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PROFESSOR: Today we're moving
on from theoretical things,
9
00:00:24,970 --> 00:00:30,570
from the mean value theorem,
to the introduction to what's
10
00:00:30,570 --> 00:00:33,120
going to occupy us for the
whole rest of the course, which
11
00:00:33,120 --> 00:00:34,850
is integration.
12
00:00:34,850 --> 00:00:38,850
So, in order to
introduce that subject,
13
00:00:38,850 --> 00:00:42,660
I need to introduce for
you a new notation, which
14
00:00:42,660 --> 00:00:52,640
is called differentials.
15
00:00:52,640 --> 00:00:55,640
I'm going to tell you
what a differential is,
16
00:00:55,640 --> 00:01:02,040
and we'll get used to
using it over time.
17
00:01:02,040 --> 00:01:10,290
If you have a function
which is y = f(x),
18
00:01:10,290 --> 00:01:27,650
then the differential of y
is going to be denoted dy,
19
00:01:27,650 --> 00:01:29,500
and it's by definition f'(x) dx.
20
00:01:34,160 --> 00:01:41,410
So here's the notation.
21
00:01:41,410 --> 00:01:44,800
And because y is really
equal to f, sometimes
22
00:01:44,800 --> 00:01:50,140
we also call it the
differential of f.
23
00:01:50,140 --> 00:01:58,700
It's also called the
differential of f.
24
00:01:58,700 --> 00:02:06,750
That's the notation,
and it's the same thing
25
00:02:06,750 --> 00:02:11,630
as what happens if you
formally just take this dx,
26
00:02:11,630 --> 00:02:14,910
act like it's a number
and divide it into dy.
27
00:02:14,910 --> 00:02:22,670
So it means the same thing
as this statement here.
28
00:02:22,670 --> 00:02:29,160
And this is more or less
the Leibniz interpretation
29
00:02:29,160 --> 00:02:38,360
of derivatives.
30
00:02:38,360 --> 00:02:50,990
Of a derivative as a ratio of
these so called differentials.
31
00:02:50,990 --> 00:03:04,290
It's a ratio of what are
known as infinitesimals.
32
00:03:04,290 --> 00:03:09,050
Now, this is kind of a vague
notion, this little bit
33
00:03:09,050 --> 00:03:12,830
here being an infinitesimal.
34
00:03:12,830 --> 00:03:16,110
It's sort of like an
infinitely small quantity.
35
00:03:16,110 --> 00:03:21,300
And Leibniz perfected
the idea of dealing
36
00:03:21,300 --> 00:03:23,050
with these intuitively.
37
00:03:23,050 --> 00:03:26,730
And subsequently, mathematicians
use them all the time.
38
00:03:26,730 --> 00:03:33,350
They're way more effective than
the notation that Newton used.
39
00:03:33,350 --> 00:03:36,250
You might think that
notations are a small matter,
40
00:03:36,250 --> 00:03:40,960
but they allow you to think
much faster, sometimes.
41
00:03:40,960 --> 00:03:43,565
When you have the right
names and the right symbols
42
00:03:43,565 --> 00:03:44,190
for everything.
43
00:03:44,190 --> 00:03:47,860
And in this case it made
it very big difference.
44
00:03:47,860 --> 00:03:52,850
Leibniz's notation was adopted
on the continent and Newton
45
00:03:52,850 --> 00:03:56,650
dominated in Britain
and, as a result,
46
00:03:56,650 --> 00:03:58,800
the British fell
behind by one or two
47
00:03:58,800 --> 00:04:01,850
hundred years in the
development of calculus.
48
00:04:01,850 --> 00:04:03,540
It was really a serious matter.
49
00:04:03,540 --> 00:04:05,860
So it's really well
worth your while to get
50
00:04:05,860 --> 00:04:08,940
used to this idea of ratios.
51
00:04:08,940 --> 00:04:12,140
And it comes up all over the
place, both in this class
52
00:04:12,140 --> 00:04:14,320
and also in
multivariable calculus.
53
00:04:14,320 --> 00:04:17,380
It's used in many contexts.
54
00:04:17,380 --> 00:04:20,030
So first of all, just
to go a little bit easy.
55
00:04:20,030 --> 00:04:25,320
We'll illustrate it by its
use in linear approximations,
56
00:04:25,320 --> 00:04:36,420
which we've already done.
57
00:04:36,420 --> 00:04:38,780
The picture here, which we've
drawn a number of times,
58
00:04:38,780 --> 00:04:41,020
is that you have some function.
59
00:04:41,020 --> 00:04:44,300
And here's a value
of the function.
60
00:04:44,300 --> 00:04:47,060
And it's coming up like that.
61
00:04:47,060 --> 00:04:48,330
So here's our function.
62
00:04:48,330 --> 00:04:51,090
And we go forward
a little increment
63
00:04:51,090 --> 00:04:56,540
to a place which is
dx further along.
64
00:04:56,540 --> 00:04:59,350
The idea of this
notation is that dx
65
00:04:59,350 --> 00:05:05,260
is going to replace
the symbol delta x,
66
00:05:05,260 --> 00:05:07,370
which is the change in x.
67
00:05:07,370 --> 00:05:10,270
And we won't think
too hard about-- well,
68
00:05:10,270 --> 00:05:12,560
this is a small quantity,
this is a small quantity,
69
00:05:12,560 --> 00:05:16,030
we're not going to think too
hard about what that means.
70
00:05:16,030 --> 00:05:20,780
Now, similarly, if you see
how much we've gone up - well,
71
00:05:20,780 --> 00:05:26,600
this is kind of low, so
it's a small bit here.
72
00:05:26,600 --> 00:05:31,240
So this distance
here is, previously
73
00:05:31,240 --> 00:05:36,040
we called it delta y.
74
00:05:36,040 --> 00:05:41,810
But now we're just
going to call it dy.
75
00:05:41,810 --> 00:05:51,200
So dy replaces delta y.
76
00:05:51,200 --> 00:05:57,450
So this is the change in
level of the function.
77
00:05:57,450 --> 00:05:59,760
And we'll represent it
symbolically this way.
78
00:05:59,760 --> 00:06:04,050
Very frequently, this just
saves a little bit of notation.
79
00:06:04,050 --> 00:06:05,930
For the purposes
of this, we'll be
80
00:06:05,930 --> 00:06:09,620
doing the same things we did
with delta x and delta y,
81
00:06:09,620 --> 00:06:12,540
but this is the way that
Leibniz thought of it.
82
00:06:12,540 --> 00:06:14,690
And he would just have
drawn it with this.
83
00:06:14,690 --> 00:06:24,670
So this distance here is dx
and this distance here is dy.
84
00:06:24,670 --> 00:06:30,320
So for an example of
linear approximation,
85
00:06:30,320 --> 00:06:35,970
we'll say what's 64.1,
say, to the 1/3 power,
86
00:06:35,970 --> 00:06:39,500
approximately equal to?
