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PROFESSOR: Now, to start out
today we're going to finish
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00:00:26 --> 00:00:27
up what we did last time.
11
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Which has to do with
partial fractions.
12
00:00:30 --> 00:00:34
I told you how to do partial
fractions in several special
13
00:00:34 --> 00:00:36
cases and everybody was trying
to figure out what the
14
00:00:36 --> 00:00:37
general picture was.
15
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But I'd like to lay that out.
16
00:00:39 --> 00:00:41
I'll still only do
it for an example.
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00:00:41 --> 00:00:44
But it will be somehow a bigger
example so that you can see
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what the general pattern is.
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00:00:53 --> 00:01:04
Partial fractions, remember,
is a method for breaking up
20
00:01:04 --> 00:01:06
so-called rational functions.
21
00:01:06 --> 00:01:09
Which are ratios
of polynomials.
22
00:01:09 --> 00:01:13
And it shows you that you
can always integrate them.
23
00:01:13 --> 00:01:14
That's really the theme here.
24
00:01:14 --> 00:01:21
And this is what's reassuring
is that it always works.
25
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That's really the bottom line.
26
00:01:23 --> 00:01:28
And that's good because there
are a lot of integrals that
27
00:01:28 --> 00:01:34
don't have formulas
and these do.
28
00:01:34 --> 00:01:35
It always works.
29
00:01:35 --> 00:01:43
But, maybe with lots of help.
30
00:01:43 --> 00:01:46
So maybe slowly.
31
00:01:46 --> 00:01:48
Now, there's a little bit of
bad news, and I have to be
32
00:01:48 --> 00:01:52
totally honest and tell you
what all the bad news is.
33
00:01:52 --> 00:01:54
Along with the good news.
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00:01:54 --> 00:02:00
The first step, which maybe I
should be calling Step 0, I
35
00:02:00 --> 00:02:08
had a Step 1, 2 and 3 last
time, is long division.
36
00:02:08 --> 00:02:12
That's the step where you take
your polynomial divided by your
37
00:02:12 --> 00:02:18
other polynomial, and you find
the quotient plus
38
00:02:18 --> 00:02:22
some remainder.
39
00:02:22 --> 00:02:24
And you do that by
long division.
40
00:02:24 --> 00:02:28
And the quotient is easy to
take the antiderivative of up
41
00:02:28 --> 00:02:30
because it's just a polynomial.
42
00:02:30 --> 00:02:33
And the key extra property here
is that the degree of the
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00:02:33 --> 00:02:37
numerator now over here, this
remainder, is strictly less
44
00:02:37 --> 00:02:40
than the degree of
the denominator.
45
00:02:40 --> 00:02:44
So that you can do
the next step.
46
00:02:44 --> 00:02:48
Now, the next step which I
called Step 1 last time, that's
47
00:02:48 --> 00:02:52
great imagination, it's right
after Step 0, Step 1 was to
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00:02:52 --> 00:02:54
factor the denominator.
49
00:02:54 --> 00:03:00
And I'm going to illustrate by
example what the setup is here.
50
00:03:00 --> 00:03:09
I don't know maybe,
we'll do this.
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00:03:09 --> 00:03:12
Some polynomial here,
maybe cube this one.
52
00:03:12 --> 00:03:21
So here I've factored
the denominator.
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That's what I called
Step 1 last time.
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00:03:24 --> 00:03:27
Now, here's the first
piece of bad news.
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00:03:27 --> 00:03:33
In reality, if somebody gave
you a multiplied out degree,
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00:03:33 --> 00:03:36
whatever polynomial here,
you would be very hard
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00:03:36 --> 00:03:40
pressed to factor it.
58
00:03:40 --> 00:03:44
A lot of them are extremely
difficult to factor.
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00:03:44 --> 00:03:45
And so that's something
you would have to give
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to a machine to do.
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00:03:47 --> 00:03:50
And it's just basically
a hard problem.
