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PROFESSOR: Now,
to start out today
9
00:00:25,240 --> 00:00:27,850
we're going to finish up
what we did last time.
10
00:00:27,850 --> 00:00:30,950
Which has to do with
partial fractions.
11
00:00:30,950 --> 00:00:33,160
I told you how to
do partial fractions
12
00:00:33,160 --> 00:00:35,080
in several special
cases and everybody
13
00:00:35,080 --> 00:00:37,920
was trying to figure out
what the general picture was.
14
00:00:37,920 --> 00:00:39,180
But I'd like to lay that out.
15
00:00:39,180 --> 00:00:41,730
I'll still only do
it for an example.
16
00:00:41,730 --> 00:00:43,810
But it will be somehow
a bigger example
17
00:00:43,810 --> 00:00:53,310
so that you can see what
the general pattern is.
18
00:00:53,310 --> 00:01:04,280
Partial fractions, remember,
is a method for breaking up
19
00:01:04,280 --> 00:01:06,730
so-called rational functions.
20
00:01:06,730 --> 00:01:09,510
Which are ratios of polynomials.
21
00:01:09,510 --> 00:01:13,110
And it shows you that you
can always integrate them.
22
00:01:13,110 --> 00:01:14,920
That's really the theme here.
23
00:01:14,920 --> 00:01:21,190
And this is what's reassuring
is that it always works.
24
00:01:21,190 --> 00:01:23,670
That's really the bottom line.
25
00:01:23,670 --> 00:01:26,550
And that's good
because there are
26
00:01:26,550 --> 00:01:34,100
a lot of integrals that don't
have formulas and these do.
27
00:01:34,100 --> 00:01:35,560
It always works.
28
00:01:35,560 --> 00:01:43,340
But, maybe with lots of help.
29
00:01:43,340 --> 00:01:46,257
So maybe slowly.
30
00:01:46,257 --> 00:01:47,840
Now, there's a little
bit of bad news,
31
00:01:47,840 --> 00:01:50,820
and I have to be totally
honest and tell you
32
00:01:50,820 --> 00:01:52,230
what all the bad news is.
33
00:01:52,230 --> 00:01:54,930
Along with the good news.
34
00:01:54,930 --> 00:02:00,070
The first step, which maybe
I should be calling Step 0,
35
00:02:00,070 --> 00:02:08,710
I had a Step 1, 2 and 3
last time, is long division.
36
00:02:08,710 --> 00:02:11,790
That's the step where you
take your polynomial divided
37
00:02:11,790 --> 00:02:17,440
by your other polynomial,
and you find the quotient
38
00:02:17,440 --> 00:02:22,470
plus some remainder.
39
00:02:22,470 --> 00:02:24,370
And you do that
by long division.
40
00:02:24,370 --> 00:02:27,520
And the quotient is easy
to take the antiderivative
41
00:02:27,520 --> 00:02:30,010
of because it's
just a polynomial.
42
00:02:30,010 --> 00:02:32,380
And the key extra
property here is
43
00:02:32,380 --> 00:02:35,680
that the degree of the numerator
now over here, this remainder,
44
00:02:35,680 --> 00:02:40,510
is strictly less than the
degree of the denominator.
45
00:02:40,510 --> 00:02:44,140
So that you can
do the next step.
46
00:02:44,140 --> 00:02:48,900
Now, the next step which I
called Step 1 last time, that's
47
00:02:48,900 --> 00:02:52,290
great imagination, it's
right after Step 0, Step 1
48
00:02:52,290 --> 00:02:54,930
was to factor the denominator.
49
00:02:54,930 --> 00:03:00,630
And I'm going to illustrate by
example what the setup is here.
50
00:03:00,630 --> 00:03:09,120
I don't know maybe,
we'll do this.
51
00:03:09,120 --> 00:03:12,920
Some polynomial here,
maybe cube this one.
52
00:03:12,920 --> 00:03:21,440
So here I've factored
the denominator.
53
00:03:21,440 --> 00:03:24,840
That's what I called
Step 1 last time.
54
00:03:24,840 --> 00:03:27,650
Now, here's the first
piece of bad news.
55
00:03:27,650 --> 00:03:31,620
In reality, if somebody
gave you a multiplied
56
00:03:31,620 --> 00:03:35,530
out degree-whatever
polynomial here,
57
00:03:35,530 --> 00:03:40,710
you would be very hard
pressed to factor it.
58
00:03:40,710 --> 00:03:44,132
A lot of them are extremely
difficult to factor.
59
00:03:44,132 --> 00:03:46,590
And so that's something you
would have to give to a machine
60
00:03:46,590 --> 00:03:47,940
to do.
61
00:03:47,940 --> 00:03:50,880
And it's just basically
a hard problem.
62
00:03:50,880 --> 00:03:54,050
So obviously, we're only
going to give you ones
63
00:03:54,050 --> 00:03:55,480
that you can do by hand.
64
00:03:55,480 --> 00:03:58,060
So very low degree examples.
65
00:03:58,060 --> 00:03:59,320
And that's just the way it is.
66
00:03:59,320 --> 00:04:03,600
So this is really a hard step
in disguise, in real life.
67
00:04:03,600 --> 00:04:06,000
Anyway, we're just going
to take it as given.
68
00:04:06,000 --> 00:04:07,820
And we have this
numerator, which
69
00:04:07,820 --> 00:04:10,720
is of degree less
than the denominator.
70
00:04:10,720 --> 00:04:14,830
So let's count up what
its degree has to be.
71
00:04:14,830 --> 00:04:18,640
This is 4 + 2 + 6.
72
00:04:18,640 --> 00:04:22,440
So this is degree 4 + 2 + 6.
73
00:04:22,440 --> 00:04:24,190
I added that up because
this is degree 4,
74
00:04:24,190 --> 00:04:28,600
this is degree 2 and
(x^2)^3 is the 6th power.
75
00:04:28,600 --> 00:04:32,330
So all together it's
this, which is 12.
76
00:04:32,330 --> 00:04:39,822
And so this thing
is of degree <= 11.
77
00:04:39,822 --> 00:04:41,280
All the way up to
degree 11, that's
78
00:04:41,280 --> 00:04:44,240
the possibilities for
the numerator here.
79
00:04:44,240 --> 00:04:49,810
Now, the extra information that
I want to impart right now,
80
00:04:49,810 --> 00:04:56,450
is just this setup.
81
00:04:56,450 --> 00:04:58,990
Which I called Step 2 last time.
82
00:04:58,990 --> 00:05:05,260
And the setup is this.
83
00:05:05,260 --> 00:05:07,590
Now, it's going to take
us a while to do this.
84
00:05:07,590 --> 00:05:10,810
We have this factor here.
85
00:05:10,810 --> 00:05:12,450
We have another factor.
86
00:05:12,450 --> 00:05:14,620
We have another term,
with the square.
87
00:05:14,620 --> 00:05:18,220
We have another
term with the cube.
88
00:05:18,220 --> 00:05:22,581
We have another term
with the fourth power.
89
00:05:22,581 --> 00:05:24,330
So this is what's going
to happen whenever
90
00:05:24,330 --> 00:05:25,530
you have linear factors.
91
00:05:25,530 --> 00:05:28,560
You'll have a collection
of terms like this.
92
00:05:28,560 --> 00:05:31,030
So you have four
constants to take care of.
93
00:05:31,030 --> 00:05:34,550
Now, with a quadratic
in the denominator,
94
00:05:34,550 --> 00:05:36,840
you need a linear
function in the numerator.
95
00:05:36,840 --> 00:05:41,230
So that's, if you like,
B_0 x + C_0 divided
96
00:05:41,230 --> 00:05:49,370
by this quadratic term here.
97
00:05:49,370 --> 00:05:52,500
And what I didn't
show you last time
98
00:05:52,500 --> 00:05:59,720
was how you deal with higher
powers of quadratic terms.
99
00:05:59,720 --> 00:06:04,090
So when you have a quadratic
term, what's going to happen
100
00:06:04,090 --> 00:06:07,370
is you're going to take
that first factor here.
101
00:06:07,370 --> 00:06:11,870
Just the way you
did in this case.
102
00:06:11,870 --> 00:06:15,500
But then you're going to
have to do the same thing
103
00:06:15,500 --> 00:06:24,120
with the next power.
104
00:06:24,120 --> 00:06:27,880
Now notice, just as in
the case of this top row,
105
00:06:27,880 --> 00:06:29,830
I have just a constant here.
106
00:06:29,830 --> 00:06:33,030
And even though I increased
the degree of the denominator
107
00:06:33,030 --> 00:06:34,540
I'm not increasing
the numerator.
108
00:06:34,540 --> 00:06:35,970
It's staying just a constant.
109
00:06:35,970 --> 00:06:38,230
It's not linear up here.