87
00:06:39,500 --> 00:06:43,470
Now, I'm going to carry this
out in this new notation here.
88
00:06:43,470 --> 00:06:47,810
The function involved is x^1/3.
89
00:06:47,810 --> 00:06:50,990
And then it's a
differential, dy.
90
00:06:50,990 --> 00:06:53,670
Now, I want to use this
rule to get used to it.
91
00:06:53,670 --> 00:06:56,870
Because this is what we're going
to be doing all of today is,
92
00:06:56,870 --> 00:07:00,010
we're differentiating, or
taking the differential of y.
93
00:07:00,010 --> 00:07:02,490
So that is going to be
just the derivative.
94
00:07:02,490 --> 00:07:11,980
That's 1/3 x^(-2/3) dx.
95
00:07:11,980 --> 00:07:18,060
And now I'm just going to
fill in exactly what this is.
96
00:07:18,060 --> 00:07:25,450
At x = 64, which is the natural
place close by where it's easy
97
00:07:25,450 --> 00:07:36,110
to do the evaluations, we have
y = 64^(1/3), which is just 4.
98
00:07:36,110 --> 00:07:39,740
And how about dy?
99
00:07:39,740 --> 00:07:42,360
Well, so this is a little
bit more complicated.
100
00:07:42,360 --> 00:07:43,540
Put it over here.
101
00:07:43,540 --> 00:07:50,690
So dy = 1/3 64^(-2/3) dx.
102
00:07:55,800 --> 00:08:16,680
And that is 1/3 * 1/16
dx, which is 1/48 dx.
103
00:08:16,680 --> 00:08:19,810
And now I'm going
to work out what
104
00:08:19,810 --> 00:08:26,320
64 to the, whatever it is
here, this strange fraction.
105
00:08:26,320 --> 00:08:31,210
I just want to be very careful
to explain to you one more
106
00:08:31,210 --> 00:08:33,170
thing.
107
00:08:33,170 --> 00:08:37,780
Which is that
we're using x = 64,
108
00:08:37,780 --> 00:08:45,240
and so we're thinking of x
+ dx is going to be 64.1.
109
00:08:45,240 --> 00:08:53,700
So that means that dx
is going to be 1/10.
110
00:08:53,700 --> 00:08:59,760
So that's the increment
that we're interested in.
111
00:08:59,760 --> 00:09:03,910
And now I can carry
out the approximation.
112
00:09:03,910 --> 00:09:11,230
The approximation says
that 64.1^(1/3) is, well,
113
00:09:11,230 --> 00:09:14,710
it's approximately what
I'm going to call y + dy.
114
00:09:14,710 --> 00:09:18,060
Because really, the dy
that I'm determining here
115
00:09:18,060 --> 00:09:26,450
is determined by this linear
relation. dy = 1/48 dx.
116
00:09:26,450 --> 00:09:29,820
And so this is only
approximately true.
117
00:09:29,820 --> 00:09:33,430
Because what's really
true is that this
118
00:09:33,430 --> 00:09:37,690
is equal to y + delta y.
119
00:09:37,690 --> 00:09:39,590
In our previous notation.
120
00:09:39,590 --> 00:09:41,600
So this is in disguise.
121
00:09:41,600 --> 00:09:42,960
What this is equal to.
122
00:09:42,960 --> 00:09:44,500
And that's the
only approximately
123
00:09:44,500 --> 00:09:47,470
equal to what the linear
approximation would give you.
124
00:09:47,470 --> 00:09:51,730
So, really, even though I wrote
dy is this increment here,
125
00:09:51,730 --> 00:09:54,830
what it really is if
dx is exactly that,
126
00:09:54,830 --> 00:09:57,410
is it's the amount
it would go up
127
00:09:57,410 --> 00:10:00,270
if you went straight
up the tangent line.
128
00:10:00,270 --> 00:10:02,300
So I'm not going to do
that because that's not
129
00:10:02,300 --> 00:10:03,370
what people write.
130
00:10:03,370 --> 00:10:06,040
And that's not even
what they think.
131
00:10:06,040 --> 00:10:08,110
They're really thinking
of both dx and dy
132
00:10:08,110 --> 00:10:10,750
as being infinitesimally small.
133
00:10:10,750 --> 00:10:15,900
And here we're going to the
finite level and doing it.
134
00:10:15,900 --> 00:10:20,350
So this is just something
you have to live with,
135
00:10:20,350 --> 00:10:28,120
is a little ambiguity
in this notation.
136
00:10:28,120 --> 00:10:29,490
This is the approximation.
137
00:10:29,490 --> 00:10:32,970
And now I can just calculate
these numbers here.
138
00:10:32,970 --> 00:10:36,130
y at this value is 4.
139
00:10:36,130 --> 00:10:43,630
And dy, as I said, is 1/48 dx.
140
00:10:43,630 --> 00:10:50,440
And that turns out to be 4
+ 1/480, because dx is 1/10.
141
00:10:50,440 --> 00:10:54,390
So that's approximately 4.002.
142
00:10:54,390 --> 00:11:04,550
And that's our approximation.
143
00:11:04,550 --> 00:11:20,760
Now, let's just compare it
to our previous notation.
144
00:11:20,760 --> 00:11:22,970
This will serve as
a review of, if you
145
00:11:22,970 --> 00:11:35,870
like, of linear approximation.
146
00:11:35,870 --> 00:11:38,990
But what I want to emphasize
is that these things
147
00:11:38,990 --> 00:11:43,210
are supposed to be the same.
148
00:11:43,210 --> 00:11:45,520
Just that it's really
the same thing.
149
00:11:45,520 --> 00:11:52,280
It's just a different
notation for the same thing.
150
00:11:52,280 --> 00:11:56,360
I remind you the basic formula
for linear approximation is
151
00:11:56,360 --> 00:12:00,980
that f(x) is approximately
f(a) + f'(a) (x-a).
152
00:12:05,160 --> 00:12:11,620
And we're applying it in the
situation that a = 64 and f(x)
153
00:12:11,620 --> 00:12:12,910
= x^(1/3).
154
00:12:17,980 --> 00:12:27,580
And so f(a), which is
f(64), is of course 4.
155
00:12:27,580 --> 00:12:43,210
And f'(a), which is 1/3
a^(-2/3), is in our case 1/16.
156
00:12:43,210 --> 00:12:49,600
No, 1/48.
157
00:12:49,600 --> 00:12:52,980
OK, that's the same
calculation as before.
158
00:12:52,980 --> 00:12:59,490
And then our relationship
becomes x^(1/3) is
159
00:12:59,490 --> 00:13:08,420
approximately equal to 4 plus
1/48 times x minus a, which is
160
00:13:08,420 --> 00:13:12,250
64.
161
00:13:12,250 --> 00:13:14,820
So look, every single number
that I've written over here
162
00:13:14,820 --> 00:13:20,030
has a corresponding number
for this other method.