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00:03:50 --> 00:03:54
So obviously, we're only
going to give you ones
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that you can do by hand.
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00:03:55 --> 00:03:58
So very low degree examples.
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00:03:58 --> 00:03:59
And that's just the way it is.
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So this is really a hard step
in disguise, in real life.
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Anyway, we're just going
to take it as given.
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And we have this numerator,
which is of degree less
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than the denominator.
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So let's count up what
its degree has to be.
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This is 4 + 2 + 6.
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So this is degree 4 + 2 + 6.
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I added that up because this is
degree 4, this is degree 2 and
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(x ^2) ^3 is the 6th power.
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So all together it's
this, which is 12.
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And so this thing is
of degree <= 11.
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All the way up to degree 11,
that's the possibilities
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for the numerator here.
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Now, the extra information
that I want to impart right
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now, is just this setup.
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Which I called
Step 2 last time.
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00:04:58 --> 00:05:05
And the setup is this.
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00:05:05 --> 00:05:07
Now, it's going to take
us a while to do this.
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00:05:07 --> 00:05:10
We have this factor here.
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00:05:10 --> 00:05:12
We have another factor.
86
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We have another term,
with the square.
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We have another term
with the cube.
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00:05:18 --> 00:05:22
We have another term
with the fourth power.
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00:05:22 --> 00:05:24
So this is what's going
to happen whenever you
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have linear factors.
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You'll have a collection
of terms like this.
92
00:05:28 --> 00:05:31
So you have four constants
to take care of.
93
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Now, with a quadratic in the
denominator, you need a linear
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00:05:35 --> 00:05:36
function in the numerator.
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So that's, if you like, B0
x + C0 divided by this
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00:05:42 --> 00:05:49
quadratic term here.
97
00:05:49 --> 00:05:54
And what I didn't show you last
time was how you deal with
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00:05:54 --> 00:05:59
higher powers of
quadratic terms.
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00:05:59 --> 00:06:04
So when you have a quadratic
term, what's going to happen
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is you're going to take
that first factor here.
101
00:06:07 --> 00:06:11
Just the way you
did in this case.
102
00:06:11 --> 00:06:15
But then you're going to
have to do the same thing
103
00:06:15 --> 00:06:24
with the next power.
104
00:06:24 --> 00:06:28
Now notice, just as in the
case of this top row, I
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00:06:28 --> 00:06:29
have just a constant here.
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00:06:29 --> 00:06:33
And even though I increased the
degree of the denominator I'm
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00:06:33 --> 00:06:34
not increasing the numerator.
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00:06:34 --> 00:06:35
It's staying just a constant.
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00:06:35 --> 00:06:38
It's not linear up here.
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00:06:38 --> 00:06:39
It's better than that.
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00:06:39 --> 00:06:41
It's just a constant.
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00:06:41 --> 00:06:44
And here it stayed a constant.
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00:06:44 --> 00:06:45
And here I stayed a constant.
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00:06:45 --> 00:06:48
Similarly here, even though I'm
increasing the degree of the
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00:06:48 --> 00:06:51
denominator, I'm leaving the
numerator, the form of
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00:06:51 --> 00:06:52
the numerator, alone.
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00:06:52 --> 00:06:55
It's just a linear factor
and a linear factor.
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00:06:55 --> 00:07:05
So that's the key
to this pattern.
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00:07:05 --> 00:07:09
I don't have quite as
jazzy a song on mine.
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00:07:09 --> 00:07:13
So this is so long that it
runs off the blackboard here.
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00:07:13 --> 00:07:15
So let's continue
it on the next.
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00:07:15 --> 00:07:21
We've got this B2 x +
C2 - sorry, B3 x +
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00:07:21 --> 00:07:26
C3 / (x ^2 + 4) ^3.
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00:07:26 --> 00:07:38
I guess I have room
for it over here.
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00:07:38 --> 00:07:41
I'm going to talk about
this in just a second.