110
00:06:38,230 --> 00:06:39,850
It's better than that.
111
00:06:39,850 --> 00:06:41,990
It's just a constant.
112
00:06:41,990 --> 00:06:44,130
And here it stayed a constant.
113
00:06:44,130 --> 00:06:45,630
And here it stayed a constant.
114
00:06:45,630 --> 00:06:48,070
Similarly here, even
though I'm increasing
115
00:06:48,070 --> 00:06:49,620
the degree of the
denominator, I'm
116
00:06:49,620 --> 00:06:52,810
leaving the numerator, the
form of the numerator, alone.
117
00:06:52,810 --> 00:06:55,150
It's just a linear factor
and a linear factor.
118
00:06:55,150 --> 00:07:05,350
So that's the key
to this pattern.
119
00:07:05,350 --> 00:07:09,850
I don't have quite as
jazzy a song on mine.
120
00:07:09,850 --> 00:07:13,410
So this is so long that it
runs off the blackboard here.
121
00:07:13,410 --> 00:07:15,820
So let's continue
it on the next.
122
00:07:15,820 --> 00:07:20,200
We've got this B_2
x + C_2-- sorry,
123
00:07:20,200 --> 00:07:23,150
(B_3 x + C_3) / (x^2 + 4)^3.
124
00:07:26,590 --> 00:07:38,590
I guess I have room
for it over here.
125
00:07:38,590 --> 00:07:41,120
I'm going to talk about
this in just a second.
126
00:07:41,120 --> 00:07:43,590
Alright, so here's the pattern.
127
00:07:43,590 --> 00:07:51,270
Now, let me just do a count
of the number of unknowns
128
00:07:51,270 --> 00:07:52,344
we have here.
129
00:07:52,344 --> 00:07:54,010
The number of unknowns
that we have here
130
00:07:54,010 --> 00:07:58,750
is 1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11, 12.
131
00:07:58,750 --> 00:08:00,930
That 12 is no coincidence.
132
00:08:00,930 --> 00:08:03,815
That's the degree
of the polynomial.
133
00:08:03,815 --> 00:08:05,690
And it's the number of
unknowns that we have.
134
00:08:05,690 --> 00:08:08,060
And it's the number
of degrees of freedom
135
00:08:08,060 --> 00:08:11,215
in a polynomial of degree 11.
136
00:08:11,215 --> 00:08:13,090
If you have all these
free coefficients here,
137
00:08:13,090 --> 00:08:17,560
you have the coefficient x^0,
x^1, all the way up to x^ 11.
138
00:08:17,560 --> 00:08:23,100
And 0 through 11 is 12
different coefficients.
139
00:08:23,100 --> 00:08:26,380
And so this is a very
complicated system
140
00:08:26,380 --> 00:08:28,150
of equations for unknowns.
141
00:08:28,150 --> 00:08:33,259
This is twelve equations
for twelve unknowns.
142
00:08:33,259 --> 00:08:34,800
So I'll get rid of
this for a second.
143
00:08:34,800 --> 00:08:41,090
So we have twelve
equations, twelve unknowns.
144
00:08:41,090 --> 00:08:43,830
So that's the other bad news.
145
00:08:43,830 --> 00:08:46,020
Machines handle this very
well, but human beings
146
00:08:46,020 --> 00:08:47,630
have a little trouble with 12.
147
00:08:47,630 --> 00:08:51,570
Now, the cover-up
method works very neatly
148
00:08:51,570 --> 00:08:53,730
and picks out this term here.
149
00:08:53,730 --> 00:08:54,470
But that's it.
150
00:08:54,470 --> 00:08:56,550
So it reduces it to an 11 by 11.
151
00:08:56,550 --> 00:09:00,170
You'll be able to
evaluate this in no time.
152
00:09:00,170 --> 00:09:00,890
But that's it.
153
00:09:00,890 --> 00:09:04,240
That's the only simplification
of your previous method.
154
00:09:04,240 --> 00:09:06,250
We don't have a method for this.
155
00:09:06,250 --> 00:09:08,510
So I'm just showing what
the whole method looks
156
00:09:08,510 --> 00:09:10,343
like but really you'd
have to have a machine
157
00:09:10,343 --> 00:09:14,760
to implement this once it
gets to be any size at all.
158
00:09:14,760 --> 00:09:15,490
Yeah, question.
159
00:09:15,490 --> 00:09:18,060
STUDENT: [INAUDIBLE]
160
00:09:18,060 --> 00:09:22,510
PROFESSOR: It's
one big equation,
161
00:09:22,510 --> 00:09:24,920
but it's a polynomial equation.
162
00:09:24,920 --> 00:09:32,530
So there's an equation, there's
this function R(x) = a_11 x^11
163
00:09:32,530 --> 00:09:37,610
+ a_10 x^10...
164
00:09:37,610 --> 00:09:41,350
and these things are known.
165
00:09:41,350 --> 00:09:43,495
This is a known expression here.
166
00:09:43,495 --> 00:09:46,620
And then when you cross-multiply
on the other side,
167
00:09:46,620 --> 00:09:51,330
what you have is,
well, it's A_1 times--
168
00:09:51,330 --> 00:09:54,400
If you cancel this
denominator with that,
169
00:09:54,400 --> 00:10:05,330
you're going to get (x + (x+2)^3
(x^2+2x+3) (x^2+4)^3 plus
170
00:10:05,330 --> 00:10:08,260
the term for A_2, etc.
171
00:10:08,260 --> 00:10:10,350
It's a monster equation.
172
00:10:10,350 --> 00:10:12,620
And then to separate it out
into separate equations,
173
00:10:12,620 --> 00:10:19,225
you take the coefficient
on x^11, x^10, ...
174
00:10:19,225 --> 00:10:21,670
all the way down to x^0.
175
00:10:21,670 --> 00:10:27,280
And all told, that means there
are a total of 12 equations.
176
00:10:27,280 --> 00:10:31,030
11 through 0 is 12 equations.
177
00:10:31,030 --> 00:10:34,727
Yeah, another question.
178
00:10:34,727 --> 00:10:35,560
STUDENT: [INAUDIBLE]
179
00:10:35,560 --> 00:10:37,777
PROFESSOR: Should I
write down rest of this?
180
00:10:37,777 --> 00:10:38,610
STUDENT: [INAUDIBLE]
181
00:10:38,610 --> 00:10:40,820
PROFESSOR: Should you
write down all this stuff?
182
00:10:40,820 --> 00:10:43,950
Well, that's a good question.
183
00:10:43,950 --> 00:10:46,070
So you notice I
didn't write it down.
184
00:10:46,070 --> 00:10:47,300
Why didn't I write it down?
185
00:10:47,300 --> 00:10:50,200
Because it's incredibly long.
186
00:10:50,200 --> 00:10:54,104
In fact, you probably-- So
how many pages of writing
187
00:10:54,104 --> 00:10:54,770
would this take?
188
00:10:54,770 --> 00:10:56,210
This is about a page of writing.
189
00:10:56,210 --> 00:10:58,950
So just think of you're
a machine, how much time
190
00:10:58,950 --> 00:11:01,660
you want to spend on this.
191
00:11:01,660 --> 00:11:05,430
So the answer is that
you have to be realistic.
192
00:11:05,430 --> 00:11:07,420
You're a human
being, not a machine.
193
00:11:07,420 --> 00:11:10,070
And so there's certain things
that you can write down
194
00:11:10,070 --> 00:11:12,510
and other things you should
not attempt to write down.
195
00:11:12,510 --> 00:11:17,770
So do not do this at home.
196
00:11:17,770 --> 00:11:21,180
So that's the first
down-side to this method.
197
00:11:21,180 --> 00:11:24,350
It gets more and more
complicated as time goes on.
198
00:11:24,350 --> 00:11:27,280
The second down-side, I
want to point out to you,
199
00:11:27,280 --> 00:11:35,100
is what's happening
with the pieces.
200
00:11:35,100 --> 00:11:42,830
So the pieces still
need to be integrated.
201
00:11:42,830 --> 00:11:48,130
We simplified this problem,
but we didn't get rid of it.
202
00:11:48,130 --> 00:11:50,890
We still have the problem
of integrating the pieces.
203
00:11:50,890 --> 00:11:52,740
Now, some of the
pieces are very easy.
204
00:11:52,740 --> 00:11:55,540
This top row here, the
antiderivatives of these,
205
00:11:55,540 --> 00:11:59,150
you can just write down.
206
00:11:59,150 --> 00:12:01,390
By advanced guessing.
207
00:12:01,390 --> 00:12:04,300
I'm going to skip over to the
most complicated one over here.
208
00:12:04,300 --> 00:12:06,220
For one second here.
209
00:12:06,220 --> 00:12:09,240
And what is it that you'd have
to deal with for that one.