163
00:13:20,030 --> 00:13:23,250
And now if I plug in the
value we happen to want,
164
00:13:23,250 --> 00:13:32,500
which is the 64.1, this
would be 4 + 1/48 1/10,
165
00:13:32,500 --> 00:13:38,580
which is just the same
thing we had before.
166
00:13:38,580 --> 00:13:45,270
So again, same answer.
167
00:13:45,270 --> 00:13:55,300
Same method, new notation.
168
00:13:55,300 --> 00:14:02,300
Well, now I get to use this
notation in a novel way.
169
00:14:02,300 --> 00:14:04,640
So again, here's the notation.
170
00:14:04,640 --> 00:14:16,410
This notation of differential.
171
00:14:16,410 --> 00:14:20,520
The way I'm going to use it
is in discussing something
172
00:14:20,520 --> 00:14:32,560
called antiderivative Again,
this is a new notation now.
173
00:14:32,560 --> 00:14:33,680
But it's also a new idea.
174
00:14:33,680 --> 00:14:37,850
It's one that we
haven't discussed yet.
175
00:14:37,850 --> 00:14:42,500
Namely, the notation that
I want to describe here
176
00:14:42,500 --> 00:14:48,090
is what's called the
integral of g(x) dx.
177
00:14:48,090 --> 00:14:51,560
And I'll denote that by a
function capital G of x.
178
00:14:51,560 --> 00:14:53,820
So it's, you start
with a function g(x)
179
00:14:53,820 --> 00:14:55,380
and you produce a
function capital
180
00:14:55,380 --> 00:15:12,190
G(x), which is called
the antiderivative of g.
181
00:15:12,190 --> 00:15:15,330
Notice there's a
differential sitting in here.
182
00:15:15,330 --> 00:15:31,380
This symbol, this guy here,
is called an integral sign.
183
00:15:31,380 --> 00:15:34,600
Or an integral, or this whole
thing is called an integral.
184
00:15:34,600 --> 00:15:37,830
And another name for
the antiderivative of g
185
00:15:37,830 --> 00:15:50,700
is the indefinite integral of g.
186
00:15:50,700 --> 00:15:58,050
And I'll explain to you why
it's indefinite in just--
187
00:15:58,050 --> 00:16:04,330
very shortly here.
188
00:16:04,330 --> 00:16:13,330
Well, so let's carry
out some examples.
189
00:16:13,330 --> 00:16:16,840
Basically what I'd like
to do is as many examples
190
00:16:16,840 --> 00:16:18,510
along the lines of
all the derivatives
191
00:16:18,510 --> 00:16:21,329
that we derived at the
beginning of the course.
192
00:16:21,329 --> 00:16:22,870
In other words, in
principle you want
193
00:16:22,870 --> 00:16:26,550
to be able to integrate as
many things as possible.
194
00:16:26,550 --> 00:16:34,680
We're going to start out with
the integral of sin x dx.
195
00:16:34,680 --> 00:16:40,670
That's a function whose
derivative is sin x.
196
00:16:40,670 --> 00:16:44,990
So what function would that be?
197
00:16:44,990 --> 00:16:48,400
Cosine x, minus, right.
198
00:16:48,400 --> 00:16:49,590
It's -cos x.
199
00:16:52,700 --> 00:16:56,240
So -cos x differentiated
gives you sin x.
200
00:16:56,240 --> 00:17:00,540
So that is an
antiderivative of sine.
201
00:17:00,540 --> 00:17:02,210
And it satisfies this property.
202
00:17:02,210 --> 00:17:09,520
So this function,
G(x) = - cos x,
203
00:17:09,520 --> 00:17:15,170
has the property that
its derivative is sin x.
204
00:17:15,170 --> 00:17:20,700
On the other hand, if you
differentiate a constant,
205
00:17:20,700 --> 00:17:22,230
you get 0.
206
00:17:22,230 --> 00:17:25,090
So this answer is what's
called indefinite.
207
00:17:25,090 --> 00:17:28,910
Because you can also
add any constant here.
208
00:17:28,910 --> 00:17:33,610
And the same thing will be true.
209
00:17:33,610 --> 00:17:38,060
So, c is constant.
210
00:17:38,060 --> 00:17:41,660
And as I said, the integral
is called indefinite.
211
00:17:41,660 --> 00:17:45,410
So that's an explanation
for this modifier
212
00:17:45,410 --> 00:17:46,810
in front of the "integral".
213
00:17:46,810 --> 00:17:48,960
It's indefinite because
we actually didn't
214
00:17:48,960 --> 00:17:50,540
specify a single function.
215
00:17:50,540 --> 00:17:52,170
We don't get a single answer.
216
00:17:52,170 --> 00:17:54,400
Whenever you take the
antiderivative of something
217
00:17:54,400 --> 00:18:08,110
it's ambiguous up to a constant.
218
00:18:08,110 --> 00:18:12,340
Next, let's do some
other standard functions
219
00:18:12,340 --> 00:18:13,890
from our repertoire.
220
00:18:13,890 --> 00:18:17,820
We have the integral of x^a dx.
221
00:18:17,820 --> 00:18:20,730
Some power, the
integral of a power.
222
00:18:20,730 --> 00:18:24,680
And if you think about it, what
you should be differentiating
223
00:18:24,680 --> 00:18:27,590
is one power larger than that.
224
00:18:27,590 --> 00:18:33,250
But then you have to
divide by 1/(a+1),
225
00:18:33,250 --> 00:18:36,780
in order that the
differentiation be correct.
226
00:18:36,780 --> 00:18:44,360
So this just is the fact
that d/dx of x^(a+1),
227
00:18:44,360 --> 00:18:46,600
or maybe I should
even say it this way.
228
00:18:46,600 --> 00:18:49,620
Maybe I'll do it in
differential notation.
229
00:18:49,620 --> 00:18:54,230
d(x^(a+1)) = (a+1) x^a dx.
230
00:18:57,370 --> 00:19:02,950
So if I divide that
through by a+1,
231
00:19:02,950 --> 00:19:06,280
then I get the relation above.
232
00:19:06,280 --> 00:19:11,800
And because this is
ambiguous up to a constant,
233
00:19:11,800 --> 00:19:15,820
it could be any
additional constant
234
00:19:15,820 --> 00:19:20,830
added to that function.
235
00:19:20,830 --> 00:19:26,290
Now, the identity that I
wrote down below is correct.
236
00:19:26,290 --> 00:19:30,700
But this one is
not always correct.
237
00:19:30,700 --> 00:19:35,200
What's the exception?
238
00:19:35,200 --> 00:19:38,211
Yeah. a equals--
239
00:19:38,211 --> 00:19:38,710
STUDENT: 0.
240
00:19:38,710 --> 00:19:42,060
PROFESSOR: Negative 1.
241
00:19:42,060 --> 00:19:47,440
So this one is OK for all a.
242
00:19:47,440 --> 00:19:52,550
But this one fails because
we've divided by 0 when a = -1.
243
00:19:52,550 --> 00:20:04,770
So this is only true when
a is not equal to -1.