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00:07:41 --> 00:07:43
Alright, so here's the pattern.
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00:07:43 --> 00:07:50
Now, let me just do a
count of the number of
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00:07:50 --> 00:07:52
unknowns we have here.
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00:07:52 --> 00:07:55
The number of unknowns that
we have here is 1, 2, 3, 4,
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00:07:55 --> 00:07:58
5, 6, 7, 8, 9, 10, 11, 12.
131
00:07:58 --> 00:08:00
That 12 is no coincidence.
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00:08:00 --> 00:08:03
That's the degree
of the polynomial.
133
00:08:03 --> 00:08:05
And it's the number of
unknowns that we have.
134
00:08:05 --> 00:08:09
And it's the number of
degrees of freedom in a
135
00:08:09 --> 00:08:11
polynomial of degree 11.
136
00:08:11 --> 00:08:13
If you have all these free
coefficients here, you have
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00:08:13 --> 00:08:17
the coefficient x^ 0, x ^
1, all the way up to x^ 11.
138
00:08:17 --> 00:08:23
And 0 through 11 is 12
different coefficients.
139
00:08:23 --> 00:08:26
And so this is a very
complicated system of
140
00:08:26 --> 00:08:28
equations for unknowns.
141
00:08:28 --> 00:08:33
This is twelve equations
for twelve unknowns.
142
00:08:33 --> 00:08:34
So I'll get rid of
this for a second.
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00:08:34 --> 00:08:41
So we have twelve equations,
twelve unknowns.
144
00:08:41 --> 00:08:43
So that's the other bad news.
145
00:08:43 --> 00:08:46
Machines handle this very
well, but human beings have
146
00:08:46 --> 00:08:47
a little trouble with 12.
147
00:08:47 --> 00:08:51
Now, the cover-up method
works very neatly and
148
00:08:51 --> 00:08:53
picks out this term here.
149
00:08:53 --> 00:08:54
But that's it.
150
00:08:54 --> 00:08:56
So it reduces it
to an 11 by 11.
151
00:08:56 --> 00:09:00
You'll be able to evaluate
this in no time.
152
00:09:00 --> 00:09:00
But that's it.
153
00:09:00 --> 00:09:04
That's the only simplification
of your previous method.
154
00:09:04 --> 00:09:06
We don't have a
method for this.
155
00:09:06 --> 00:09:08
So I'm just showing what the
whole method looks like but
156
00:09:08 --> 00:09:11
really you'd have to have a
machine to implement this once
157
00:09:11 --> 00:09:14
it gets to be any size at all.
158
00:09:14 --> 00:09:15
Yeah, question.
159
00:09:15 --> 00:09:18
STUDENT: [INAUDIBLE]
160
00:09:18 --> 00:09:23
PROFESSOR: It's one big
equation, but it's a
161
00:09:23 --> 00:09:24
polynomial equation.
162
00:09:24 --> 00:09:32
So there's an equation, there's
this function r ( x) = a11
163
00:09:32 --> 00:09:41
x^2 11 + a10 x^ 10... and
these things are known.
164
00:09:41 --> 00:09:43
This is a known
expression here.
165
00:09:43 --> 00:09:45
And then when you
cross-multiply on the other
166
00:09:45 --> 00:09:52
side, what you have is, well,
it's a1 times, if you cancel
167
00:09:52 --> 00:09:58
this denominator with that,
you're going to get (x + 2) ^3
168
00:09:58 --> 00:10:08
( x ^2 + 2x + 3)( x^2 + 4)^3 +
the term for a2, etc.
169
00:10:08 --> 00:10:10
It's a monster equation.
170
00:10:10 --> 00:10:12
And then to separate it out
into separate equations, you
171
00:10:12 --> 00:10:19
take the coefficient on x^
11th, x ^ 10, ... all
172
00:10:19 --> 00:10:20
the way down to x^ 0.
173
00:10:21 --> 00:10:27
And all told, that means there
are a total of 12 equations.
