compute cos 𝑥+ 𝑥 3 𝑑𝑥 because the factor 1+3 𝑥 2 does not appear.

42. Theorem – The Substitution Method
If 𝐹 ′ 𝑥 =𝑓 𝑥 , then
𝑓 𝑢 𝑥 𝑢 ′ 𝑥 𝑑𝑥 =𝐹 𝑢 𝑥 +𝐶

43. Equivalently, 𝑑𝑢 and 𝑑𝑥 are related by 𝑑𝑢= 𝑢 ′ 𝑥 𝑑𝑥 For example
Before proceeding to the examples, we discuss the procedure for carrying out substitution using differentials.
Differentials are symbols such as 𝑑𝑢 or 𝑑𝑥 that occur in the Leibniz notations.
In our calculators, we shall manipulate them as thorugh they are related by an equation in which the 𝑑𝑥 “cancels”: 𝑑𝑢= 𝑑𝑢 𝑑𝑥 𝑑𝑥
Equivalently, 𝑑𝑢 and 𝑑𝑥 are related by
𝑑𝑢= 𝑢 ′ 𝑥 𝑑𝑥
For example
If 𝑢= 𝑥 2 then 𝑑𝑢=2𝑥𝑑𝑥
If 𝑢= cos 𝑥 3 then 𝑑𝑢=−3 𝑥 2 sin 𝑥 3 𝑑𝑥

44. Example
Evalaute 3 𝑥 2 sin 𝑥 3 𝑑𝑥

45. Example
Evalaute 3 𝑥 2 sin 𝑥 3 𝑑𝑥

46. Example
Evalaute 𝑥 𝑥 𝑑𝑥

47. Example
Evaluate 𝑥 2 +2𝑥 𝑑𝑥 𝑥 3 +3 𝑥

48. Example
Evalaute sin 7𝜃+5 𝑑𝜃

49. Example
Evaluate 𝑒 −9𝑡 𝑑𝑡

50. Change of Variables for Definite Integrals
𝑎 𝑏 𝑓 𝑢 𝑥 𝑢 ′ 𝑥 𝑑𝑥 = 𝑢 𝑎 𝑢 𝑏 𝑓 𝑢 𝑑𝑢

51. Example
Evaluate 𝑥 2 𝑥 3 +1 𝑑𝑥

52. example
Evaluate 𝑥+1 𝑥 2 +2𝑥 5 𝑑𝑥

53. example
Evalaute 0 𝜋/4 tan 3 𝜃 sec 2 𝜃 𝑑𝜃

54. example
Calculate the area under the graph of 𝑦= 𝑥 𝑥 over 1, 3

55. Homework
Text pages #s even, 48, 50, 80, 82, 86