244
00:20:04,770 --> 00:20:07,790
And in fact, of course,
what's happening when a = 0,
245
00:20:07,790 --> 00:20:11,670
you're getting 0 when you
differentiate the constant.
246
00:20:11,670 --> 00:20:15,510
So there's a third case
that we have to carry out.
247
00:20:15,510 --> 00:20:25,230
Which is the exceptional case,
namely the integral of dx/x.
248
00:20:25,230 --> 00:20:31,265
And this time, if
we just think back
249
00:20:31,265 --> 00:20:32,640
to what our-- So
what we're doing
250
00:20:32,640 --> 00:20:35,080
is thinking backwards here,
which a very important thing
251
00:20:35,080 --> 00:20:38,690
to do in math at all stages.
252
00:20:38,690 --> 00:20:41,800
We got all of our formulas, now
we're reading them backwards.
253
00:20:41,800 --> 00:20:49,890
And so this one, you
may remember, is ln x.
254
00:20:49,890 --> 00:20:53,370
The reason why I want to do
this carefully and slowly now,
255
00:20:53,370 --> 00:20:57,299
is right now I also want to
write the more standard form
256
00:20:57,299 --> 00:20:58,090
which is presented.
257
00:20:58,090 --> 00:21:01,540
So first of all, first we
have to add a constant.
258
00:21:01,540 --> 00:21:04,160
And please don't put
the parentheses here.
259
00:21:04,160 --> 00:21:10,360
The parentheses go there.
260
00:21:10,360 --> 00:21:14,700
But there's another formula
hiding in the woodwork
261
00:21:14,700 --> 00:21:16,500
here behind this one.
262
00:21:16,500 --> 00:21:19,260
Which is that you can also
get the correct formula
263
00:21:19,260 --> 00:21:20,870
when x is negative.
264
00:21:20,870 --> 00:21:27,130
And that turns out
to be this one here.
265
00:21:27,130 --> 00:21:32,670
So I'm treating the case, x
positive, as being something
266
00:21:32,670 --> 00:21:34,460
that you know.
267
00:21:34,460 --> 00:21:43,530
But let's check the
case, x negative.
268
00:21:43,530 --> 00:21:45,810
In order to check
the case x negative,
269
00:21:45,810 --> 00:21:51,750
I have to differentiate the
logarithm of the absolute value
270
00:21:51,750 --> 00:21:55,680
of x in that case.
271
00:21:55,680 --> 00:21:57,612
And that's the
same thing, again,
272
00:21:57,612 --> 00:22:02,170
for x negative as the derivative
of the logarithm of negative x.
273
00:22:02,170 --> 00:22:08,410
That's the formula,
when x is negative.
274
00:22:08,410 --> 00:22:10,980
And if you carry
that out, what you
275
00:22:10,980 --> 00:22:18,400
get, maybe I'll put
this over here, is,
276
00:22:18,400 --> 00:22:20,800
well, it's the chain rule.
277
00:22:20,800 --> 00:22:27,160
It's 1/(-x) times the derivative
with respect to x of -x.
278
00:22:27,160 --> 00:22:30,730
So see that there
are two minus signs.
279
00:22:30,730 --> 00:22:32,750
There's a -x in the
denominator and then
280
00:22:32,750 --> 00:22:35,610
there's the derivative
of -x in the numerator.
281
00:22:35,610 --> 00:22:38,070
That's just -1.
282
00:22:38,070 --> 00:22:39,160
This part is -1.
283
00:22:39,160 --> 00:22:43,100
So this -1 over
-x, which is 1/x.
284
00:22:43,100 --> 00:22:53,200
So the negative signs cancel.
285
00:22:53,200 --> 00:23:00,090
If you just keep track of this
in terms of ln(-x) and its
286
00:23:00,090 --> 00:23:05,480
graph, that's a function
that looks like this.
287
00:23:05,480 --> 00:23:08,370
For x negative.
288
00:23:08,370 --> 00:23:14,170
And its derivative
is 1/x, I claim.
289
00:23:14,170 --> 00:23:17,120
And if you just look at
it a little bit carefully,
290
00:23:17,120 --> 00:23:23,431
you see that the slope
is always negative.
291
00:23:23,431 --> 00:23:23,930
Right?
292
00:23:23,930 --> 00:23:26,950
So here the slope is negative.
293
00:23:26,950 --> 00:23:30,480
So it's going to
be below the axis.
294
00:23:30,480 --> 00:23:32,890
And, in fact, it's getting
steeper and steeper negative
295
00:23:32,890 --> 00:23:34,980
as we go down.
296
00:23:34,980 --> 00:23:37,980
And it's getting less and less
negative as we go horizontally.
297
00:23:37,980 --> 00:23:40,790
So it's going like
this, which is indeed
298
00:23:40,790 --> 00:23:43,770
the graph of this
function, for x negative.
299
00:23:43,770 --> 00:23:53,320
Again, x negative.
300
00:23:53,320 --> 00:23:56,850
So that's one other
standard formula.
301
00:23:56,850 --> 00:24:00,501
And very quickly, very often,
we won't put the absolute value
302
00:24:00,501 --> 00:24:01,000
signs.
303
00:24:01,000 --> 00:24:03,180
We'll only consider the
case x positive here.
304
00:24:03,180 --> 00:24:06,280
But I just want you to
have the tools to do it
305
00:24:06,280 --> 00:24:08,710
in case we want to
use, we want to handle,
306
00:24:08,710 --> 00:24:14,040
both positive and negative x.
307
00:24:14,040 --> 00:24:28,620
Now, let's do two more examples.
308
00:24:28,620 --> 00:24:35,870
The integral of sec^2 x dx.
309
00:24:35,870 --> 00:24:38,270
These are supposed to
get you to remember
310
00:24:38,270 --> 00:24:41,130
all of your differentiation
formulas, the standard ones.
311
00:24:41,130 --> 00:24:48,200
So this one, integral of
sec^2 dx is what? tan x.
312
00:24:48,200 --> 00:24:50,690
And here we have + c, all right?
313
00:24:50,690 --> 00:24:55,520
And then the last one of, a
couple of, this type would be,
314
00:24:55,520 --> 00:24:56,800
let's see.
315
00:24:56,800 --> 00:25:04,139
I should do at least this one
here, square root of 1 - x^2.
316
00:25:04,139 --> 00:25:05,680
This is another
notation, by the way,
317
00:25:05,680 --> 00:25:07,240
which is perfectly acceptable.
318
00:25:07,240 --> 00:25:10,280
Notice I've put the
dx in the numerator
319
00:25:10,280 --> 00:25:13,400
and the function in
the denominator here.
320
00:25:13,400 --> 00:25:18,880
So this one turns
out to be sin^(-1) x.
321
00:25:18,880 --> 00:25:23,710
And, finally, let's see.
322
00:25:23,710 --> 00:25:28,470
About the integral
of dx / (1 + x^2).
323
00:25:28,470 --> 00:25:41,320
That one is tan^(-1) x.