174
00:10:27 --> 00:10:31
11 through 0 is 12 equations.
175
00:10:31 --> 00:10:35
Yeah, another question.
176
00:10:35 --> 00:10:35
STUDENT: [INAUDIBLE]
177
00:10:35 --> 00:10:38
PROFESSOR: Should I write
down rest of this?
178
00:10:38 --> 00:10:38
STUDENT: [INAUDIBLE]
179
00:10:38 --> 00:10:40
PROFESSOR: Should you write
down all this stuff?
180
00:10:40 --> 00:10:43
Well, that's a good question.
181
00:10:43 --> 00:10:46
So you notice I didn't
write it down.
182
00:10:46 --> 00:10:47
Why didn't I write it down?
183
00:10:47 --> 00:10:50
Because it's incredibly long.
184
00:10:50 --> 00:10:53
In fact, you probably,
so how many pages of
185
00:10:53 --> 00:10:54
writing would this take?
186
00:10:54 --> 00:10:56
This is about a
page of writing.
187
00:10:56 --> 00:10:59
So just think of your machine,
how much time you want
188
00:10:59 --> 00:11:01
to spend on this.
189
00:11:01 --> 00:11:05
So the answer is that you
have to be realistic.
190
00:11:05 --> 00:11:07
You're a human being,
not a machine.
191
00:11:07 --> 00:11:10
And so there's certain things
that you can write down and
192
00:11:10 --> 00:11:12
other things you should
attempt to write down.
193
00:11:12 --> 00:11:17
So do not do this at home.
194
00:11:17 --> 00:11:21
So that's the first
down-side to this method.
195
00:11:21 --> 00:11:24
It gets more and more
complicated as time goes on.
196
00:11:24 --> 00:11:28
The second down-side, I want to
point out to you, is what's
197
00:11:28 --> 00:11:35
happening with the pieces.
198
00:11:35 --> 00:11:42
So the pieces still
need to be integrated.
199
00:11:42 --> 00:11:48
We simplified this problem,
but we didn't get rid of it.
200
00:11:48 --> 00:11:50
We still have the problem
of integrating the pieces.
201
00:11:50 --> 00:11:52
Now, some of the
pieces are very easy.
202
00:11:52 --> 00:11:55
This top row here, the
antiderivatives of these,
203
00:11:55 --> 00:11:59
you can just write down.
204
00:11:59 --> 00:12:01
By advanced guessing.
205
00:12:01 --> 00:12:04
I'm going to skip over to the
most complicated one over here.
206
00:12:04 --> 00:12:06
For one second here.
207
00:12:06 --> 00:12:09
And what is it that you'd have
to deal with for that one.
208
00:12:09 --> 00:12:13
You'd have to deal with,
for example, so e.g., for
209
00:12:13 --> 00:12:21
example, I need to deal
with the this guy.
210
00:12:21 --> 00:12:26
I've got to get this
antiderivative here.
211
00:12:26 --> 00:12:29
Now, this one you're supposed
to be able to know.
212
00:12:29 --> 00:12:30
So this is why I'm
mentioning this.
213
00:12:30 --> 00:12:33
Because this kind of
ingredient is something
214
00:12:33 --> 00:12:34
you already covered.
215
00:12:34 --> 00:12:35
And what is it?
216
00:12:35 --> 00:12:39
Well, you do this one by
advanced guessing, although
217
00:12:39 --> 00:12:42
you it as the method
of substitution.
218
00:12:42 --> 00:12:46
You realize that it's going
to be of the form (x^2 + 4)
219
00:12:46 --> 00:12:49
^ - 2, roughly speaking.
220
00:12:49 --> 00:12:51
And now we're going
to fix that.
221
00:12:51 --> 00:12:54
Because if you differentiate
it you get 2x (-2), that's
222
00:12:54 --> 00:12:56
- 4 (x)(x) times this.
223
00:12:56 --> 00:12:58
There's an x in the
numerator here.