210
00:12:09,240 --> 00:12:11,810
You'd have to deal
with, for example,
211
00:12:11,810 --> 00:12:21,660
so e.g., for example, I
need to deal with this guy.
212
00:12:21,660 --> 00:12:26,590
I've got to get this
antiderivative here.
213
00:12:26,590 --> 00:12:28,972
Now, this one you're
supposed to be able to know.
214
00:12:28,972 --> 00:12:30,430
So this is why I'm
mentioning this.
215
00:12:30,430 --> 00:12:33,270
Because this kind of
ingredient is something
216
00:12:33,270 --> 00:12:34,860
you already covered.
217
00:12:34,860 --> 00:12:35,740
And what is it?
218
00:12:35,740 --> 00:12:39,060
Well, you do this one
by advanced guessing,
219
00:12:39,060 --> 00:12:42,000
although you learned it as
the method of substitution.
220
00:12:42,000 --> 00:12:47,900
You realize that it's going to
be of the form (x^2 + 4)^(-2),
221
00:12:47,900 --> 00:12:49,400
roughly speaking.
222
00:12:49,400 --> 00:12:51,170
And now we're going to fix that.
223
00:12:51,170 --> 00:12:53,950
Because if you differentiate
it you get 2x times the -2,
224
00:12:53,950 --> 00:12:56,410
that's -4 times x times this.
225
00:12:56,410 --> 00:12:58,370
There's an x in
the numerator here.
226
00:12:58,370 --> 00:13:02,600
So it's -1/4 of that
will fix the factor.
227
00:13:02,600 --> 00:13:06,550
And here's the
answer for that one.
228
00:13:06,550 --> 00:13:10,560
So that's one you can do.
229
00:13:10,560 --> 00:13:19,020
The second piece is this guy.
230
00:13:19,020 --> 00:13:20,480
This is the other piece.
231
00:13:20,480 --> 00:13:25,680
Now, this was the piece
that came from B_3.
232
00:13:25,680 --> 00:13:27,170
This is the one
that came from B_3.
233
00:13:27,170 --> 00:13:30,430
And this is the one
that's coming from C_3.
234
00:13:30,430 --> 00:13:32,360
This is coming from C_3.
235
00:13:32,360 --> 00:13:35,000
We still need to get
this one out there.
236
00:13:35,000 --> 00:13:37,630
So C_3 times that will
be the correct answer,
237
00:13:37,630 --> 00:13:40,960
once we've found these numbers.
238
00:13:40,960 --> 00:13:44,160
So how do we do this?
239
00:13:44,160 --> 00:13:45,490
How's this one integrated?
240
00:13:45,490 --> 00:13:49,690
STUDENT: Trig substitution?
241
00:13:49,690 --> 00:13:51,640
PROFESSOR: Trig substitution.
242
00:13:51,640 --> 00:13:57,630
So the trig substitution
here is x = 2 tan u.
243
00:13:57,630 --> 00:14:00,800
Or 2 tan theta.
244
00:14:00,800 --> 00:14:03,860
And when you do that, there are
a couple of simplifications.
245
00:14:03,860 --> 00:14:06,450
Well, I wouldn't call
this a simplification.
246
00:14:06,450 --> 00:14:14,830
This is just the differentiation
formula. dx = 2 sec^2 u du.
247
00:14:14,830 --> 00:14:19,030
And then you have to plug in,
and you're using the fact that
248
00:14:19,030 --> 00:14:22,960
when you plug in the
tan^2, 4 tan ^2 + 4,
249
00:14:22,960 --> 00:14:24,370
you'll get a secant squared.
250
00:14:24,370 --> 00:14:32,240
So altogether, this
thing is, 2 sec^2 u du.
251
00:14:32,240 --> 00:14:40,300
And then there's a (4 sec^2
u)^3, in the denominator.
252
00:14:40,300 --> 00:14:44,340
So that's what happens when
you change variables here.
253
00:14:44,340 --> 00:14:46,790
And now look, this
keeps on going.
254
00:14:46,790 --> 00:14:49,120
This is not the
end of the problem.
255
00:14:49,120 --> 00:14:50,790
Because what does
that simplify to?
256
00:14:50,790 --> 00:14:57,860
That is, let's see, it's
2/64, the integral of sec^6
257
00:14:57,860 --> 00:14:58,620
and sec^2.
258
00:14:58,620 --> 00:15:00,090
That's the same as cos^4.
259
00:15:04,300 --> 00:15:06,370
And now, you did a
trig substitution
260
00:15:06,370 --> 00:15:11,140
but you still have
a trig integral.
261
00:15:11,140 --> 00:15:15,540
The trig integral now,
there's a method for this.
262
00:15:15,540 --> 00:15:18,730
The method for this is
when it's an even power,
263
00:15:18,730 --> 00:15:22,280
you have to use the
double angle formula.
264
00:15:22,280 --> 00:15:31,890
So that's this guy here.
265
00:15:31,890 --> 00:15:33,470
And you're still not done.
266
00:15:33,470 --> 00:15:35,040
You have to square
this thing out.
267
00:15:35,040 --> 00:15:37,120
And then you'll still
get a cos^2 (2u).
268
00:15:37,120 --> 00:15:38,160
And it keeps on going.
269
00:15:38,160 --> 00:15:41,737
So this thing goes
on for a long time.
270
00:15:41,737 --> 00:15:43,320
But I'm not even
going to finish this,
271
00:15:43,320 --> 00:15:44,780
but I just want to show you.
272
00:15:44,780 --> 00:15:46,450
The point is, we're
not showing you how
273
00:15:46,450 --> 00:15:48,270
to do any complicated problem.
274
00:15:48,270 --> 00:15:50,550
We're just showing you all
the little ingredients.
275
00:15:50,550 --> 00:15:52,050
And you have to
string them together
276
00:15:52,050 --> 00:15:56,170
a long, long, long process to
get to the final answer of one
277
00:15:56,170 --> 00:15:57,680
of these questions.
278
00:15:57,680 --> 00:16:07,300
So it always works,
but maybe slowly.
279
00:16:07,300 --> 00:16:13,440
By the way, there's even another
horrible thing that happens.
280
00:16:13,440 --> 00:16:22,669
Which is, if you handle this
guy here, what's the technique.
281
00:16:22,669 --> 00:16:24,460
This is another technique
that you learned,
282
00:16:24,460 --> 00:16:28,770
supposedly within
the last few days.
283
00:16:28,770 --> 00:16:30,810
Completing the square.
284
00:16:30,810 --> 00:16:39,020
So this, it turns out, you
have to rewrite it this way.
285
00:16:39,020 --> 00:16:42,530
And then the evaluation is going
to be expressed in terms of,
286
00:16:42,530 --> 00:16:44,440
I'm going to jump to the end.
287
00:16:44,440 --> 00:16:49,310
It's going to turn out to be
expressed in terms of this.
288
00:16:49,310 --> 00:16:53,940
That's what will eventually
show up in the formula.
289
00:16:53,940 --> 00:16:56,370
And not only that,
but if you deal
290
00:16:56,370 --> 00:16:59,890
with ones involving
x as well, you'll
291
00:16:59,890 --> 00:17:07,420
also need to deal with something
like log of this denominator
292
00:17:07,420 --> 00:17:09,560
here.
293
00:17:09,560 --> 00:17:13,120
So all of these things
will be involved.
294
00:17:13,120 --> 00:17:16,700
So now, the last message that
I have for you is just this.
295
00:17:16,700 --> 00:17:18,150
This thing is very complicated.
296
00:17:18,150 --> 00:17:20,150
We're certainly never
going to ask you to do it.
297
00:17:20,150 --> 00:17:23,160
But you should just be aware
that this level of complexity,
298
00:17:23,160 --> 00:17:26,690
we are absolutely stuck
with in this problem.
299
00:17:26,690 --> 00:17:29,580
And the reason why
we're stuck with it
300
00:17:29,580 --> 00:17:36,160
is that this is what the
formulas look like in the end.
301
00:17:36,160 --> 00:17:39,130
If the answers look
like this, the formulas
302
00:17:39,130 --> 00:17:41,045
have to be this complicated.
303
00:17:41,045 --> 00:17:43,170
If you differentiate this,
you get your polynomial,
304
00:17:43,170 --> 00:17:44,254
your ratio of polynomials.
305
00:17:44,254 --> 00:17:46,794
If you differentiate this, you
get some ratio of polynomials.
306
00:17:46,794 --> 00:17:48,480
These are the
things that come up
307
00:17:48,480 --> 00:17:51,710
when you take antiderivatives
of those rational functions.
308
00:17:51,710 --> 00:17:56,080
So we're just stuck
with these guys.