324
00:25:41,320 --> 00:25:43,652
For a little while, because
you're reading these things
325
00:25:43,652 --> 00:25:45,110
backwards and
forwards, you'll find
326
00:25:45,110 --> 00:25:46,970
this happens to you on exams.
327
00:25:46,970 --> 00:25:49,540
It gets slightly worse
for a little while.
328
00:25:49,540 --> 00:25:53,120
You will antidifferentiate when
you meant to differentiate.
329
00:25:53,120 --> 00:25:54,620
And you'll differentiate
when you're
330
00:25:54,620 --> 00:25:57,230
meant to antidifferentiate.
331
00:25:57,230 --> 00:26:00,190
Don't get too
frustrated by that.
332
00:26:00,190 --> 00:26:05,270
But eventually, you'll
get them squared away.
333
00:26:05,270 --> 00:26:08,130
And it actually helps
to do a lot of practice
334
00:26:08,130 --> 00:26:15,410
with antidifferentiations,
or integrations,
335
00:26:15,410 --> 00:26:17,100
as they're sometimes called.
336
00:26:17,100 --> 00:26:19,790
Because that will
solidify your remembering
337
00:26:19,790 --> 00:26:25,820
all of the
differentiation formulas.
338
00:26:25,820 --> 00:26:30,080
So, last bit of
information that I
339
00:26:30,080 --> 00:26:32,730
want to emphasize before
we go on some more
340
00:26:32,730 --> 00:26:44,130
complicated examples is this.
341
00:26:44,130 --> 00:26:49,250
It's obvious because the
derivative of a constant is 0.
342
00:26:49,250 --> 00:26:55,330
That the antiderivative is
ambiguous up to a constant.
343
00:26:55,330 --> 00:26:56,920
But it's very
important to realize
344
00:26:56,920 --> 00:27:01,410
that this is the only
ambiguity that there is.
345
00:27:01,410 --> 00:27:04,390
So the last thing that
I want to tell you about
346
00:27:04,390 --> 00:27:24,070
is uniqueness of antiderivatives
up to a constant.
347
00:27:24,070 --> 00:27:30,290
The theorem is the following.
348
00:27:30,290 --> 00:27:41,230
The theorem is if F' =
G', then F equals G--
349
00:27:41,230 --> 00:27:43,620
so F(x) equals G(x)
plus a constant.
350
00:27:48,420 --> 00:27:54,050
But that means, not only that
these are antiderivatives,
351
00:27:54,050 --> 00:27:56,640
all these things with these
plus c's are antiderivatives.
352
00:27:56,640 --> 00:28:02,080
But these are the only ones.
353
00:28:02,080 --> 00:28:03,300
Which is very reassuring.
354
00:28:03,300 --> 00:28:06,020
And that's a kind of uniqueness,
although its uniqueness up
355
00:28:06,020 --> 00:28:09,660
to a constant, it's
acceptable to us.
356
00:28:09,660 --> 00:28:12,850
Now, the proof of
this is very quick.
357
00:28:12,850 --> 00:28:18,216
But this is a fundamental fact.
358
00:28:18,216 --> 00:28:19,340
The proof is the following.
359
00:28:19,340 --> 00:28:29,340
If F' = G', then if you take
the difference between the two
360
00:28:29,340 --> 00:28:33,520
functions, its derivative,
which of course is F' -
361
00:28:33,520 --> 00:28:40,110
G', is equal to 0.
362
00:28:40,110 --> 00:28:55,300
Hence, F(x) - G(x)
is a constant.
363
00:28:55,300 --> 00:28:58,940
Now, this is a key fact.
364
00:28:58,940 --> 00:28:59,870
Very important fact.
365
00:28:59,870 --> 00:29:03,840
We deduced it last time
from the mean value theorem.
366
00:29:03,840 --> 00:29:05,370
It's not a small matter.
367
00:29:05,370 --> 00:29:06,850
It's a very, very
important thing.
368
00:29:06,850 --> 00:29:08,690
It's the basis for calculus.
369
00:29:08,690 --> 00:29:11,550
It's the reason why
calculus make sense.
370
00:29:11,550 --> 00:29:14,120
If we didn't have the fact
that the derivative is
371
00:29:14,120 --> 00:29:18,890
0 implied that the function
was constant, we would be done.
372
00:29:18,890 --> 00:29:23,012
We would have-- Calculus
would be just useless for us.
373
00:29:23,012 --> 00:29:24,470
The point is, the
rate of change is
374
00:29:24,470 --> 00:29:27,330
supposed to determine
the function up
375
00:29:27,330 --> 00:29:29,650
to this starting value.
376
00:29:29,650 --> 00:29:32,330
So this conclusion
is very important.
377
00:29:32,330 --> 00:29:35,010
And we already checked it
last time, this conclusion.
378
00:29:35,010 --> 00:29:37,710
And now just by
algebra, I can rearrange
379
00:29:37,710 --> 00:30:03,340
this to say that F(x) is
equal to G(x) plus a constant.
380
00:30:03,340 --> 00:30:07,530
Now, maybe I should leave
differentials up here.
381
00:30:07,530 --> 00:30:21,390
Because I want to
illustrate-- So let's
382
00:30:21,390 --> 00:30:29,050
go on to some trickier,
slightly trickier, integrals.
383
00:30:29,050 --> 00:30:35,740
Here's an example.
384
00:30:35,740 --> 00:30:42,010
The integral of, say,
x^3 (x^4 + 2)^5 dx.
385
00:30:51,210 --> 00:30:53,480
This is a function
which you actually
386
00:30:53,480 --> 00:30:56,500
do know how to integrate,
because we already
387
00:30:56,500 --> 00:30:59,840
have a formula for all powers.
388
00:30:59,840 --> 00:31:03,280
Namely, the integral of
x^a is equal to this.
389
00:31:03,280 --> 00:31:06,520
And even if it were a negative
power, we could do it.
390
00:31:06,520 --> 00:31:08,630
So it's OK.
391
00:31:08,630 --> 00:31:14,290
On the other hand, to expand the
5th power here is quite a mess.
392
00:31:14,290 --> 00:31:18,480
And this is just a
very, very bad idea.
393
00:31:18,480 --> 00:31:21,320
There's another trick for
doing this that evaluates this
394
00:31:21,320 --> 00:31:23,060
much more efficiently.
395
00:31:23,060 --> 00:31:26,950
And it's the only
device that we're going
396
00:31:26,950 --> 00:31:31,550
to learn now for integrating.
397
00:31:31,550 --> 00:31:36,690
Integration actually is much
harder than differentiation,
398
00:31:36,690 --> 00:31:37,520
symbolically.
399
00:31:37,520 --> 00:31:42,610
It's quite difficult. And
occasionally impossible.
400
00:31:42,610 --> 00:31:45,550
And so we have to
go about it gently.
401
00:31:45,550 --> 00:31:47,450
But for the purposes
of this unit,
402
00:31:47,450 --> 00:31:50,069
we're only going
to use one method.