224
00:12:58 --> 00:13:02
So it's - 1/4 of that
will fix the factor.
225
00:13:02 --> 00:13:06
And here's the answer
for that one.
226
00:13:06 --> 00:13:10
So that's one you can do.
227
00:13:10 --> 00:13:19
The second piece is this guy.
228
00:13:19 --> 00:13:20
This is the other piece.
229
00:13:20 --> 00:13:25
Now, this was the piece
that came from B3.
230
00:13:25 --> 00:13:27
This is the one
that came from B3.
231
00:13:27 --> 00:13:30
And this is the one
that's coming from C3.
232
00:13:30 --> 00:13:32
This is coming from C3.
233
00:13:32 --> 00:13:35
We still need to get
this one out there.
234
00:13:35 --> 00:13:37
So C3 times that will be the
correct answer, once we've
235
00:13:37 --> 00:13:40
found these numbers.
236
00:13:40 --> 00:13:44
So how do we do this?
237
00:13:44 --> 00:13:45
How's this one integrated?
238
00:13:45 --> 00:13:49
STUDENT: Trig substitution?
239
00:13:49 --> 00:13:51
PROFESSOR: Trig substitution.
240
00:13:51 --> 00:13:57
So the trig substitution
here is x = 2 tan u.
241
00:13:57 --> 00:14:00
Or 2 tan theta.
242
00:14:00 --> 00:14:03
And when you do that, there are
a couple of simplifications.
243
00:14:03 --> 00:14:06
Well, I wouldn't call
this a simplification.
244
00:14:06 --> 00:14:10
This is just the
differentiation formula.
245
00:14:10 --> 00:14:14
dx = 2 sec^2 u du.
246
00:14:14 --> 00:14:19
And then you have to plug in,
and you're using the fact that
247
00:14:19 --> 00:14:23
when you plug in the tan^2, 4
tan ^2 + 4, you'll
248
00:14:23 --> 00:14:24
get a sec ^2.
249
00:14:24 --> 00:14:32
So altogether, this
thing is, 2 sec^2 u du.
250
00:14:32 --> 00:14:40
And then there's a (4 sec^2
u) ^3, in the denominator.
251
00:14:40 --> 00:14:44
So that's what happens when
you change variables here.
252
00:14:44 --> 00:14:46
And now look, this
keeps on going.
253
00:14:46 --> 00:14:49
This is not the end
of the problem.
254
00:14:49 --> 00:14:50
Because what does
that simplify to?
255
00:14:50 --> 00:14:55
That is, let's see, it's
2/64, the integral
256
00:14:55 --> 00:14:58
of sec^6 and sec^2.
257
00:14:58 --> 00:15:00
That's the same as cos^4.
258
00:15:00 --> 00:15:04
259
00:15:04 --> 00:15:06
And now, you did a trig
substitution but you still
260
00:15:06 --> 00:15:11
have a trig integral.
261
00:15:11 --> 00:15:15
The trig integral now,
there's a method for this.
262
00:15:15 --> 00:15:19
The method for this is when
it's an even power, you have to
263
00:15:19 --> 00:15:22
use the double angle formula.
264
00:15:22 --> 00:15:31
So that's this guy here.
265
00:15:31 --> 00:15:33
And you're still not done.
266
00:15:33 --> 00:15:35
You have to square
this thing out.
267
00:15:35 --> 00:15:37
And then you'll still
get a cos^2 2u.
268
00:15:37 --> 00:15:38
And it keeps on going.
269
00:15:38 --> 00:15:41
So this thing goes
on for a long time.
270
00:15:41 --> 00:15:43
But I'm not even going
to finish this, but I
271
00:15:43 --> 00:15:44
just want to show you.
272
00:15:44 --> 00:15:47
The point is, we're not
showing you how to do
273
00:15:47 --> 00:15:48
any complicated problem.
274
00:15:48 --> 00:15:50
We're just showing you all
the little ingredients.