309
00:17:56,080 --> 00:17:58,770
And so don't let it
get to you too much.
310
00:17:58,770 --> 00:17:59,770
I mean, it's not so bad.
311
00:17:59,770 --> 00:18:01,510
In fact, there are
computer programs
312
00:18:01,510 --> 00:18:03,510
that will do this for
you anytime you want.
313
00:18:03,510 --> 00:18:05,800
And you just have to be
not intimidated by them.
314
00:18:05,800 --> 00:18:10,260
They're like other functions.
315
00:18:10,260 --> 00:18:20,600
OK, that's it for the general
comments on partial fractions.
316
00:18:20,600 --> 00:18:24,215
Now we're going to change
subjects to our last technique.
317
00:18:24,215 --> 00:18:25,840
This is one more
technical thing to get
318
00:18:25,840 --> 00:18:27,540
you familiar with functions.
319
00:18:27,540 --> 00:18:32,260
And this technique is
called integration by parts.
320
00:18:32,260 --> 00:18:34,580
Please, just because
its name sort
321
00:18:34,580 --> 00:18:35,957
of sounds like
partial fractions,
322
00:18:35,957 --> 00:18:37,290
don't think it's the same thing.
323
00:18:37,290 --> 00:18:38,450
It's totally different.
324
00:18:38,450 --> 00:18:44,340
It's not the same.
325
00:18:44,340 --> 00:19:06,640
So this one is called
integration by parts.
326
00:19:06,640 --> 00:19:09,570
Now, unlike the previous case,
where I couldn't actually
327
00:19:09,570 --> 00:19:12,736
justify to you that the
linear algebra always works.
328
00:19:12,736 --> 00:19:14,860
I claimed it worked, but
I wasn't able to prove it.
329
00:19:14,860 --> 00:19:17,390
That's a complicated
theorem which I'm not
330
00:19:17,390 --> 00:19:19,560
able to do in this class.
331
00:19:19,560 --> 00:19:22,270
Here I can explain to
you what's going on
332
00:19:22,270 --> 00:19:24,200
with integration by parts.
333
00:19:24,200 --> 00:19:26,600
It's just the fundamental
theorem of calculus,
334
00:19:26,600 --> 00:19:30,430
if you like, coupled
with the product formula.
335
00:19:30,430 --> 00:19:33,740
Sort of unwound and
read in reverse.
336
00:19:33,740 --> 00:19:35,610
And here's how that works.
337
00:19:35,610 --> 00:19:38,480
If you take the product of two
functions and you differentiate
338
00:19:38,480 --> 00:19:41,910
them, then we know that the
product rule says that this is
339
00:19:41,910 --> 00:19:45,790
u'v + uv'.
340
00:19:45,790 --> 00:19:50,400
And now I'm just going to
rearrange in the following way.
341
00:19:50,400 --> 00:19:53,370
I'm going to solve for uv'.
342
00:19:53,370 --> 00:19:54,710
That is, this term here.
343
00:19:54,710 --> 00:19:56,370
So what is this term?
344
00:19:56,370 --> 00:19:59,990
It's this other term, (uv)'.
345
00:19:59,990 --> 00:20:04,520
Minus the other piece.
346
00:20:04,520 --> 00:20:08,360
So I just rewrote this equation.
347
00:20:08,360 --> 00:20:10,900
And now I'm going
to integrate it.
348
00:20:10,900 --> 00:20:11,860
So here's the formula.
349
00:20:11,860 --> 00:20:15,160
The integral of
the left-hand side
350
00:20:15,160 --> 00:20:17,200
is equal to the integral
of the right-hand side.
351
00:20:17,200 --> 00:20:18,670
Well when I integrate
a derivative,
352
00:20:18,670 --> 00:20:21,070
of I get back the
function itself.
353
00:20:21,070 --> 00:20:27,010
That's the fundamental theorem.
354
00:20:27,010 --> 00:20:27,600
So this is it.
355
00:20:27,600 --> 00:20:30,500
Sorry, I missed the
dx, which is important.
356
00:20:30,500 --> 00:20:32,460
I apologize.
357
00:20:32,460 --> 00:20:35,410
Let's put that in there.
358
00:20:35,410 --> 00:20:41,540
So this is the integration
by parts formula.
359
00:20:41,540 --> 00:20:46,760
I'm going to write it one more
time with the limits stuck in.
360
00:20:46,760 --> 00:21:02,170
It's also written this way, when
you have a definite integral.
361
00:21:02,170 --> 00:21:13,260
Just the same formula,
written twice.
362
00:21:13,260 --> 00:21:14,910
Alright, now I'm
going to show you
363
00:21:14,910 --> 00:21:24,360
how it works on a few examples.
364
00:21:24,360 --> 00:21:29,470
And I have to give you a
flavor for how it works.
365
00:21:29,470 --> 00:21:34,230
But it'll grow as we get
more and more experience.
366
00:21:34,230 --> 00:21:38,360
The first example
that I'm going to take
367
00:21:38,360 --> 00:21:43,040
is one that looks intractable
on the face of it.
368
00:21:43,040 --> 00:21:49,740
Which is the
integral of ln x dx.
369
00:21:49,740 --> 00:21:52,560
Now, it looks like there's sort
of nothing we can do with this.
370
00:21:52,560 --> 00:21:55,310
And we don't know
what the solution is.
371
00:21:55,310 --> 00:21:59,480
However, I claim that if
we fit it into this form,
372
00:21:59,480 --> 00:22:03,100
we can figure out what the
integral is relatively easily.
373
00:22:03,100 --> 00:22:07,160
By some little magic of
cancellation, it happens.
374
00:22:07,160 --> 00:22:08,960
The idea is the following.
375
00:22:08,960 --> 00:22:13,130
If I consider this
function to be u,
376
00:22:13,130 --> 00:22:15,680
then what's going to
appear on the other side
377
00:22:15,680 --> 00:22:19,540
in the integrated form
is the function u', which
378
00:22:19,540 --> 00:22:22,680
is-- so, if you like, u = ln x.
379
00:22:22,680 --> 00:22:25,620
So u' = 1 / x.
380
00:22:25,620 --> 00:22:28,680
Now, 1 / x is a more
manageable function than ln x.
381
00:22:28,680 --> 00:22:31,370
What we're using is that when
we differentiate the function,
382
00:22:31,370 --> 00:22:33,100
it's getting nicer.
383
00:22:33,100 --> 00:22:36,830
It's getting more
tractable for us.
384
00:22:36,830 --> 00:22:38,860
In order for this to
fit into this pattern,
385
00:22:38,860 --> 00:22:45,410
however, I need a function
v. So what in the world
386
00:22:45,410 --> 00:22:48,220
am I going to put here for v?
387
00:22:48,220 --> 00:22:51,920
The answer is, well, dx is
almost the right answer.
388
00:22:51,920 --> 00:22:53,860
The answer turns out to be x.
389
00:22:53,860 --> 00:23:01,260
And the reason is that
that makes v' = 1.
390
00:23:01,260 --> 00:23:02,630
It makes v' = 1.
391
00:23:02,630 --> 00:23:05,560
So that means that this
is u, but it's also uv'.
392
00:23:05,560 --> 00:23:11,240
Which was what I had
on the left-hand side.
393
00:23:11,240 --> 00:23:13,470
So it's both u and uv'.
394
00:23:13,470 --> 00:23:14,710
So this is the setup.
395
00:23:14,710 --> 00:23:19,300
And now all I'm going to do is
read off what the formula says.
396
00:23:19,300 --> 00:23:24,190
What it says is, this is
equal to u times v. So u
397
00:23:24,190 --> 00:23:25,270
is this and v is that.
398
00:23:25,270 --> 00:23:32,440
So it's x ln x minus, so
that again, this is uv.
399
00:23:32,440 --> 00:23:37,170
Except in the other order, vu.
400
00:23:37,170 --> 00:23:40,510
And then I'm integrating, and
what do I have to integrate?
401
00:23:40,510 --> 00:23:42,540
u'v. So look up there.
402
00:23:42,540 --> 00:23:47,760
u'v with a minus sign here.
u' = 1 / x, and v = x.
403
00:23:47,760 --> 00:23:50,890
So it's 1 / x, that's u'.
404
00:23:50,890 --> 00:23:56,070
And here is x, that's v, dx.
405
00:23:56,070 --> 00:23:58,270
Now, that one is
easy to integrate.
406
00:23:58,270 --> 00:24:00,870
Because (1/x) x = 1.
407
00:24:00,870 --> 00:24:07,630
And the integral of 1 dx
is x, plus c, if you like.
408
00:24:07,630 --> 00:24:10,510
So the antiderivative of 1 is x.
409
00:24:10,510 --> 00:24:11,610
And so here's our answer.
410
00:24:11,610 --> 00:24:34,480
Our answer is that
this is x ln x - x + c.
