1
00:00:00,025 --> 00:00:08,266
[music] I must now unveil a secret which
is rarely revealed when people are first

2
00:00:08,266 --> 00:00:15,062
learning about antiderivatives.
Consider for example the function f of x

3
00:00:15,062 --> 00:00:19,445
equals 1 over x.
What is the general antiderivative for

4
00:00:19,445 --> 00:00:24,205
this function, right?
What's the most general thing that

5
00:00:24,205 --> 00:00:28,749
differentiates to 1 over x?
Well, I know what anti derivative log of

6
00:00:28,749 --> 00:00:31,889
the absolute value of x.
And that's right.

7
00:00:31,889 --> 00:00:37,646
Because the derivative of log absolute
value x is equal to 1 over x.

8
00:00:37,646 --> 00:00:43,606
So, an anti derivative of 1 over x is log
of the absolute value of x.

9
00:00:43,607 --> 00:00:46,849
In fact I know some other anti derivatives
as well.

10
00:00:46,849 --> 00:00:53,686
For example, the derivative of log of the
absolute value of x plus 17 is also equal

11
00:00:53,686 --> 00:00:58,160
to 1 over x, right, because the derivative
of 17 is 0.

12
00:00:58,160 --> 00:01:04,298
More generally the derivative of log of
absolute value of x plus some fixed

13
00:01:04,298 --> 00:01:10,154
constant is equal to 1 over x.
So this is a very general anti-derivative

14
00:01:10,154 --> 00:01:13,279
for 1 over x.
But there are many, many more anti

15
00:01:13,279 --> 00:01:17,520
derivatives where those came from.
For example, here's another one that

16
00:01:17,520 --> 00:01:20,248
doesn't fit into the pattern that we've
seen thus far.

17
00:01:20,248 --> 00:01:24,187
With the final function big F.
And I'll define it using this, piece wise

18
00:01:24,187 --> 00:01:28,176
notation, right?
So if x is positive the function big F

19
00:01:28,176 --> 00:01:33,260
will be defined to be log x plus 17, and
if x is negative, the function will be

20
00:01:33,260 --> 00:01:39,092
defined to be log of negative x plus 20.
And in fact, the derivative of this

21
00:01:39,092 --> 00:01:44,665
function is 1 over x, so this is an
antiderivative for 1 over x.

22
00:01:44,666 --> 00:01:50,042
The upshot here is that the constant could
be different to the left and to the right

23
00:01:50,042 --> 00:01:53,247
of zero.
So the most general anti derivative of 1

24
00:01:53,247 --> 00:01:59,500
over x is a function of this form, right?
Big F, log of x plus c if x is positive.

25
00:01:59,500 --> 00:02:05,194
And log of negative x plus d if x is
negative, just for some constants c and d.

26
00:02:05,195 --> 00:02:09,865
We can see this in a graph.
Well here I've got a graph of y equals log

27
00:02:09,865 --> 00:02:12,815
x.
And here I've got a graph of log negative

28
00:02:12,815 --> 00:02:15,449
x.
And the point, is that I can move these

29
00:02:15,449 --> 00:02:20,502
two graphs up and down, independently,
without affecting the derivative.

30
00:02:20,502 --> 00:02:25,180
No matter how I move these graphs up and
down The derivative is always 1 over x.

31
00:02:25,180 --> 00:02:29,275
I mean, if I just move this thing up and
down, it doesn't affect the slopes of the

32
00:02:29,275 --> 00:02:32,251
tangent lines at all.
If I move this piece up and down, it

33
00:02:32,251 --> 00:02:35,123
doesn't affect the slopes of the tangent
lines at all.

34
00:02:35,123 --> 00:02:37,502
And I can move them up and down
independently.

35
00:02:37,502 --> 00:02:40,186
'Because I don't really have to line up at
all, right?

36
00:02:40,186 --> 00:02:44,218
I mean, since the function's not defined
at zero, I don't have to worry about

37
00:02:44,218 --> 00:02:48,234
making them agree anywhere.
I just slide them up and down Fully

38
00:02:48,234 --> 00:02:53,167
independently and nevertheless the
derivative is always 1 over x.

39
00:02:53,168 --> 00:02:58,142
One way to summarize this is as follows.
So suppose that f is a function with an

40
00:02:58,142 --> 00:03:04,300
antiderivative big F.
Then any other antiderivative for little f

41
00:03:04,300 --> 00:03:07,673
Has the form big F of x plus c of x.
Right?

42
00:03:07,673 --> 00:03:13,757
C's not a single constant anymore.
C is some locally constant function.

43
00:03:13,758 --> 00:03:19,980
The point is that this so called constant
could be different on different pieces of

44
00:03:19,980 --> 00:03:23,405
the domain.
Just like in this example with 1 over x.

45
00:03:23,406 --> 00:03:28,627
The constant can change on the right and
the left hand side of zero.

46
00:03:28,628 --> 00:03:33,756
Here's the sad, sad truth.
In spite of all of this, some textbooks

47
00:03:33,756 --> 00:03:39,652
nevertheless write that the most general
antiderivative of 1 over x is log of the

48
00:03:39,652 --> 00:03:45,919
absolute value of x plus some constant c.
All of these provides a sneaky way out.

49
00:03:45,920 --> 00:03:51,125
This is perfectly fine, right?
So this is fine provided what?

50
00:03:51,125 --> 00:04:13,745
Provided that c isn't a constant but c is
a locally constant function of x.