An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion. This is the slideshow version from class.

1. Section 4.7
Antiderivatives
V63.0121.006/016, Calculus I
New York University
April 8, 2010
Announcements
Quiz April 16 on §§4.1–4.4
Final Exam: Monday, May 10, 10:00am
.
.
Image credit: Ian Hampton
. . . . . .

2. Announcements
Quiz April 16 on §§4.1–4.4
Final Exam: Monday, May 10, 10:00am
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 2 / 32

3. Outline
What is an antiderivative?
Tabulating Antiderivatives
Power functions
Combinations
Exponential functions
Trigonometric functions
Finding Antiderivatives Graphically
Rectilinear motion
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 3 / 32

4. Objectives
Given an expression for
function f, find a
differentiable function F
such that F′ = f (F is called
an antiderivative for f).
Given the graph of a
function f, find a
differentiable function F
such that F′ = f
Use antiderivatives to
solve problems in
rectilinear motion
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 4 / 32

5. Hard problem, easy check
Example
Find an antiderivative for f(x) = ln x.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 5 / 32

6. Hard problem, easy check
Example
Find an antiderivative for f(x) = ln x.
Solution
???
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 5 / 32

7. Hard problem, easy check
Example
Find an antiderivative for f(x) = ln x.
Solution
???
Example
is F(x) = x ln x − x an antiderivative for f(x) = ln x?
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 5 / 32

8. Hard problem, easy check
Example
Find an antiderivative for f(x) = ln x.
Solution
???
Example
is F(x) = x ln x − x an antiderivative for f(x) = ln x?
Solution
d
(x ln x − x)
dx
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 5 / 32

9. Hard problem, easy check
Example
Find an antiderivative for f(x) = ln x.
Solution
???
Example
is F(x) = x ln x − x an antiderivative for f(x) = ln x?
Solution
d 1
(x ln x − x) = 1 · ln x + x · − 1
dx x
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 5 / 32

10. Hard problem, easy check
Example
Find an antiderivative for f(x) = ln x.
Solution
???
Example
is F(x) = x ln x − x an antiderivative for f(x) = ln x?
Solution
d 1
(x ln x − x) = 1 · ln x + x · − 1 = ln x
dx x
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 5 / 32

11. Hard problem, easy check
Example
Find an antiderivative for f(x) = ln x.
Solution
???
Example
is F(x) = x ln x − x an antiderivative for f(x) = ln x?
Solution
d
dx
1
(x ln x − x) = 1 · ln x + x · − 1 = ln x
x
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 5 / 32

12. Why the MVT is the MITC
Most Important Theorem In Calculus!
Theorem
Let f′ = 0 on an interval (a, b). Then f is constant on (a, b).
Proof.
Pick any points x and y in (a, b) with x y. Then f is continuous on
[x, y] and differentiable on (x, y). By MVT there exists a point z in (x, y)
such that
f(y) − f(x)
= f′ (z) =⇒ f(y) = f(x) + f′ (z)(y − x)
y−x
But f′ (z) = 0, so f(y) = f(x). Since this is true for all x and y in (a, b),
then f is constant.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 6 / 32

13. When two functions have the same derivative
Theorem
Suppose f and g are two differentiable functions on (a, b) with f′ = g′ .
Then f and g differ by a constant. That is, there exists a constant C
such that f(x) = g(x) + C.
Proof.
Let h(x) = f(x) − g(x)
Then h′ (x) = f′ (x) − g′ (x) = 0 on (a, b)
So h(x) = C, a constant
This means f(x) − g(x) = C on (a, b)
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 7 / 32

14. Outline
What is an antiderivative?
Tabulating Antiderivatives
Power functions
Combinations
Exponential functions
Trigonometric functions
Finding Antiderivatives Graphically
Rectilinear motion
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 8 / 32

15. Antiderivatives of power functions
y
.
.(x) = x2
f
Recall that the derivative of a
power function is a power
function.
Fact (The Power Rule)
If f(x) = xr , then f′ (x) = rxr−1 .
.
x
.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 9 / 32

16. Antiderivatives of power functions
′
y f
. . (x) = 2x
.(x) = x2
f
Recall that the derivative of a
power function is a power
function.
Fact (The Power Rule)
If f(x) = xr , then f′ (x) = rxr−1 .
.
x
.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 9 / 32