275
00:15:50 --> 00:15:53
And you have to string them
together a long, long, long
276
00:15:53 --> 00:15:56
process to get to the final
answer of one of
277
00:15:56 --> 00:15:57
these questions.
278
00:15:57 --> 00:16:07
So it always works,
but maybe slowly.
279
00:16:07 --> 00:16:10
By the way, there's
even another horrible
280
00:16:10 --> 00:16:13
thing that happens.
281
00:16:13 --> 00:16:22
Which is, if you handle this
guy here, what's the technique.
282
00:16:22 --> 00:16:25
This is another technique
that you learned, supposedly
283
00:16:25 --> 00:16:28
within the last few days.
284
00:16:28 --> 00:16:30
Completing the square.
285
00:16:30 --> 00:16:39
So this, it turns out, you
have to rewrite it this way.
286
00:16:39 --> 00:16:41
And then the evaluation is
going to be expressed in
287
00:16:41 --> 00:16:44
terms of, I'm going
to jump to the end.
288
00:16:44 --> 00:16:49
It's going to turn out to be
expressed in terms of this.
289
00:16:49 --> 00:16:53
That's what will eventually
show up in the formula.
290
00:16:53 --> 00:16:58
And not only that, but if you
deal with ones involving x as
291
00:16:58 --> 00:17:06
well, you'll also need to deal
with something like ln of
292
00:17:06 --> 00:17:09
this denominator here.
293
00:17:09 --> 00:17:13
So all of these things
will be involved.
294
00:17:13 --> 00:17:16
So now, the last message that
I have for you is just this.
295
00:17:16 --> 00:17:18
This thing is very complicated.
296
00:17:18 --> 00:17:20
We're certainly never going
to ask you to do it.
297
00:17:20 --> 00:17:23
But you should just be aware
that this level of complexity,
298
00:17:23 --> 00:17:26
we are absolutely stuck
with in this problem.
299
00:17:26 --> 00:17:32
And the reason why we're stuck
with it is that this is what
300
00:17:32 --> 00:17:36
the formulas look
like in the end.
301
00:17:36 --> 00:17:39
If the answers look like
this, the formulas have
302
00:17:39 --> 00:17:41
to be this complicated.
303
00:17:41 --> 00:17:43
If you differentiate this, you
get your polynomial, your
304
00:17:43 --> 00:17:44
ratio of polynomials.
305
00:17:44 --> 00:17:46
If you differentiate this, you
get some ratio of polynomials.
306
00:17:46 --> 00:17:48
These are the things that
come up when you take
307
00:17:48 --> 00:17:51
antiderivatives of those
rational functions.
308
00:17:51 --> 00:17:56
So we're just stuck
with these guys.
309
00:17:56 --> 00:17:58
And so don't let it
get to you too much.
310
00:17:58 --> 00:17:59
I mean, it's not so bad.
311
00:17:59 --> 00:18:02
In fact, there are computer
programs that will do this
312
00:18:02 --> 00:18:03
for you anytime you want.
313
00:18:03 --> 00:18:05
And you just have to be
not intimidated by them.
314
00:18:05 --> 00:18:10
They're like other functions.
315
00:18:10 --> 00:18:20
OK, that's it for the general
comments on partial fractions.
316
00:18:20 --> 00:18:24
Now we're going to change
subjects to our last technique.
317
00:18:24 --> 00:18:25
This is one more technical
thing to get you
318
00:18:25 --> 00:18:27
familiar with functions.
319
00:18:27 --> 00:18:32
And this technique is called
integration by parts.
320
00:18:32 --> 00:18:35
Please, just because its name
sort of sounds like partial
321
00:18:35 --> 00:18:37
fractions, don't think
it's the same thing.
322
00:18:37 --> 00:18:38
It's totally different.
323
00:18:38 --> 00:18:44
It's not the same.
324
00:18:44 --> 00:19:06
So this one is called
integration by parts.