403
00:31:50,069 --> 00:31:50,860
Which is very good.
404
00:31:50,860 --> 00:31:52,526
That means whenever
you see an integral,
405
00:31:52,526 --> 00:31:56,180
either you'll be able to divine
immediately what the answer is,
406
00:31:56,180 --> 00:31:57,830
or you'll use this method.
407
00:31:57,830 --> 00:31:59,090
So this is it.
408
00:31:59,090 --> 00:32:09,470
The trick is called the
method of substitution.
409
00:32:09,470 --> 00:32:17,860
And it is tailor-made for
notion of differentials.
410
00:32:17,860 --> 00:32:36,510
So tailor-made for
differential notation.
411
00:32:36,510 --> 00:32:37,840
The idea is the following.
412
00:32:37,840 --> 00:32:40,260
I'm going to to
define a new function.
413
00:32:40,260 --> 00:32:43,200
And it's the messiest
function that I see here.
414
00:32:43,200 --> 00:32:50,290
It's u = x^4 + 2.
415
00:32:50,290 --> 00:32:56,300
And then, I'm going to take
its differential and what
416
00:32:56,300 --> 00:32:58,840
I discover, if I
look at its formula,
417
00:32:58,840 --> 00:33:02,570
and the rule for differentials,
which is right here.
418
00:33:02,570 --> 00:33:06,070
Its formula is what?
419
00:33:06,070 --> 00:33:10,180
4x^3 dx.
420
00:33:10,180 --> 00:33:14,000
Now, lo and behold with
these two quantities,
421
00:33:14,000 --> 00:33:17,940
I can substitute, I can
plug in to this integral.
422
00:33:17,940 --> 00:33:21,760
And I will simplify
it considerably.
423
00:33:21,760 --> 00:33:23,350
So how does that happen?
424
00:33:23,350 --> 00:33:34,740
Well, this integral is the
same thing as, well, really
425
00:33:34,740 --> 00:33:36,350
I should combine
it the other way.
426
00:33:36,350 --> 00:33:41,420
So let me move this over.
427
00:33:41,420 --> 00:33:43,340
So there are two pieces here.
428
00:33:43,340 --> 00:33:46,440
And this one is u^5.
429
00:33:46,440 --> 00:33:54,990
And this one is 1/4 du.
430
00:33:54,990 --> 00:34:01,840
Now, that makes it the
integral of u^5 du / 4.
431
00:34:01,840 --> 00:34:04,040
And that's relatively
easy to integrate.
432
00:34:04,040 --> 00:34:05,460
That is just a power.
433
00:34:05,460 --> 00:34:06,410
So let's see.
434
00:34:06,410 --> 00:34:11,250
It's just 1/20 u to
the-- whoops, not 1/20.
435
00:34:11,250 --> 00:34:15,480
The antiderivative
of u^5 is u^6.
436
00:34:15,480 --> 00:34:25,480
With the 1/6, so
it's 1/24 u^6 + c.
437
00:34:25,480 --> 00:34:29,050
Now, that's not the
answer to the question.
438
00:34:29,050 --> 00:34:32,260
It's almost the answer
to the question.
439
00:34:32,260 --> 00:34:33,287
Why isn't it the answer?
440
00:34:33,287 --> 00:34:35,120
It isn't the answer
because now the answer's
441
00:34:35,120 --> 00:34:37,480
expressed in terms of u.
442
00:34:37,480 --> 00:34:41,750
Whereas the problem was posed
in terms of this variable x.
443
00:34:41,750 --> 00:34:45,960
So we must change back
to our variable here.
444
00:34:45,960 --> 00:34:47,990
And we do that just
by writing it in.
445
00:34:47,990 --> 00:34:56,190
So it's 1/24 (x^4 + 2)^6 + c.
446
00:34:56,190 --> 00:35:02,120
And this is the
end of the problem.
447
00:35:02,120 --> 00:35:02,990
Yeah, question.
448
00:35:02,990 --> 00:35:16,350
STUDENT: [INAUDIBLE]
449
00:35:16,350 --> 00:35:19,330
PROFESSOR: The question is,
can you see it directly?
450
00:35:19,330 --> 00:35:20,160
Yeah.
451
00:35:20,160 --> 00:35:23,820
And we're going to talk about
that in just one second.
452
00:35:23,820 --> 00:35:30,290
OK.
453
00:35:30,290 --> 00:35:35,500
Now, I'm going to
do one more example
454
00:35:35,500 --> 00:35:44,310
and illustrate this method.
455
00:35:44,310 --> 00:35:45,405
Here's another example.
456
00:35:45,405 --> 00:35:51,430
The integral of x dx over
the square root of 1 + x^2.
457
00:35:51,430 --> 00:35:56,610
Now, here's another example.
458
00:35:56,610 --> 00:36:03,475
Now, the method of substitution
leads us to the idea u = 1 +
459
00:36:03,475 --> 00:36:05,190
x^2.
460
00:36:05,190 --> 00:36:11,540
du = 2x dx, etc.
461
00:36:11,540 --> 00:36:14,450
It takes about as long as
this other problem did.
462
00:36:14,450 --> 00:36:15,720
To figure out what's going on.
463
00:36:15,720 --> 00:36:17,440
It's a very similar
sort of thing.
464
00:36:17,440 --> 00:36:20,870
You end up integrating u^(-1/2).
465
00:36:20,870 --> 00:36:23,480
It leads to the
integral of u^(-1/2) du.
466
00:36:28,350 --> 00:36:31,630
Is everybody seeing
where this...?
467
00:36:31,630 --> 00:36:37,540
However, there is a
slightly better method.
468
00:36:37,540 --> 00:36:46,070
So recommended method.
469
00:36:46,070 --> 00:36:59,250
And I call this method
advanced guessing.
470
00:36:59,250 --> 00:37:01,077
What advanced guessing
means is that you've
471
00:37:01,077 --> 00:37:02,660
done enough of these
problems that you
472
00:37:02,660 --> 00:37:04,750
can see two steps ahead.
473
00:37:04,750 --> 00:37:08,030
And you know what's
going to happen.
474
00:37:08,030 --> 00:37:10,440
So the advanced
guessing leads you
475
00:37:10,440 --> 00:37:12,817
to believe that here
you had a power -1/2,
476
00:37:12,817 --> 00:37:14,650
here you have the
differential of the thing.
477
00:37:14,650 --> 00:37:16,790
So it's going to
work out somehow.
478
00:37:16,790 --> 00:37:19,670
And the advanced guessing allows
you to guess that the answer
479
00:37:19,670 --> 00:37:23,520
should be something like
this. (1 + x^2)^(1/2).
480
00:37:26,050 --> 00:37:27,800
So this is your advanced guess.
481
00:37:27,800 --> 00:37:31,670
And now you just differentiate
it, and see whether it works.
482
00:37:31,670 --> 00:37:32,550
Well, here it is.