17. Antiderivatives of power functions
′
y f
. . (x) = 2x
.(x) = x2
f
Recall that the derivative of a
power function is a power
function. F
. (x) = ?
Fact (The Power Rule)
If f(x) = xr , then f′ (x) = rxr−1 .
.
x
.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 9 / 32

18. Antiderivatives of power functions
′
y f
. . (x) = 2x
.(x) = x2
f
Recall that the derivative of a
power function is a power
function. F
. (x) = ?
Fact (The Power Rule)
If f(x) = xr , then f′ (x) = rxr−1 .
So in looking for antiderivatives
.
of power functions, try power x
.
functions!
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 9 / 32

19. Example
Find an antiderivative for the function f(x) = x3 .
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 10 / 32

20. Example
Find an antiderivative for the function f(x) = x3 .
Solution
Try a power function F(x) = axr
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 10 / 32

21. Example
Find an antiderivative for the function f(x) = x3 .
Solution
Try a power function F(x) = axr
Then F′ (x) = arxr−1 , so we want arxr−1 = x3 .
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 10 / 32

22. Example
Find an antiderivative for the function f(x) = x3 .
Solution
Try a power function F(x) = axr
Then F′ (x) = arxr−1 , so we want arxr−1 = x3 .
r − 1 = 3 =⇒ r = 4
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 10 / 32

23. Example
Find an antiderivative for the function f(x) = x3 .
Solution
Try a power function F(x) = axr
Then F′ (x) = arxr−1 , so we want arxr−1 = x3 .
1
r − 1 = 3 =⇒ r = 4, and ar = 1 =⇒ a = .
4
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 10 / 32

24. Example
Find an antiderivative for the function f(x) = x3 .
Solution
Try a power function F(x) = axr
Then F′ (x) = arxr−1 , so we want arxr−1 = x3 .
1
r − 1 = 3 =⇒ r = 4, and ar = 1 =⇒ a = .
4
1 4
So F(x) = x is an antiderivative.
4
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 10 / 32

25. Example
Find an antiderivative for the function f(x) = x3 .
Solution
Try a power function F(x) = axr
Then F′ (x) = arxr−1 , so we want arxr−1 = x3 .
1
r − 1 = 3 =⇒ r = 4, and ar = 1 =⇒ a = .
4
1 4
So F(x) = x is an antiderivative.
4
Check: ( )
d 1 4 1
x = 4 · x4−1 = x3
dx 4 4
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 10 / 32

26. Example
Find an antiderivative for the function f(x) = x3 .
Solution
Try a power function F(x) = axr
Then F′ (x) = arxr−1 , so we want arxr−1 = x3 .
1
r − 1 = 3 =⇒ r = 4, and ar = 1 =⇒ a = .
4
1 4
So F(x) = x is an antiderivative.
4
Check: ( )
d 1 4
dx 4
1
x = 4 · x4−1 = x3
4
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 10 / 32

27. Example
Find an antiderivative for the function f(x) = x3 .
Solution
Try a power function F(x) = axr
Then F′ (x) = arxr−1 , so we want arxr−1 = x3 .
1
r − 1 = 3 =⇒ r = 4, and ar = 1 =⇒ a = .
4
1 4
So F(x) = x is an antiderivative.
4
Check: ( )
d 1 4
dx 4
1
x = 4 · x4−1 = x3
4
Any others?
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 10 / 32

28. Example
Find an antiderivative for the function f(x) = x3 .
Solution
Try a power function F(x) = axr
Then F′ (x) = arxr−1 , so we want arxr−1 = x3 .
1
r − 1 = 3 =⇒ r = 4, and ar = 1 =⇒ a = .
4
1 4
So F(x) = x is an antiderivative.
4
Check: ( )
d 1 4
dx 4
1
x = 4 · x4−1 = x3
4
1 4
Any others? Yes, F(x) = x + C is the most general form.
4
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 10 / 32

29. Fact (The Power Rule for antiderivatives)
If f(x) = xr , then
1 r+1
F(x) = x
r+1
is an antiderivative for f…
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 11 / 32

30. Fact (The Power Rule for antiderivatives)
If f(x) = xr , then
1 r+1
F(x) = x
r+1
is an antiderivative for f as long as r ̸= −1.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 11 / 32

31. Fact (The Power Rule for antiderivatives)
If f(x) = xr , then
1 r+1
F(x) = x
r+1
is an antiderivative for f as long as r ̸= −1.
Fact
1
If f(x) = x−1 = , then
x
F(x) = ln |x| + C
is an antiderivative for f.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 11 / 32