411
00:24:34,480 --> 00:24:37,610
I'm going to do two
more slightly more
412
00:24:37,610 --> 00:24:39,380
complicated examples.
413
00:24:39,380 --> 00:24:42,590
And then really,
the main thing is
414
00:24:42,590 --> 00:24:44,580
to get yourself
used to this method.
415
00:24:44,580 --> 00:24:47,930
And there's no one
way of doing that.
416
00:24:47,930 --> 00:24:49,970
Just practice makes perfect.
417
00:24:49,970 --> 00:24:53,070
And so we'll just do
a few more examples.
418
00:24:53,070 --> 00:24:55,870
And illustrate them.
419
00:24:55,870 --> 00:24:59,960
The second example that I'm
going to use is the integral
420
00:24:59,960 --> 00:25:03,590
of (ln x)^2 dx.
421
00:25:03,590 --> 00:25:08,510
And this is just slightly
more recalcitrant.
422
00:25:08,510 --> 00:25:13,270
Namely, I'm going to
let u be (ln x)^2.
423
00:25:17,740 --> 00:25:20,440
And again, v = x.
424
00:25:20,440 --> 00:25:21,890
So that matches up here.
425
00:25:21,890 --> 00:25:23,730
That is, v' = 1.
426
00:25:23,730 --> 00:25:28,390
So this is uv'.
427
00:25:28,390 --> 00:25:31,440
So this thing is uv'.
428
00:25:31,440 --> 00:25:33,510
And then we'll just
see what happens.
429
00:25:33,510 --> 00:25:38,060
Now, the game that we get
is that when I differentiate
430
00:25:38,060 --> 00:25:42,870
the logarithm squared, I'm going
to to get something simpler.
431
00:25:42,870 --> 00:25:46,850
It's not going to win
us the whole battle,
432
00:25:46,850 --> 00:25:49,860
but it will get us started.
433
00:25:49,860 --> 00:25:51,770
So here we get u'.
434
00:25:51,770 --> 00:25:56,650
And that's 2 ln x times 1/x.
435
00:25:56,650 --> 00:26:00,470
Applying the chain rule.
436
00:26:00,470 --> 00:26:06,020
And so the formula is
that this is x (ln x)^2,
437
00:26:06,020 --> 00:26:11,710
minus the integral of,
well it's u'v, right,
438
00:26:11,710 --> 00:26:13,210
that's what I have
to put over here.
439
00:26:13,210 --> 00:26:22,190
So u' = 2 ln x 1/x and v = x.
440
00:26:22,190 --> 00:26:25,880
And so now, you notice something
interesting happening here.
441
00:26:25,880 --> 00:26:28,960
So let me just demarcate
this a little bit.
442
00:26:28,960 --> 00:26:34,680
And let you see what it
is that I'm doing here.
443
00:26:34,680 --> 00:26:36,820
So notice, this is
the same integral.
444
00:26:36,820 --> 00:26:38,650
So here we have x (ln x)^2.
445
00:26:38,650 --> 00:26:41,050
We've already solved that part.
446
00:26:41,050 --> 00:26:43,870
But now know notice that
the 1/x and the x cancel.
447
00:26:43,870 --> 00:26:46,890
So we're back to
the previous case.
448
00:26:46,890 --> 00:26:49,570
We didn't win all the way, but
actually we reduced ourselves
449
00:26:49,570 --> 00:26:51,350
to this integral.
450
00:26:51,350 --> 00:26:56,630
To the integral of ln x,
which we already know.
451
00:26:56,630 --> 00:26:58,700
So here, I can copy that down.
452
00:26:58,700 --> 00:27:04,450
That's - -2(x ln x - x),
and then I have to throw
453
00:27:04,450 --> 00:27:05,460
in a constant, c.
454
00:27:05,460 --> 00:27:07,270
And that's the end
of the problem here.
455
00:27:07,270 --> 00:27:10,260
That's it.
456
00:27:10,260 --> 00:27:26,400
So this piece, I
got from Example 1.
457
00:27:26,400 --> 00:27:34,350
Now, this illustrates
a principle
458
00:27:34,350 --> 00:27:36,200
which is a little
bit more complicated
459
00:27:36,200 --> 00:27:40,390
than just the one of
integration by parts.
460
00:27:40,390 --> 00:27:43,630
Which is a sort of a
general principle which
461
00:27:43,630 --> 00:27:48,150
I'll call my Example 3,
which is something which
462
00:27:48,150 --> 00:27:56,520
is called a reduction formula.
463
00:27:56,520 --> 00:28:00,540
A reduction formula is a
case where we apply some rule
464
00:28:00,540 --> 00:28:03,220
and we figure out one
of these integrals
465
00:28:03,220 --> 00:28:05,440
in terms of something else.
466
00:28:05,440 --> 00:28:07,320
Which is a little bit simpler.
467
00:28:07,320 --> 00:28:09,070
And eventually we'll
get down to the end,
468
00:28:09,070 --> 00:28:12,810
but it may take us n
steps from the beginning.
469
00:28:12,810 --> 00:28:17,541
So the example is (ln x)^n dx.
470
00:28:17,541 --> 00:28:18,040
.
471
00:28:18,040 --> 00:28:21,710
And the claim is that if I
do what I did in Example 2,
472
00:28:21,710 --> 00:28:26,500
to this case, I'll get a
simpler one which will involve
473
00:28:26,500 --> 00:28:28,224
the (n-1)st power.
474
00:28:28,224 --> 00:28:29,640
And that way I can
get all the way
475
00:28:29,640 --> 00:28:32,440
back down to the final answer.
476
00:28:32,440 --> 00:28:34,160
So here's what happens.
477
00:28:34,160 --> 00:28:37,090
We take u as (ln x)^n.
478
00:28:37,090 --> 00:28:40,980
This is the same discussion
as before, v = x.
479
00:28:40,980 --> 00:28:44,850
And then u' is n n
(ln x)^(n-1) 1/x.
480
00:28:47,440 --> 00:28:50,020
And v' is 1.
481
00:28:50,020 --> 00:28:52,800
And so the setup is similar.
482
00:28:52,800 --> 00:28:59,320
We have here x (ln x)^n
minus the integral.
483
00:28:59,320 --> 00:29:05,020
And there's n times, it
turns out to be (ln x)^(n-1).
484
00:29:05,020 --> 00:29:26,410
And then there's a 1/x
and an x, which cancel.
485
00:29:26,410 --> 00:29:31,010
So I'm going to explain
this also abstractly
486
00:29:31,010 --> 00:29:35,450
a little bit just to show
you what's happening here.
487
00:29:35,450 --> 00:29:44,360
If you use the notation F_n(x)
is the integral of (ln x)^n dx,
488
00:29:44,360 --> 00:29:46,780
and we're going to
forget the constant here.
489
00:29:46,780 --> 00:29:51,710
Then the relationship that we
have here is that F_n(x) is
490
00:29:51,710 --> 00:29:56,590
equal to n ln-- I'm
sorry, x (ln x)^n.
491
00:29:56,590 --> 00:29:59,490
That's the first term over here.
492
00:29:59,490 --> 00:30:03,650
Minus n times the preceding one.
493
00:30:03,650 --> 00:30:07,670
This one here.
494
00:30:07,670 --> 00:30:11,060
And the idea is that
eventually we can get down.
495
00:30:11,060 --> 00:30:15,010
If we start with the nth one, we
have a formula that includes--
496
00:30:15,010 --> 00:30:17,440
So the reduction is
to the n (n-1)st.
497
00:30:17,440 --> 00:30:21,280
Then we can reduce to
the (n-2)nd and so on.
498
00:30:21,280 --> 00:30:23,610
Until we reduce to
the 1, the first one.
499
00:30:23,610 --> 00:30:29,390
And then in fact we can
even go down to the 0th one.
500
00:30:29,390 --> 00:30:32,510
So this is the idea of
a reduction formula.
501
00:30:32,510 --> 00:30:37,320
And let me illustrate it exactly
in the context of Examples 1
502
00:30:37,320 --> 00:30:38,870
and 2.
503
00:30:38,870 --> 00:30:44,670
So the first step would be
to evaluate the first one.
504
00:30:44,670 --> 00:30:48,190
Which is, if you
like, (ln x)^0 dx.
505
00:30:48,190 --> 00:30:52,370
That's very easy, that's x.
506
00:30:52,370 --> 00:31:01,000
And then F_1(x) =
x ln x - F_0(x).
507
00:31:01,000 --> 00:31:03,240
Now, that's applying this rule.
508
00:31:03,240 --> 00:31:06,830
So let me just put
it in a box here.
509
00:31:06,830 --> 00:31:09,380
This is the method of induction.
510
00:31:09,380 --> 00:31:13,510
Here's the rule.