483
00:37:32,550 --> 00:37:39,330
It's 1/2 (1 + x^2)^(-1/2) 2x,
that's the chain rule here.
484
00:37:39,330 --> 00:37:44,770
Which, sure enough, gives you
x over square root of 1 + x^2.
485
00:37:44,770 --> 00:37:45,480
So we're done.
486
00:37:45,480 --> 00:37:56,960
And so the answer is square
root of (1 + x^2) + c.
487
00:37:56,960 --> 00:38:02,160
Let me illustrate this
further with another example.
488
00:38:02,160 --> 00:38:06,010
I strongly recommend
that you do this,
489
00:38:06,010 --> 00:38:09,270
but you have to get used to it.
490
00:38:09,270 --> 00:38:11,490
So here's another example.
491
00:38:11,490 --> 00:38:18,900
e^(6x) dx.
492
00:38:18,900 --> 00:38:26,360
My advanced guess is e^(6x).
493
00:38:26,360 --> 00:38:30,020
And if I check, when
I differentiate it,
494
00:38:30,020 --> 00:38:33,180
I get 6e^(6x).
495
00:38:33,180 --> 00:38:35,020
That's the derivative.
496
00:38:35,020 --> 00:38:38,030
And so I know that
the answer, so now I
497
00:38:38,030 --> 00:38:39,030
know what the answer is.
498
00:38:39,030 --> 00:38:46,300
It's 1/6 e^(6x) + c.
499
00:38:46,300 --> 00:38:54,510
Now, OK, you could,
it's also OK, but slow,
500
00:38:54,510 --> 00:39:02,550
to use a substitution,
to use u = 6x.
501
00:39:02,550 --> 00:39:07,920
Then you're going to get
du = 6dx, dot, dot, dot.
502
00:39:07,920 --> 00:39:23,220
It's going to work, it's
just a waste of time.
503
00:39:23,220 --> 00:39:26,760
Well, I'm going to give
you a couple more examples.
504
00:39:26,760 --> 00:39:27,910
So how about this one.
505
00:39:33,560 --> 00:39:35,120
x e^(-x^2) dx.
506
00:39:41,340 --> 00:39:45,600
What's the guess?
507
00:39:45,600 --> 00:39:51,270
Anybody have a guess?
508
00:39:51,270 --> 00:39:52,480
Well, you could also correct.
509
00:39:52,480 --> 00:39:54,438
So I don't want you to
bother - yeah, go ahead.
510
00:39:54,438 --> 00:39:56,787
STUDENT: [INAUDIBLE]
511
00:39:56,787 --> 00:39:59,120
PROFESSOR: Yeah, so you're
already one step ahead of me.
512
00:39:59,120 --> 00:40:02,050
Because this is too easy.
513
00:40:02,050 --> 00:40:04,110
When they get more
complicated, you just
514
00:40:04,110 --> 00:40:05,690
want to make this guess here.
515
00:40:05,690 --> 00:40:08,884
So various people have said
1/2, and they understand
516
00:40:08,884 --> 00:40:10,050
that there's 1/2 going here.
517
00:40:10,050 --> 00:40:13,890
But let me just show
you what happens, OK?
518
00:40:13,890 --> 00:40:19,140
If you make this guess
and you differentiate it,
519
00:40:19,140 --> 00:40:23,940
what you get here is
e^(-x^2) times the derivative
520
00:40:23,940 --> 00:40:28,130
of negative 2x, so that's -2x.
521
00:40:28,130 --> 00:40:30,400
x^2, so it's -2x.
522
00:40:30,400 --> 00:40:37,970
So now you see that you're off
by a factor of not 2, but -2.
523
00:40:37,970 --> 00:40:39,820
So a number of you
were saying that.
524
00:40:39,820 --> 00:40:43,120
So the answer is
-1/2 e^(-x^2) + c.
525
00:40:46,510 --> 00:40:49,090
And I can guarantee
you, having watched
526
00:40:49,090 --> 00:40:55,290
this on various problems, that
people who don't write this out
527
00:40:55,290 --> 00:40:57,360
make arithmetic mistakes.
528
00:40:57,360 --> 00:41:00,140
In other words, there
is a limit to how much
529
00:41:00,140 --> 00:41:02,760
people can think ahead
and guess correctly.
530
00:41:02,760 --> 00:41:04,760
Another way of doing
it, by the way,
531
00:41:04,760 --> 00:41:06,680
is simply to write
this thing in and then
532
00:41:06,680 --> 00:41:10,160
fix the coefficient by doing
the differentiation here.
533
00:41:10,160 --> 00:41:14,850
That's perfectly OK as well.
534
00:41:14,850 --> 00:41:18,920
All right, one more example.
535
00:41:18,920 --> 00:41:30,840
We're going to integrate
sin x cos x dx.
536
00:41:30,840 --> 00:41:33,750
So what's a good
guess for this one?
537
00:41:33,750 --> 00:41:36,520
STUDENT: [INAUDIBLE]
538
00:41:36,520 --> 00:41:38,850
PROFESSOR: Someone
suggesting sin^2 x.
539
00:41:38,850 --> 00:41:41,490
So let's try that.
540
00:41:41,490 --> 00:41:45,020
Over 2 - well, we'll get the
coefficient in just a second.
541
00:41:45,020 --> 00:41:50,970
So sin^2 x, if I differentiate,
I get 2 sin x cos x.
542
00:41:50,970 --> 00:41:53,380
So that's off by a factor of 2.
543
00:41:53,380 --> 00:42:04,540
So the answer is 1/2 sin^2 x.
544
00:42:04,540 --> 00:42:12,320
But now I want to
point out to you
545
00:42:12,320 --> 00:42:17,120
that there's another way
of doing this problem.
546
00:42:17,120 --> 00:42:31,240
It's also true that if
you differentiate cos^2 x,
547
00:42:31,240 --> 00:42:34,600
you get 2 cos x (-sin x).
548
00:42:38,130 --> 00:42:51,030
So another answer is that the
integral of sin x cos x dx is
549
00:42:51,030 --> 00:43:01,840
equal to -1/2 cos^2 x + c.
550
00:43:01,840 --> 00:43:03,740
So what is going on here?
551
00:43:03,740 --> 00:43:06,890
What's the problem with this?
552
00:43:06,890 --> 00:43:10,785
STUDENT: [INAUDIBLE]
553
00:43:10,785 --> 00:43:11,660
PROFESSOR: Pardon me?
554
00:43:11,660 --> 00:43:15,060
STUDENT: [INAUDIBLE]
555
00:43:15,060 --> 00:43:18,130
PROFESSOR: Integrals
aren't unique.
556
00:43:18,130 --> 00:43:21,390
That's part of the-- but
somehow these two answers still
557
00:43:21,390 --> 00:43:22,320
have to be the same.
558
00:43:22,320 --> 00:43:32,910
STUDENT: [INAUDIBLE]
559
00:43:32,910 --> 00:43:35,910
PROFESSOR: OK.