32. What's with the absolute value?
{
ln(x) if x 0;
F(x) = ln |x| =
ln(−x) if x 0.
The domain of F is all nonzero numbers, while ln x is only defined
on positive numbers.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 12 / 32

33. What's with the absolute value?
{
ln(x) if x 0;
F(x) = ln |x| =
ln(−x) if x 0.
The domain of F is all nonzero numbers, while ln x is only defined
on positive numbers.
If x 0,
d
ln |x|
dx
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 12 / 32

34. What's with the absolute value?
{
ln(x) if x 0;
F(x) = ln |x| =
ln(−x) if x 0.
The domain of F is all nonzero numbers, while ln x is only defined
on positive numbers.
If x 0,
d d
ln |x| = ln(x)
dx dx
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 12 / 32

35. What's with the absolute value?
{
ln(x) if x 0;
F(x) = ln |x| =
ln(−x) if x 0.
The domain of F is all nonzero numbers, while ln x is only defined
on positive numbers.
If x 0,
d d 1
ln |x| = ln(x) =
dx dx x
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 12 / 32

36. What's with the absolute value?
{
ln(x) if x 0;
F(x) = ln |x| =
ln(−x) if x 0.
The domain of F is all nonzero numbers, while ln x is only defined
on positive numbers.
If x 0,
d
dx
ln |x| =
d
dx
ln(x) =
1
x
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 12 / 32

37. What's with the absolute value?
{
ln(x) if x 0;
F(x) = ln |x| =
ln(−x) if x 0.
The domain of F is all nonzero numbers, while ln x is only defined
on positive numbers.
If x 0,
d
dx
ln |x| =
d
dx
ln(x) =
1
x
If x 0,
d
ln |x|
dx
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 12 / 32

38. What's with the absolute value?
{
ln(x) if x 0;
F(x) = ln |x| =
ln(−x) if x 0.
The domain of F is all nonzero numbers, while ln x is only defined
on positive numbers.
If x 0,
d
dx
ln |x| =
d
dx
ln(x) =
1
x
If x 0,
d d
ln |x| = ln(−x)
dx dx
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 12 / 32

39. What's with the absolute value?
{
ln(x) if x 0;
F(x) = ln |x| =
ln(−x) if x 0.
The domain of F is all nonzero numbers, while ln x is only defined
on positive numbers.
If x 0,
d
dx
ln |x| =
d
dx
ln(x) =
1
x
If x 0,
d d 1
ln |x| = ln(−x) = · (−1)
dx dx −x
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 12 / 32

40. What's with the absolute value?
{
ln(x) if x 0;
F(x) = ln |x| =
ln(−x) if x 0.
The domain of F is all nonzero numbers, while ln x is only defined
on positive numbers.
If x 0,
d
dx
ln |x| =
d
dx
ln(x) =
1
x
If x 0,
d d 1 1
ln |x| = ln(−x) = · (−1) =
dx dx −x x
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 12 / 32

41. What's with the absolute value?
{
ln(x) if x 0;
F(x) = ln |x| =
ln(−x) if x 0.
The domain of F is all nonzero numbers, while ln x is only defined
on positive numbers.
If x 0,
d
dx
ln |x| =
d
dx
ln(x) =
1
x
If x 0,
d
dx
ln |x| =
d
dx
ln(−x) =
1
−x
· (−1) =
1
x
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 12 / 32

42. What's with the absolute value?
{
ln(x) if x 0;
F(x) = ln |x| =
ln(−x) if x 0.
The domain of F is all nonzero numbers, while ln x is only defined
on positive numbers.
If x 0,
d
dx
ln |x| =
d
dx
ln(x) =
1
x
If x 0,
d
dx
ln |x| =
d
dx
ln(−x) =
1
−x
· (−1) =
1
x
We prefer the antiderivative with the larger domain.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 12 / 32

43. Graph of ln |x|
y
.
. f
.(x) = 1/x
x
.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 13 / 32

44. Graph of ln |x|
y
.
F
. (x) = ln(x)
. f
.(x) = 1/x
x
.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 13 / 32

45. Graph of ln |x|
y
.
. (x) = ln |x|
F
. f
.(x) = 1/x
x
.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 13 / 32

46. Combinations of antiderivatives
Fact (Sum and Constant Multiple Rule for Antiderivatives)
If F is an antiderivative of f and G is an antiderivative of g, then
F + G is an antiderivative of f + g.
If F is an antiderivative of f and c is a constant, then cF is an
antiderivative of cf.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 14 / 32