511
00:31:13,510 --> 00:31:21,930
And I'm applying it for n = 1.
512
00:31:21,930 --> 00:31:23,810
I plugged in n = 1 here.
513
00:31:23,810 --> 00:31:26,720
So here, I have x
(ln x)^1 - 1*F_0(x).
514
00:31:32,430 --> 00:31:39,240
And that's what I put right
here, on the right-hand side.
515
00:31:39,240 --> 00:31:42,440
And that's going to generate
for me the formula that I want,
516
00:31:42,440 --> 00:31:44,920
which is x ln x - x.
517
00:31:44,920 --> 00:31:49,160
That's the answer to
this problem over here.
518
00:31:49,160 --> 00:31:51,230
This was Example 1.
519
00:31:51,230 --> 00:31:52,980
Notice I dropped the
constants because I
520
00:31:52,980 --> 00:31:54,880
can add them in at the end.
521
00:31:54,880 --> 00:31:57,590
So I'll put in
parentheses here, plus c.
522
00:31:57,590 --> 00:32:01,850
That's what would happen
at the end of the problem.
523
00:32:01,850 --> 00:32:10,320
The next step, so that was
Example 1, and now Example 2
524
00:32:10,320 --> 00:32:12,190
works more or less the same way.
525
00:32:12,190 --> 00:32:14,590
I'm just summarizing what
I did on that blackboard
526
00:32:14,590 --> 00:32:16,640
right up here.
527
00:32:16,640 --> 00:32:21,030
The same thing, but in
much more compact notation.
528
00:32:21,030 --> 00:32:29,950
If I take F_2(x), that's going
to be equal to x (ln x)^2 -
529
00:32:29,950 --> 00:32:31,820
2 F_1(x).
530
00:32:31,820 --> 00:32:41,550
Again, this is box for n = 2.
531
00:32:41,550 --> 00:32:46,730
And if I plug it in, what I'm
getting here is x (ln x)^2
532
00:32:46,730 --> 00:32:49,570
minus twice this stuff here.
533
00:32:49,570 --> 00:32:55,780
Which is right here. x ln x - x.
534
00:32:55,780 --> 00:32:58,580
If you like, plus c.
535
00:32:58,580 --> 00:33:07,360
So I'll leave the c off.
536
00:33:07,360 --> 00:33:12,170
So this is how reduction
formulas work in general.
537
00:33:12,170 --> 00:33:22,620
I'm going to give you one more
example of a reduction formula.
538
00:33:22,620 --> 00:33:30,560
So I guess we have to
call this Example 4.
539
00:33:30,560 --> 00:33:34,050
Let's be fancy, let's
make it the sine.
540
00:33:34,050 --> 00:33:35,950
No no, no, let's
be fancier still.
541
00:33:35,950 --> 00:33:48,790
Let's make it e^x So this would
also work for cos x and sin x.
542
00:33:48,790 --> 00:33:50,110
The same sort of thing.
543
00:33:50,110 --> 00:33:52,840
And I should mention
that on your homework,
544
00:33:52,840 --> 00:33:54,300
you have to do it for cos x.
545
00:33:54,300 --> 00:33:56,550
I decided to change my mind
on the spur of the moment.
546
00:33:56,550 --> 00:33:57,924
I'm not going to
do it for cosine
547
00:33:57,924 --> 00:34:00,530
because you have to work it out
on your homework for cosine.
548
00:34:00,530 --> 00:34:03,100
In a later homework
you'll even do this case.
549
00:34:03,100 --> 00:34:05,190
So it's fine.
550
00:34:05,190 --> 00:34:07,400
You need the practice.
551
00:34:07,400 --> 00:34:10,240
OK, so how am I going
to do it this time.
552
00:34:10,240 --> 00:34:13,970
This is again, a
reduction formula.
553
00:34:13,970 --> 00:34:19,420
And the trick here is to pick
u to be this function here.
554
00:34:19,420 --> 00:34:20,780
And the reason is the following.
555
00:34:20,780 --> 00:34:23,029
So it's very important to
pick which function is the u
556
00:34:23,029 --> 00:34:26,450
and which function is the v.
That's the only decision you
557
00:34:26,450 --> 00:34:30,020
have to make if you're going
to apply integration by parts.
558
00:34:30,020 --> 00:34:34,420
When I pick this function as
the u, the advantage that I have
559
00:34:34,420 --> 00:34:38,150
is that u' is simpler.
560
00:34:38,150 --> 00:34:39,630
How is it simpler?
561
00:34:39,630 --> 00:34:42,820
It's simpler because
it's one degree down.
562
00:34:42,820 --> 00:34:45,420
So that's making
progress for us.
563
00:34:45,420 --> 00:34:48,820
On the other hand,
this function here
564
00:34:48,820 --> 00:34:52,530
is going to be what
I'll use for v.
565
00:34:52,530 --> 00:34:55,279
And if I differentiated that,
if I did it the other way around
566
00:34:55,279 --> 00:34:57,070
and I differentiated
that, I would just get
567
00:34:57,070 --> 00:34:58,900
the same level of complexity.
568
00:34:58,900 --> 00:35:01,120
Differentiating e^x
just gives you back e^x.
569
00:35:01,120 --> 00:35:02,000
So that's boring.
570
00:35:02,000 --> 00:35:05,750
It doesn't make any
progress in this process.
571
00:35:05,750 --> 00:35:11,460
And so I'm going to instead
let v = e^x and-- Sorry,
572
00:35:11,460 --> 00:35:12,830
this is v'.
573
00:35:12,830 --> 00:35:14,380
Make it v' = e^x.
574
00:35:14,380 --> 00:35:15,950
And then v = e^x.
575
00:35:15,950 --> 00:35:20,640
At least it isn't any worse
when I went backwards like that.
576
00:35:20,640 --> 00:35:28,150
So now, I have u and v',
and now I get x^n e^x.
577
00:35:28,150 --> 00:35:31,490
This again is u, and this is v.
So it happens that v is equal
578
00:35:31,490 --> 00:35:34,200
to v ' so it's a
little confusing here.
579
00:35:34,200 --> 00:35:37,640
But this is the one
we're calling v'.
580
00:35:37,640 --> 00:35:41,510
And here's v. And now minus
the integral and I have here
581
00:35:41,510 --> 00:35:43,760
nx^(n-1).
582
00:35:43,760 --> 00:35:45,120
And I have here e^x.
583
00:35:45,120 --> 00:35:52,060
So this is u' and this is v dx.
584
00:35:52,060 --> 00:35:55,180
So this recurrence
is a new recurrence.
585
00:35:55,180 --> 00:35:57,050
And let me summarize it here.
586
00:35:57,050 --> 00:36:02,270
It's saying that G_n(x) should
be the integral of x^n e^x dx.
587
00:36:05,210 --> 00:36:06,810
Again, I'm dropping the c.
588
00:36:06,810 --> 00:36:17,060
And then the reduction formula
is that G_n(x) is equal to this
589
00:36:17,060 --> 00:36:25,300
expression here: x^n
e^x - n*G_(n-1)(x).
590
00:36:25,300 --> 00:36:32,830
So here's our reduction formula.
591
00:36:32,830 --> 00:36:37,912
And to illustrate
this, if I take G_0(x),
592
00:36:37,912 --> 00:36:39,870
if you think about it
for a second that's just,
593
00:36:39,870 --> 00:36:40,744
there's nothing here.
594
00:36:40,744 --> 00:36:44,680
The antiderivative of e^x,
that's going to be e^x,
595
00:36:44,680 --> 00:36:48,220
that's getting started at
the real basement here.
596
00:36:48,220 --> 00:36:52,000
Again, as always, 0
is my favorite number.
597
00:36:52,000 --> 00:36:52,820
Not 1.
598
00:36:52,820 --> 00:36:55,850
I always start with the
easiest one, if possible.
599
00:36:55,850 --> 00:37:00,150
And now G_1, applying
this formula,
600
00:37:00,150 --> 00:37:06,830
is going to be equal
to x e^x - G_0(x).
601
00:37:06,830 --> 00:37:11,180
Which is just-- Right,
because n is 1 and n - 1 is 0.
602
00:37:11,180 --> 00:37:13,970
And so that's just
^ x e^x - e^x.
603
00:37:17,220 --> 00:37:19,770
So this is a very,
very fancy way
604
00:37:19,770 --> 00:37:22,620
of saying the following fact.
605
00:37:22,620 --> 00:37:32,210
I'll put it over on
this other board.
606
00:37:32,210 --> 00:37:38,270
Which is that the integral of
x e^x dx is equal to x e^x -
607
00:37:38,270 --> 00:37:44,600
x + c.
608
00:37:44,600 --> 00:37:45,270
Yeah, question.