325
00:19:06 --> 00:19:09
Now, unlike the previous case,
where I couldn't actually
326
00:19:09 --> 00:19:13
justify to you that the
linear algebra always works.
327
00:19:13 --> 00:19:14
I claimed it worked, but I
wasn't able to prove it.
328
00:19:14 --> 00:19:17
That's a complicated
theorem which I'm not
329
00:19:17 --> 00:19:19
able to do in this class.
330
00:19:19 --> 00:19:22
Here I can explain to you
what's going on with
331
00:19:22 --> 00:19:24
integration by parts.
332
00:19:24 --> 00:19:26
It's just the fundamental
theorem of calculus, if
333
00:19:26 --> 00:19:30
you like, coupled with
the product formula.
334
00:19:30 --> 00:19:33
Sort of unwound and
read in reverse.
335
00:19:33 --> 00:19:35
And here's how that works.
336
00:19:35 --> 00:19:38
If you take the product of two
functions and you differentiate
337
00:19:38 --> 00:19:40
them, then we know that the
product rule says that
338
00:19:40 --> 00:19:43
this is u' v + uv'.
339
00:19:45 --> 00:19:50
And now I'm just going to
rearrange in the following way.
340
00:19:50 --> 00:19:53
I'm going to solve for uv'.
341
00:19:53 --> 00:19:54
That is, this term here.
342
00:19:54 --> 00:19:56
So what is this term?
343
00:19:56 --> 00:19:58
It's this other term, (uv)'.
344
00:19:59 --> 00:20:04
Minus the other piece.
345
00:20:04 --> 00:20:08
So I just rewrote
this equation.
346
00:20:08 --> 00:20:10
And now I'm going
to integrate it.
347
00:20:10 --> 00:20:11
So here's the formula.
348
00:20:11 --> 00:20:16
The integral of the left-hand
side is equal to the integral
349
00:20:16 --> 00:20:17
of the right-hand side.
350
00:20:17 --> 00:20:19
Well when I integrate a
derivative, of I get back
351
00:20:19 --> 00:20:21
the function itself.
352
00:20:21 --> 00:20:27
That's the fundamental theorem.
353
00:20:27 --> 00:20:27
So this it.
354
00:20:27 --> 00:20:30
Sorry, I missed the dx,
which is important.
355
00:20:30 --> 00:20:32
I apologize.
356
00:20:32 --> 00:20:35
Let's put that in there.
357
00:20:35 --> 00:20:41
So this is the integration
by parts formula.
358
00:20:41 --> 00:20:46
I'm going to write it one more
time with the limits stuck in.
359
00:20:46 --> 00:20:49
It's also written this
way, when you have a
360
00:20:49 --> 00:21:02
definite integral.
361
00:21:02 --> 00:21:13
Just the same formula,
written twice.
362
00:21:13 --> 00:21:18
Alright, now I'm going to
show you how it works
363
00:21:18 --> 00:21:24
on a few examples.
364
00:21:24 --> 00:21:29
And I have to give you a
flavor for how it works.
365
00:21:29 --> 00:21:34
But it'll grow as we get
more and more experience.
366
00:21:34 --> 00:21:40
The first example that I'm
going to take is one that looks
367
00:21:40 --> 00:21:43
intractable on the face of it.
368
00:21:43 --> 00:21:49
Which is the integral
of ln x dx.
369
00:21:49 --> 00:21:52
Now, it looks like there's sort
of nothing we can do with this.
370
00:21:52 --> 00:21:55
And we don't know what
the solution is.
371
00:21:55 --> 00:21:59
However, I claim that if we fit
it into this form, we can
372
00:21:59 --> 00:22:03
figure out what the integral
is relatively easily.
373
00:22:03 --> 00:22:07
By some little magic of
cancellation it happens.
374
00:22:07 --> 00:22:08
The idea is the following.