47. Combinations of antiderivatives
Fact (Sum and Constant Multiple Rule for Antiderivatives)
If F is an antiderivative of f and G is an antiderivative of g, then
F + G is an antiderivative of f + g.
If F is an antiderivative of f and c is a constant, then cF is an
antiderivative of cf.
Proof.
These follow from the sum and constant multiple rule for derivatives:
If F′ = f and G′ = g, then
(F + G)′ = F′ + G′ = f + g
Or, if F′ = f,
(cF)′ = cF′ = cf
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 14 / 32

48. Antiderivatives of Polynomials
Example
Find an antiderivative for f(x) = 16x + 5.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 15 / 32

49. Antiderivatives of Polynomials
Example
Find an antiderivative for f(x) = 16x + 5.
Solution
1 2
The expression x is an antiderivative for x, and x is an antiderivative
2
for 1. So
( )
1 2
F(x) = 16 · x + 5 · x + C = 8x2 + 5x + C
2
is the antiderivative of f.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 15 / 32

50. Antiderivatives of Polynomials
Example
Find an antiderivative for f(x) = 16x + 5.
Solution
1 2
The expression x is an antiderivative for x, and x is an antiderivative
2
for 1. So
( )
1 2
F(x) = 16 · x + 5 · x + C = 8x2 + 5x + C
2
is the antiderivative of f.
Question
Why do we not need two C’s?
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 15 / 32

51. Antiderivatives of Polynomials
Example
Find an antiderivative for f(x) = 16x + 5.
Solution
( )
1 2
F(x) = 16 · x + 5 · x + C = 8x2 + 5x + C
2
Question
Why do we not need two C’s?
Answer
A combination of two arbitrary constants is still an arbitrary constant.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 15 / 32

52. Exponential Functions
Fact
If f(x) = ax , f′ (x) = (ln a)ax .
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 16 / 32

53. Exponential Functions
Fact
If f(x) = ax , f′ (x) = (ln a)ax .
Accordingly,
Fact
1 x
If f(x) = ax , then F(x) = a + C is the antiderivative of f.
ln a
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 16 / 32

54. Exponential Functions
Fact
If f(x) = ax , f′ (x) = (ln a)ax .
Accordingly,
Fact
1 x
If f(x) = ax , then F(x) = a + C is the antiderivative of f.
ln a
Proof.
Check it yourself.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 16 / 32

55. Exponential Functions
Fact
If f(x) = ax , f′ (x) = (ln a)ax .
Accordingly,
Fact
1 x
If f(x) = ax , then F(x) = a + C is the antiderivative of f.
ln a
Proof.
Check it yourself.
In particular,
Fact
If f(x) = ex , then F(x) = ex + C is the antiderivative of f.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 16 / 32

56. Logarithmic functions?
Remember we found
F(x) = x ln x − x
is an antiderivative of f(x) = ln x.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 17 / 32

57. Logarithmic functions?
Remember we found
F(x) = x ln x − x
is an antiderivative of f(x) = ln x.
This is not obvious. See Calc II for the full story.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 17 / 32

58. Logarithmic functions?
Remember we found
F(x) = x ln x − x
is an antiderivative of f(x) = ln x.
This is not obvious. See Calc II for the full story.
ln x
However, using the fact that loga x = , we get:
ln a
Fact
If f(x) = loga (x)
1 1
F(x) = (x ln x − x) + C = x loga x − x+C
ln a ln a
is the antiderivative of f(x).
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 17 / 32

59. Trigonometric functions
Fact
d d
sin x = cos x cos x = − sin x
dx dx
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 18 / 32

60. Trigonometric functions
Fact
d d
sin x = cos x cos x = − sin x
dx dx
So to turn these around,
Fact
The function F(x) = − cos x + C is the antiderivative of f(x) = sin x.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 18 / 32

61. Trigonometric functions
Fact
d d
sin x = cos x cos x = − sin x
dx dx
So to turn these around,
Fact
The function F(x) = − cos x + C is the antiderivative of f(x) = sin x.
The function F(x) = sin x + C is the antiderivative of f(x) = cos x.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 18 / 32