609
00:37:45,270 --> 00:37:50,950
STUDENT: [INAUDIBLE]
610
00:37:50,950 --> 00:37:53,020
PROFESSOR: The question
is, why is this true.
611
00:37:53,020 --> 00:37:54,830
Why is this statement true.
612
00:37:54,830 --> 00:37:56,420
Why is G_0 equal to e^x.
613
00:37:56,420 --> 00:37:58,410
I did that in my head.
614
00:37:58,410 --> 00:38:02,910
What I did was, I first wrote
down the formula for G_0.
615
00:38:02,910 --> 00:38:08,100
Which was G_0 is equal to
the integral of e^x dx.
616
00:38:11,076 --> 00:38:12,950
Because there's an x to
the 0 power in there,
617
00:38:12,950 --> 00:38:15,010
which is just 1.
618
00:38:15,010 --> 00:38:17,780
And then I know the
antiderivative of e^x.
619
00:38:17,780 --> 00:38:23,230
It's e^x.
620
00:38:23,230 --> 00:38:30,780
STUDENT: [INAUDIBLE]
621
00:38:30,780 --> 00:38:33,030
PROFESSOR: How do you know
when this method will work?
622
00:38:33,030 --> 00:38:37,370
The answer is only
by experience.
623
00:38:37,370 --> 00:38:40,091
You must get
practice doing this.
624
00:38:40,091 --> 00:38:41,590
If you look in your
textbook, you'll
625
00:38:41,590 --> 00:38:44,430
see hints as to what to do.
626
00:38:44,430 --> 00:38:46,050
The other hint
that I want to say
627
00:38:46,050 --> 00:38:48,180
is that if you
find that you have
628
00:38:48,180 --> 00:38:51,000
one factor in your expression
which when you differentiate
629
00:38:51,000 --> 00:38:52,590
it, it gets easier.
630
00:38:52,590 --> 00:38:55,140
And when you antidifferentiate
the other half,
631
00:38:55,140 --> 00:38:57,780
it doesn't get any
worse, then that's
632
00:38:57,780 --> 00:39:01,790
when this method has
a chance of helping.
633
00:39:01,790 --> 00:39:04,430
And there is-- there's
no general thing.
634
00:39:04,430 --> 00:39:09,330
The thing is, though, if you
do it with x^n e^x, x^n cos x,
635
00:39:09,330 --> 00:39:11,970
x^n sin x, those are
examples where it works.
636
00:39:11,970 --> 00:39:15,600
This power of the log.
637
00:39:15,600 --> 00:39:19,150
I'll give you one
more example here.
638
00:39:19,150 --> 00:39:26,519
So this was G_1(x), right.
639
00:39:26,519 --> 00:39:28,310
I'll give you one more
example in a second.
640
00:39:28,310 --> 00:39:29,330
Yeah.
641
00:39:29,330 --> 00:39:33,220
STUDENT: [INAUDIBLE]
642
00:39:33,220 --> 00:39:35,680
PROFESSOR: Thank you.
643
00:39:35,680 --> 00:39:38,490
There's a mistake here.
644
00:39:38,490 --> 00:39:39,240
That's bad.
645
00:39:39,240 --> 00:39:45,910
I was thinking in the back of my
head of the following formula.
646
00:39:45,910 --> 00:39:51,159
Which is another one
which we've just done.
647
00:39:51,159 --> 00:39:53,450
So these are the types of
formulas that you can get out
648
00:39:53,450 --> 00:39:57,620
of integration by parts.
649
00:39:57,620 --> 00:40:00,650
There's also another way of
getting these, which I'm not
650
00:40:00,650 --> 00:40:02,240
going to say anything about.
651
00:40:02,240 --> 00:40:04,282
Which is called
advance guessing.
652
00:40:04,282 --> 00:40:06,740
You guess in advance what the
form is, you differentiate it
653
00:40:06,740 --> 00:40:08,160
and you check.
654
00:40:08,160 --> 00:40:14,250
That does work too, with
many of these cases.
655
00:40:14,250 --> 00:40:21,580
I want to give you
an illustration.
656
00:40:21,580 --> 00:40:30,760
Just because, you know, these
formulas are somewhat dry.
657
00:40:30,760 --> 00:40:34,570
So I want to give you just
at least one application.
658
00:40:34,570 --> 00:40:42,230
We're almost done with the
idea of these formulas.
659
00:40:42,230 --> 00:40:44,880
And we're going to
get back now to being
660
00:40:44,880 --> 00:40:47,990
able to handle lots more
integrals than we could before.
661
00:40:47,990 --> 00:40:49,810
And what's satisfying
is that now we
662
00:40:49,810 --> 00:40:53,830
can get numbers out instead
of being stuck and hamstrung
663
00:40:53,830 --> 00:40:55,120
with only a few techniques.
664
00:40:55,120 --> 00:40:57,680
Now we have all of the
techniques of integration
665
00:40:57,680 --> 00:40:59,250
that anybody has.
666
00:40:59,250 --> 00:41:01,820
And so we can do
pretty much anything
667
00:41:01,820 --> 00:41:04,430
we want that's possible to do.
668
00:41:04,430 --> 00:41:14,250
So here's, if you like,
an application that
669
00:41:14,250 --> 00:41:18,890
illustrates how integration
by parts can be helpful.
670
00:41:18,890 --> 00:41:27,030
And we're going to find the
volume of an exponential wine
671
00:41:27,030 --> 00:41:34,290
glass here.
672
00:41:34,290 --> 00:41:38,350
Again, don't try
this at home, but.
673
00:41:38,350 --> 00:41:40,500
So let's see.
674
00:41:40,500 --> 00:41:44,660
It's going to be this
beautiful guy here.
675
00:41:44,660 --> 00:41:46,930
I think.
676
00:41:46,930 --> 00:41:49,060
OK, so what's it going to be.
677
00:41:49,060 --> 00:41:52,780
This graph is going
to be y = e^x.
678
00:41:52,780 --> 00:42:04,030
Then we're going to rotate
it around the y-axis.
679
00:42:04,030 --> 00:42:10,290
And this level here
is the height y = 1.
680
00:42:10,290 --> 00:42:12,990
And the top, let's
say, is y = e.
681
00:42:12,990 --> 00:42:22,160
So that the horizontal
here, coming down, is x = 1.
682
00:42:22,160 --> 00:42:35,050
Now, there are two ways
to set up this problem.
683
00:42:35,050 --> 00:42:40,050
And so there are two methods.
684
00:42:40,050 --> 00:42:44,110
And this is also a good
review because, of course,
685
00:42:44,110 --> 00:42:46,330
we did this in the last unit.
686
00:42:46,330 --> 00:42:58,480
The two methods are horizontal
and vertical slices.
687
00:42:58,480 --> 00:43:00,660
Those are the two
ways we can do this.
688
00:43:00,660 --> 00:43:03,710
Now, if we do it with--
So let's start out
689
00:43:03,710 --> 00:43:09,370
with the horizontal ones.
690
00:43:09,370 --> 00:43:12,370
That's this shape here.
691
00:43:12,370 --> 00:43:15,370
And we're going like that.
692
00:43:15,370 --> 00:43:19,900
And the horizontal slices
mean that this little bit here
693
00:43:19,900 --> 00:43:22,842
is of thickness dy.
694
00:43:22,842 --> 00:43:24,550
And then we're going
to wrap that around.
695
00:43:24,550 --> 00:43:30,810
So this is going
to become a disk.
696
00:43:30,810 --> 00:43:34,070
This is the method of disks.
697
00:43:34,070 --> 00:43:35,830
And what's this distance here?
698
00:43:35,830 --> 00:43:37,770
Well, this place is x.
699
00:43:37,770 --> 00:43:40,680
And so the disk has area pi x^2.
700
00:43:40,680 --> 00:43:42,840
And we're going to
add up the thickness
701
00:43:42,840 --> 00:43:45,730
of the disks and we're going
to integrate from 1 to e.
702
00:43:45,730 --> 00:43:51,930
So here's our volume.
703
00:43:51,930 --> 00:43:54,510
And now we have one last
little item of business
704
00:43:54,510 --> 00:43:56,230
before we can evaluate
this integral.
705
00:43:56,230 --> 00:43:58,480
And that is that we need to
know the relationship here
706
00:43:58,480 --> 00:44:01,360
on the curve, that y = e^x.
707
00:44:01,360 --> 00:44:07,490
So that means x = ln y.
708
00:44:07,490 --> 00:44:09,180
And in order to
evaluate this integral,
709
00:44:09,180 --> 00:44:13,050
we have to evaluate x
correctly as a function of y.
710
00:44:13,050 --> 00:44:26,200
So that's the integral from 1
to e of (ln y)^2, times pi, dy.