560
00:43:35,910 --> 00:43:36,660
What do you think?
561
00:43:36,660 --> 00:43:38,743
STUDENT: If you add them
together, you just get c.
562
00:43:38,743 --> 00:43:40,900
PROFESSOR: If you add
them together you get c.
563
00:43:40,900 --> 00:43:44,185
Well, actually,
that's almost right.
564
00:43:44,185 --> 00:43:45,810
That's not what you
want to do, though.
565
00:43:45,810 --> 00:43:47,620
You don't want to add them.
566
00:43:47,620 --> 00:43:50,930
You want to subtract them.
567
00:43:50,930 --> 00:43:53,750
So let's see what happens
when you subtract them.
568
00:43:53,750 --> 00:43:56,840
I'm going to ignore the
c, for the time being.
569
00:43:56,840 --> 00:44:05,520
I get sin^2 x, 1/2 sin^2
x - (-1/2 cos^2 x).
570
00:44:05,520 --> 00:44:08,880
So the difference between
them, we hope to be 0.
571
00:44:08,880 --> 00:44:10,760
But actually of
course it's not 0.
572
00:44:10,760 --> 00:44:17,680
What it is, is it's 1/2 sine
squared plus cosine squared,
573
00:44:17,680 --> 00:44:18,850
which is 1/2.
574
00:44:18,850 --> 00:44:24,200
It's not 0, it's a constant.
575
00:44:24,200 --> 00:44:27,310
So what's really going on here
is that these two formulas
576
00:44:27,310 --> 00:44:29,290
are the same.
577
00:44:29,290 --> 00:44:31,740
But you have to understand
how to interpret them.
578
00:44:31,740 --> 00:44:34,450
The two constants, here's
a constant up here.
579
00:44:34,450 --> 00:44:37,900
There's a constant, c_1
associated to this one.
580
00:44:37,900 --> 00:44:43,250
There's a different constant,
c_2 associated to this one.
581
00:44:43,250 --> 00:44:45,960
And this family of functions
for all possible c_1's
582
00:44:45,960 --> 00:44:49,860
and all possible c_2's, is
the same family of functions.
583
00:44:49,860 --> 00:44:52,940
Now, what's the relationship
between c_1 and c_2?
584
00:44:52,940 --> 00:44:57,210
Well, if you do the
subtraction, c_1 - c_2
585
00:44:57,210 --> 00:44:59,240
has to be equal to 1/2.
586
00:44:59,240 --> 00:45:06,610
They're both constants,
but they differ by 1/2.
587
00:45:06,610 --> 00:45:08,404
So this explains,
when you're dealing
588
00:45:08,404 --> 00:45:10,820
with families of things, they
don't have to look the same.
589
00:45:10,820 --> 00:45:12,560
And there are lots
of trig functions
590
00:45:12,560 --> 00:45:16,280
which look a little different.
591
00:45:16,280 --> 00:45:19,050
So there can be several formulas
that actually are the same.
592
00:45:19,050 --> 00:45:21,960
And it's hard to check that
they're actually the same.
593
00:45:21,960 --> 00:45:28,950
You need some trig
identities to do it.
594
00:45:28,950 --> 00:45:55,210
Let's do one more example here.
595
00:45:55,210 --> 00:46:06,250
Here's another one.
596
00:46:06,250 --> 00:46:13,510
Now, you may be thinking,
and a lot of people
597
00:46:13,510 --> 00:46:22,250
are, thinking ugh,
it's got a ln in it.
598
00:46:22,250 --> 00:46:24,400
If you're experienced,
you actually
599
00:46:24,400 --> 00:46:26,130
can read off the
answer just the way
600
00:46:26,130 --> 00:46:28,088
there were several people
who were shouting out
601
00:46:28,088 --> 00:46:31,000
the answers when we were doing
the rest of these problems.
602
00:46:31,000 --> 00:46:32,970
But, you do need to relax.
603
00:46:32,970 --> 00:46:35,790
Because in this case, now
this is definitely not
604
00:46:35,790 --> 00:46:37,490
true in general when
we do integrals.
605
00:46:37,490 --> 00:46:39,080
But, for now, when
we do integrals,
606
00:46:39,080 --> 00:46:40,570
they'll all be manageable.
607
00:46:40,570 --> 00:46:42,670
And there's only one method.
608
00:46:42,670 --> 00:46:47,390
Which is substitution.
609
00:46:47,390 --> 00:46:49,680
And in the substitution
method, you
610
00:46:49,680 --> 00:46:52,200
want to go for the
trickiest part.
611
00:46:52,200 --> 00:46:55,220
And substitute for that.
612
00:46:55,220 --> 00:46:57,630
So the substitution
that I proposed
613
00:46:57,630 --> 00:47:02,200
to you is that this should
be, u should be ln x.
614
00:47:02,200 --> 00:47:06,270
And the advantage that that
has is that its differential
615
00:47:06,270 --> 00:47:08,720
is simpler than itself.
616
00:47:08,720 --> 00:47:15,570
So du = dx / x.
617
00:47:15,570 --> 00:47:18,500
Remember, we use that in
logarithmic differentiation,
618
00:47:18,500 --> 00:47:21,550
too.
619
00:47:21,550 --> 00:47:28,810
So now we can express this
using this substitution.
620
00:47:28,810 --> 00:47:32,290
And what we get is,
the integral of,
621
00:47:32,290 --> 00:47:33,990
so I'll divide the
two parts here.
622
00:47:33,990 --> 00:47:36,515
It's 1 / ln x, and
then it's dx / x.
623
00:47:36,515 --> 00:47:43,370
And this part is 1 /
u, and this part is du.
624
00:47:43,370 --> 00:47:49,260
So it's the integral of du / u.
625
00:47:49,260 --> 00:47:58,490
And that is ln u + c.
626
00:47:58,490 --> 00:48:11,030
Which altogether, if I put back
in what u is, is ln (ln x) + c.
627
00:48:11,030 --> 00:48:14,290
And now we see
some uglier things.
628
00:48:14,290 --> 00:48:15,900
In fact, technically
speaking, we
629
00:48:15,900 --> 00:48:18,730
could take the
absolute value here.
630
00:48:18,730 --> 00:48:28,130
And then this would be
absolute values there.
631
00:48:28,130 --> 00:48:33,090
So this is the type of
example where I really
632
00:48:33,090 --> 00:48:35,740
would recommend that you
actually use the substitution,
633
00:48:35,740 --> 00:48:39,030
at least for now.
634
00:48:39,030 --> 00:48:41,820
All right, tomorrow
we're going to be
635
00:48:41,820 --> 00:48:43,080
doing differential equations.
636
00:48:43,080 --> 00:48:45,130
And we're going to
review for the test.
637
00:48:45,130 --> 00:48:47,649
I'm going to give you a handout
telling you just exactly
638
00:48:47,649 --> 00:48:48,940
what's going to be on the test.
639
00:48:48,940 --> 00:48:52,298
So, see you tomorrow.