62. More Trig
Example
Find an antiderivative of f(x) = tan x.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 19 / 32

63. More Trig
Example
Find an antiderivative of f(x) = tan x.
Solution
???
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 19 / 32

64. More Trig
Example
Find an antiderivative of f(x) = tan x.
Solution
???
Answer
F(x) = ln(sec x).
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 19 / 32

65. More Trig
Example
Find an antiderivative of f(x) = tan x.
Solution
???
Answer
F(x) = ln(sec x).
Check
d 1 d
= · sec x
dx sec x dx
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 19 / 32

66. More Trig
Example
Find an antiderivative of f(x) = tan x.
Solution
???
Answer
F(x) = ln(sec x).
Check
d 1 d 1
= · sec x = · sec x tan x
dx sec x dx sec x
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 19 / 32

67. More Trig
Example
Find an antiderivative of f(x) = tan x.
Solution
???
Answer
F(x) = ln(sec x).
Check
d 1 d 1
= · sec x = · sec x tan x = tan x
dx sec x dx sec x
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 19 / 32

68. More Trig
Example
Find an antiderivative of f(x) = tan x.
Solution
???
Answer
F(x) = ln(sec x).
Check
d
dx
=
1
·
d
sec x dx
sec x =
1
sec x
· sec x tan x = tan x
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 19 / 32

69. More Trig
Example
Find an antiderivative of f(x) = tan x.
Solution
???
Answer
F(x) = ln(sec x).
Check
d
dx
=
1
·
d
sec x dx
sec x =
1
sec x
· sec x tan x = tan x
More about this later. . . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 19 / 32

70. Outline
What is an antiderivative?
Tabulating Antiderivatives
Power functions
Combinations
Exponential functions
Trigonometric functions
Finding Antiderivatives Graphically
Rectilinear motion
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 20 / 32

71. Problem
Below is the graph of a function f. Draw the graph of an antiderivative
for F.
y
.
.
. . . = f(x)
y
. . . . . . .
x
.
1
. 2
. 3
. 4
. 5
. 6
.
.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 21 / 32

72. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
′
. . . . . . .. = F
f
y
. 1
. 2
. 3
. 4
. 5
. 6F
..
.
. .
. . . . . . .
x
.
1 2 3 4 5 6
. . . . . .
.
. . . . . .

73. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
′
. .. .
+ . . . .. = F
f
y
. 1
. 2
. 3
. 4
. 5
. 6F
..
.
. .
. . . . . . .
x
.
1 2 3 4 5 6
. . . . . .
.
. . . . . .

74. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
′
. .. .. .
+ + . . .. = F
f
y
. 1
. 2
. 3
. 4
. 5
. 6F
..
.
. .
. . . . . . .
x
.
1 2 3 4 5 6
. . . . . .
.
. . . . . .

75. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − ′
. .. .. .. . . .. = F
f
y
. 1
. 2
. 3
. 4
. 5
. 6F
..
.
. .
. . . . . . .
x
.
1 2 3 4 5 6
. . . . . .
.
. . . . . .

76. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − ′
. .. .. .. .. . .. = F
f
y
. 1
. 2
. 3
. 4
. 5
. 6F
..
.
. .
. . . . . . .
x
.
1 2 3 4 5 6
. . . . . .
.
. . . . . .

77. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1
. 2
. 3
. 4
. 5
. 6F
..
.
. .
. . . . . . .
x
.
1 2 3 4 5 6
. . . . . .
.
. . . . . .

78. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1↗2
. . . 3
. 4
. 5
. 6F
..
.
. .
. . . . . . .
x
.
1 2 3 4 5 6
. . . . . .
.
. . . . . .

79. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1↗2↗3
. . . . . 4
. 5
. 6F
..
.
. .
. . . . . . .
x
.
1 2 3 4 5 6
. . . . . .
.
. . . . . .

80. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1↗2↗3↘4
. . . . . . . 5
. 6F
..
.
. .
. . . . . . .
x
.
1 2 3 4 5 6
. . . . . .
.
. . . . . .

81. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1↗2↗3↘4↘5
. . . . . . . . . 6F
..
.
. .
. . . . . . .
x
.
1 2 3 4 5 6
. . . . . .
.
. . . . . .

82. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1 ↗ 2 ↗ 3 ↘ 4 ↘ 5 ↗ 6F
. . . . . . . . . . ..
.
. .
. . . . . . .
x
.
1 2 3 4 5 6
. . . . . .
.
. . . . . .

83. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1 ↗ 2 ↗ 3 ↘ 4 ↘ 5 ↗ 6F
. . .. . . . . . . . . .
max
.
. .
. . . . . . .
x
.
1 2 3 4 5 6
. . . . . .
.
. . . . . .

84. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1 ↗ 2 ↗ 3 ↘ 4 ↘ 5 ↗ 6F
. . .. . . . .. . . . . .
max min
.
. .
. . . . . . .
x
.
1 2 3 4 5 6
. . . . . .
.
. . . . . .

85. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1 ↗ 2 ↗ 3 ↘ 4 ↘ 5 ↗ 6F
. . .. . . . .. . . . . .
max min
.
. . ′ ′′
. . . . . . . . . . . . .. = F
f
1 2 3 4 5 6
. . . . . .
x
. 1
. 2
. 3
. 4
. 5
. 6F
..
.
. . . . . .

86. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1 ↗ 2 ↗ 3 ↘ 4 ↘ 5 ↗ 6F
. . .. . . . .. . . . . .
max min
.
. . ′ ′′
. . . . . . . .. + .
+ . . . .. = F
f
1 2 3 4 5 6
. . . . . .
x
. 1
. 2
. 3
. 4
. 5
. 6F
..
.
. . . . . .

87. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1 ↗ 2 ↗ 3 ↘ 4 ↘ 5 ↗ 6F
. . .. . . . .. . . . . .
max min
.
. . ′ ′′
. . . . . . . .. + .. − .
+ − . . .. = F
f
1 2 3 4 5 6
. . . . . .
x
. 1
. 2
. 3
. 4
. 5
. 6F
..
.
. . . . . .

88. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1 ↗ 2 ↗ 3 ↘ 4 ↘ 5 ↗ 6F
. . .. . . . .. . . . . .
max min
.
. . ′ ′′
. . . . . . . .. + .. − .. − .
+ − − . .. = F
f
1 2 3 4 5 6
. . . . . .
x
. 1
. 2
. 3
. 4
. 5
. 6F
..
.
. . . . . .

89. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1 ↗ 2 ↗ 3 ↘ 4 ↘ 5 ↗ 6F
. . .. . . . .. . . . . .
max min
.
. . ′ ′′
. . . . . . . .. + .. − .. − .. + .
+ − − + .. = F
f
1 2 3 4 5 6
. . . . . .
x
. 1
. 2
. 3
. 4
. 5
. 6F
..
.
. . . . . .

90. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1 ↗ 2 ↗ 3 ↘ 4 ↘ 5 ↗ 6F
. . .. . . . .. . . . . .
max min
.
. . + − − + + f′ ′′
. . . . . . . .. + .. − .. − .. + .. + . . = F
1 2 3 4 5 6
. . . . . .
x
. 1
. 2
. 3
. 4
. 5
. 6F
..
.
. . . . . .

91. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1 ↗ 2 ↗ 3 ↘ 4 ↘ 5 ↗ 6F
. . .. . . . .. . . . . .
max min
.
. . + − − + + f′ ′′
. . . . . . . .. + .. − .. − .. + .. + . . = F
.
⌣
1 2 3 4 5 6
. . . . . .
x
. 1
. 2
. 3
. 4
. 5
. 6F
..
.
. . . . . .

92. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1 ↗ 2 ↗ 3 ↘ 4 ↘ 5 ↗ 6F
. . .. . . . .. . . . . .
max min
.
. . + − − + + f′ ′′
. . . . . . . .. + .. − .. − .. + .. + . . = F
⌣ .
. ⌢
1 2 3 4 5 6
. . . . . .
x
. 1
. 2
. 3
. 4
. 5
. 6F
..
.
. . . . . .

93. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1 ↗ 2 ↗ 3 ↘ 4 ↘ 5 ↗ 6F
. . .. . . . .. . . . . .
max min
.
. . + − − + + f′ ′′
. . . . . . . .. + .. − .. − .. + .. + . . = F
⌣ .
. ⌢ . ⌢
1 2 3 4 5 6
. . . . . .
x
. 1
. 2
. 3
. 4
. 5
. 6F
..
.
. . . . . .

94. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1 ↗ 2 ↗ 3 ↘ 4 ↘ 5 ↗ 6F
. . .. . . . .. . . . . .
max min
.
. . + − − + + f′ ′′
. . . . . . . .. + .. − .. − .. + .. + . . = F
⌣ .
. ⌢ . ⌢ . ⌣
1 2 3 4 5 6
. . . . . .
x
. 1
. 2
. 3
. 4
. 5
. 6F
..
.
. . . . . .

95. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1 ↗ 2 ↗ 3 ↘ 4 ↘ 5 ↗ 6F
. . .. . . . .. . . . . .
max min
.
. . + − − + + f′ ′′
. . . . . . . .. + .. − .. − .. + .. + . . = F
⌣ .
. ⌢ . ⌢ . ⌣ . ⌣ .
1 2 3 4 5 6
. . . . . .
x
. 1
. 2
. 3
. 4
. 5
. .F
6
.
. . . . . .

96. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1 ↗ 2 ↗ 3 ↘ 4 ↘ 5 ↗ 6F
. . .. . . . .. . . . . .
max min
.
. . + − − + + f′ ′′
. . . . . . . .. + .. − .. − .. + .. + . . = F
⌣ .
. ⌢ . ⌢ . ⌣ . ⌣ .
1 2 3 4 5 6
. . . . . .
x
. ..
1 2
. 3
. 4
. 5
. .F
6
IP
.
. . . . . .

97. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1 ↗ 2 ↗ 3 ↘ 4 ↘ 5 ↗ 6F
. . .. . . . .. . . . . .
max min
.
. . + − − + + f′ ′′
. . . . . . . .. + .. − .. − .. + .. + . . = F
⌣ .
. ⌢ . ⌢ . ⌣ . ⌣ .
1 2 3 4 5 6
. . . . . .
x
. ..
1 2 ..
. 3 4
. 5
. .F
6
IP IP
.
. . . . . .

98. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1 ↗ 2 ↗ 3 ↘ 4 ↘ 5 ↗ 6F
. . .. . . . .. . . . . .
max min
.
. . + − − + + f′ ′′
. . . . . . . .. + .. − .. − .. + .. + . . = F
⌣ .
. ⌢ . ⌢ . ⌣ . ⌣ .
1 2 3 4 5 6
. . . . . .
x
. ..
1 2 ..
. 3 4
. 5
. .F
6
IP IP
.
. . . . . ..
F
1
. 2
. 3
. 4
. 5
. 6s
. . hape
. . . . . .

99. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1 ↗ 2 ↗ 3 ↘ 4 ↘ 5 ↗ 6F
. . .. . . . .. . . . . .
max min
.
. . + − − + + f′ ′′
. . . . . . . .. + .. − .. − .. + .. + . . = F
⌣ .
. ⌢ . ⌢ . ⌣ . ⌣ .
1 2 3 4 5 6
. . . . . .
x
. ..
1 2 ..
. 3 4
. 5
. .F
6
IP IP
.
. . . . . ..
F
.
1
. 2
. 3
. 4
. 5
. 6s
. . hape
. . . . . .

100. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1 ↗ 2 ↗ 3 ↘ 4 ↘ 5 ↗ 6F
. . .. . . . .. . . . . .
max min
.
. . + − − + + f′ ′′
. . . . . . . .. + .. − .. − .. + .. + . . = F
⌣ .
. ⌢ . ⌢ . ⌣ . ⌣ .
1 2 3 4 5 6
. . . . . .
x
. ..
1 2 ..
. 3 4
. 5
. .F
6
IP IP
.
. . . . . ..
F
. .
1
. 2
. 3
. 4
. 5
. 6s
. . hape
. . . . . .

101. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1 ↗ 2 ↗ 3 ↘ 4 ↘ 5 ↗ 6F
. . .. . . . .. . . . . .
max min
.
. . + − − + + f′ ′′
. . . . . . . .. + .. − .. − .. + .. + . . = F
⌣ .
. ⌢ . ⌢ . ⌣ . ⌣ .
1 2 3 4 5 6
. . . . . .
x
. ..
1 2 ..
. 3 4
. 5
. .F
6
IP IP
.
. . . . . ..
F
. . .
1
. 2
. 3
. 4
. 5
. 6s
. . hape
. . . . . .

102. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1 ↗ 2 ↗ 3 ↘ 4 ↘ 5 ↗ 6F
. . .. . . . .. . . . . .
max min
.
. . + − − + + f′ ′′
. . . . . . . .. + .. − .. − .. + .. + . . = F
⌣ .
. ⌢ . ⌢ . ⌣ . ⌣ .
1 2 3 4 5 6
. . . . . .
x
. ..
1 2 ..
. 3 4
. 5
. .F
6
IP IP
.
. . . . . ..
F
. . . .
1
. 2
. 3
. 4
. 5
. 6s
. . hape
. . . . . .

103. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1 ↗ 2 ↗ 3 ↘ 4 ↘ 5 ↗ 6F
. . .. . . . .. . . . . .
max min
.
. . + − − + + f′ ′′
. . . . . . . .. + .. − .. − .. + .. + . . = F
⌣ .
. ⌢ . ⌢ . ⌣ . ⌣ .
1 2 3 4 5 6
. . . . . .
x
. ..
1 2 ..
. 3 4
. 5
. .F
6
IP IP
.
. . . . . ..F
. . . . . . hape
1
. 2
. 3
. 4
. 5
. .s
6
. . . . . .

104. Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for for F:
+ + − − + f ′
. . . . . . . . . . . .. = F
y
. 1 ↗ 2 ↗ 3 ↘ 4 ↘ 5 ↗ 6F
. . .. . . . .. . . . . .
max min
.
. . + − − + + f′ ′′
. . . . . . . .. + .. − .. − .. + .. + . . = F
⌣ .
. ⌢ . ⌢ . ⌣ . ⌣ .
1 2 3 4 5 6
. . . . . .
x
. ..
1 2 ..
. 3 4
. 5
. .F
6
IP IP
.
?
.. ?
.. ?
.. ?
.. ?
.. ?F
.. .
. . . . . . hape
1
. 2
. 3
. 4
. 5
. .s
6
The only question left is: What are the function values?
. . . . . .
V63.0121, Calculus I (NYU) Section 4.7 Antiderivatives April 8, 2010 22 / 32

105. Could you repeat the question?
Problem
Below is the graph of a function f. Draw the graph of the antiderivative
for F with F(1) = 0.
y
.
Solution
.
. ..
f
. . . . . . .
x
.
1 2 3 4 5 6
. . . . . .
.
. . . . . ..
F
. . . . .
1 2 3 4 5
. . . . . 6s
. . hape
IP
.
max
.
IP
.
min
.
. . . . . .

106. Could you repeat the question?
Problem
Below is the graph of a function f. Draw the graph of the antiderivative
for F with F(1) = 0.
y
.
Solution
.
We start with F(1) = 0. . ..
f
. . . . . . .
x
.
1 2 3 4 5 6
. . . . . .
.
. . . . . ..
F
. . . . .
1 2 3 4 5
. . . . . 6s
. . hape
IP
.
max
.
IP
.
min
.
. . . . . .

107. Could you repeat the question?
Problem
Below is the graph of a function f. Draw the graph of the antiderivative
for F with F(1) = 0.
y
.
Solution
.
We start with F(1) = 0. . ..
f
. . . . . . .
Using the sign chart, we x
.
draw arcs with the 1 2 3 4 5 6
. . . . . .
specified monotonicity and
concavity .
. . . . . ..
F
. . . . .
1 2 3 4 5
. . . . . 6s
. . hape
IP
.
max
.
IP
.
min
.
. . . . . .

108. Could you repeat the question?
Problem
Below is the graph of a function f. Draw the graph of the antiderivative
for F with F(1) = 0.
y
.
Solution
.
.
We start with F(1) = 0. . ..
f
. . . . . . .
Using the sign chart, we x
.
draw arcs with the 1 2 3 4 5 6
. . . . . .
specified monotonicity and
concavity .
. . . . . ..
F
. . . . .
1 2 3 4 5
. . . . . 6s
. . hape
IP
.
max
.
IP
.
min
.
. . . . . .

109. Could you repeat the question?
Problem
Below is the graph of a function f. Draw the graph of the antiderivative
for F with F(1) = 0.
y
.
Solution
.
.
We start with F(1) = 0. . ..
f
. . . . . . .
Using the sign chart, we x
.
draw arcs with the 1 2 3 4 5 6
. . . . . .
specified monotonicity and
concavity .
. . . . . ..
F
. . . . .
1 2 3 4 5
. . . . . 6s
. . hape
IP
.
max
.
IP
.
min
.
. . . . . .