711
00:44:26,200 --> 00:44:27,860
So now you see that
this is an integral
712
00:44:27,860 --> 00:44:30,140
that we did calculate already.
713
00:44:30,140 --> 00:44:34,280
And in fact, it's
sitting right here.
714
00:44:34,280 --> 00:44:37,030
Except with the variable x
instead of the variable y.
715
00:44:37,030 --> 00:44:44,830
So the answer, which we already
had, is this F_2(y) here.
716
00:44:44,830 --> 00:44:47,820
So maybe I'll write it that way.
717
00:44:47,820 --> 00:44:52,010
So this is F_2(y)
between 1 and e.
718
00:44:52,010 --> 00:45:00,040
And now let's figure
out what it is.
719
00:45:00,040 --> 00:45:02,060
It's written over there.
720
00:45:02,060 --> 00:45:15,100
It's y (ln y)^2 - 2(y ln y - y).
721
00:45:15,100 --> 00:45:24,460
The whole thing
evaluated at 1, e.
722
00:45:24,460 --> 00:45:29,130
And that is, if I plug
in e here, I get e.
723
00:45:29,130 --> 00:45:32,150
Except there's a factor
of pi there, sorry.
724
00:45:32,150 --> 00:45:36,360
Missed the pi factor.
725
00:45:36,360 --> 00:45:38,780
So there's an e here.
726
00:45:38,780 --> 00:45:43,050
And then I subtract off,
well, at 1 this is e - e.
727
00:45:43,050 --> 00:45:44,330
So it cancels.
728
00:45:44,330 --> 00:45:45,440
There's nothing left.
729
00:45:45,440 --> 00:45:50,200
And then at 1, I
get ln 1 is 0, ln 1
730
00:45:50,200 --> 00:45:53,500
is 0, there's only one
term left, which is 2.
731
00:45:53,500 --> 00:45:55,720
So it's -2.
732
00:45:55,720 --> 00:46:03,790
That's the answer.
733
00:46:03,790 --> 00:46:10,830
Now we get to compare
that with what happens
734
00:46:10,830 --> 00:46:15,950
if we do it the other way.
735
00:46:15,950 --> 00:46:19,870
So what's the vertical?
736
00:46:19,870 --> 00:46:31,830
So by vertical
slicing, we get shells.
737
00:46:31,830 --> 00:46:38,530
And that starts-- That's
in the x variable.
738
00:46:38,530 --> 00:46:43,480
It starts at 0 and
ends at 1 and it's dx.
739
00:46:43,480 --> 00:46:46,300
And what are the shells?
740
00:46:46,300 --> 00:46:51,750
Well, the shells are, if I
can draw the picture again,
741
00:46:51,750 --> 00:46:55,120
they start-- the top value is e.
742
00:46:55,120 --> 00:47:02,530
And the bottom value is, I need
a little bit of room for this.
743
00:47:02,530 --> 00:47:06,810
The bottom value is y.
744
00:47:06,810 --> 00:47:12,670
And then we have 2 pi
x is the circumference,
745
00:47:12,670 --> 00:47:15,970
as we sweep it around dx.
746
00:47:15,970 --> 00:47:18,260
So here's our new volume.
747
00:47:18,260 --> 00:47:23,600
Expressed in this different way.
748
00:47:23,600 --> 00:47:26,220
So now I'm going to
plug in what this is.
749
00:47:26,220 --> 00:47:30,380
It's the integral from
0 to 1 of e minus e^x,
750
00:47:30,380 --> 00:47:36,530
that's the formula
for y, 2 pi x dx.
751
00:47:36,530 --> 00:47:39,670
And what you see is that
you get the integral
752
00:47:39,670 --> 00:47:45,150
from 0 to 1 of 2 pi e x dx.
753
00:47:45,150 --> 00:47:46,540
That's easy, right?
754
00:47:46,540 --> 00:47:51,940
That's just 2 pi e times 1/2.
755
00:47:51,940 --> 00:47:54,820
This one is just the
area of a triangle.
756
00:47:54,820 --> 00:47:56,890
If I factor out the 2 pi e.
757
00:47:56,890 --> 00:48:03,230
And then the other piece is
the integral of 2 pi x e^x dx.
758
00:48:03,230 --> 00:48:08,610
From 0 to 1.
759
00:48:08,610 --> 00:48:12,480
STUDENT: [INAUDIBLE] PROFESSOR:
Are you asking me whether I
760
00:48:12,480 --> 00:48:14,370
need an x^2 here?
761
00:48:14,370 --> 00:48:15,960
I just evaluated the integral.
762
00:48:15,960 --> 00:48:17,290
I just did it geometrically.
763
00:48:17,290 --> 00:48:19,570
I said, this is the
area of a triangle.
764
00:48:19,570 --> 00:48:21,850
I didn't antidifferentiate
and evaluate it,
765
00:48:21,850 --> 00:48:23,880
I just told you the number.
766
00:48:23,880 --> 00:48:27,580
Because it's a
definite integral.
767
00:48:27,580 --> 00:48:31,650
So now, this one here, I
can read off from right up
768
00:48:31,650 --> 00:48:33,980
here, above it.
769
00:48:33,980 --> 00:48:37,050
This is G_1.
770
00:48:37,050 --> 00:48:42,060
So this is equal to,
let's check it out here.
771
00:48:42,060 --> 00:48:49,170
So this is pi e, right,
minus 2 pi G_1(x),
772
00:48:49,170 --> 00:48:52,280
evaluated at 0 and 1.
773
00:48:52,280 --> 00:48:54,990
So let's make sure that it's
the same as what we had before.
774
00:48:54,990 --> 00:48:59,860
It's pi e minus 2 pi
times-- here's G_1.
775
00:48:59,860 --> 00:49:03,230
So it's x e^x - e^x.
776
00:49:03,230 --> 00:49:05,720
So at x = 1, that cancels.
777
00:49:05,720 --> 00:49:08,260
But at the bottom end, it's e^0.
778
00:49:08,260 --> 00:49:12,040
So it's -1 here.
779
00:49:12,040 --> 00:49:13,130
Is that right?
780
00:49:13,130 --> 00:49:13,710
Yep.
781
00:49:13,710 --> 00:49:17,060
So it's pi e - 2.
782
00:49:17,060 --> 00:49:21,650
It's the same.
783
00:49:21,650 --> 00:49:22,150
Question.
784
00:49:22,150 --> 00:49:28,030
STUDENT: [INAUDIBLE]
785
00:49:28,030 --> 00:49:33,380
PROFESSOR: From here to
here, is that the question?
786
00:49:33,380 --> 00:49:39,710
STUDENT: [INAUDIBLE]
787
00:49:39,710 --> 00:49:43,730
PROFESSOR: So the step here
is just the distributive law.
788
00:49:43,730 --> 00:49:46,810
This is e 2 pi x,
that's this term.
789
00:49:46,810 --> 00:49:49,550
And the other terms, the
minus sign is outside.
790
00:49:49,550 --> 00:49:51,320
The 2 pi I factored out.
791
00:49:51,320 --> 00:49:56,980
And the x and the e^x stayed
inside the integral sign.
792
00:49:56,980 --> 00:49:59,140
Thank you.
793
00:49:59,140 --> 00:50:01,670
The correction is that
there was a missing minus
794
00:50:01,670 --> 00:50:03,780
sign, last time.
795
00:50:03,780 --> 00:50:13,100
When I integrated from 0 to 1,
x e^x dx, I had a x e^x - e^x.
796
00:50:13,100 --> 00:50:15,090
Evaluated at 0 and 1.
797
00:50:15,090 --> 00:50:18,350
And that's equal to +1.
798
00:50:18,350 --> 00:50:21,580
I was missing this minus sign.
799
00:50:21,580 --> 00:50:30,840
The place where it came in
was in this wineglass example.
800
00:50:30,840 --> 00:50:36,210
We had the integral of
2 pi x (e - e^x) dx.
801
00:50:39,340 --> 00:50:48,900
And that was 2 pi e integral
of x dx, from 0 to 1, -2 pi,
802
00:50:48,900 --> 00:50:52,810
integral from 0
to 1 of x e^x dx.
803
00:50:52,810 --> 00:50:58,400
And then I worked this
out and it was pi e.
804
00:50:58,400 --> 00:51:03,030
And then this one was -2 pi,
and what I wrote down was -1.
805
00:51:03,030 --> 00:51:05,410
But there should have been
an extra minus sign there.
806
00:51:05,410 --> 00:51:08,430
So it's this.
807
00:51:08,430 --> 00:51:11,930
The final answer was
correct, but this minus sign
808
00:51:11,930 --> 00:51:13,590
was missing.
809
00:51:13,590 --> 00:51:16,930
Right there.
810
00:51:16,930 --> 00:51:20,460
So just, right there.