375
00:22:08 --> 00:22:13
If I consider this function to
be u, then what's going to
376
00:22:13 --> 00:22:17
appear on the other side in the
integrated form is the function
377
00:22:17 --> 00:22:22
u', which is -- so, if
you like, u = ln x.
378
00:22:22 --> 00:22:25
So u' = 1 / x.
379
00:22:25 --> 00:22:28
Now, 1 / x is a more manageable
function than ln x.
380
00:22:28 --> 00:22:31
What we're using is that when
we differentiate the function,
381
00:22:31 --> 00:22:33
it's getting nicer.
382
00:22:33 --> 00:22:36
It's getting more
tractable for us.
383
00:22:36 --> 00:22:39
In order for this to fit
into this pattern, however,
384
00:22:39 --> 00:22:44
I need a function v.
385
00:22:44 --> 00:22:48
So what in the world am I
going to put here for v?
386
00:22:48 --> 00:22:51
The answer is, well, dx is
almost the right answer.
387
00:22:51 --> 00:22:53
The answer turns out to be x.
388
00:22:53 --> 00:23:01
And the reason is that
that makes v' = 1.
389
00:23:01 --> 00:23:02
It makes v' = 1.
390
00:23:02 --> 00:23:05
So that means that this
is u, but it's also uv'.
391
00:23:05 --> 00:23:11
Which was what I had on
the left-hand side.
392
00:23:11 --> 00:23:12
So it's both u and uv'.
393
00:23:13 --> 00:23:14
So this is the setup.
394
00:23:14 --> 00:23:19
And now all I'm going to do is
read off what the formula says.
395
00:23:19 --> 00:23:23
What it says is, this
is equal to u v.
396
00:23:23 --> 00:23:25
So u is this and v is that.
397
00:23:25 --> 00:23:32
So it's x ln x, minus, so
that again, this is uv.
398
00:23:32 --> 00:23:37
Except in the other order, vu.
399
00:23:37 --> 00:23:38
And then I'm integrating,
and what do I have
400
00:23:38 --> 00:23:40
to integrate? u ' v.
401
00:23:40 --> 00:23:45
So look up there. u' v with
a minus sign here. u'
402
00:23:45 --> 00:23:47
= 1 / x, and v = x.
403
00:23:47 --> 00:23:49
So it's 1 / x, that's u'.
404
00:23:50 --> 00:23:56
And here is x, that's v, dx.
405
00:23:56 --> 00:23:58
Now, that one is
easy to integrate.
406
00:23:58 --> 00:24:00
Because (1/x) x = 1.
407
00:24:00 --> 00:24:07
And the integral of 1 dx
is x + c, if you like.
408
00:24:07 --> 00:24:10
So the antiderivative
of 1 is x.
409
00:24:10 --> 00:24:11
And so here's our answer.
410
00:24:11 --> 00:24:34
Our answer is that this
is x ln x - x + c.
411
00:24:34 --> 00:24:37
I'm going to do two
more slightly more
412
00:24:37 --> 00:24:39
complicated examples.
413
00:24:39 --> 00:24:43
And then really, the main
thing is to get yourself
414
00:24:43 --> 00:24:44
used to this method.
415
00:24:44 --> 00:24:47
And there's no one
way of doing that.
416
00:24:47 --> 00:24:49
Just practice makes perfect.
417
00:24:49 --> 00:24:53
And so we'll just do
a few more examples.
418
00:24:53 --> 00:24:55
And illustrate them.
419
00:24:55 --> 00:24:59
The second example that
I'm going to use is the
420
00:24:59 --> 00:25:03
integral of (ln x) ^2 dx.
421
00:25:03 --> 00:25:08
And this is just slightly
more recalcitrant.
422
00:25:08 --> 00:25:13
Namely, I'm going to
let u be (ln x)^2.
423
00:25:13 --> 00:25:17
424
00:25:17 --> 00:25:20
And again, v = to x.
425
00:25:20 --> 00:25:21
So that matches up here.
426
00:25:21 --> 00:25:23
That is, v' = 1.
427
00:25:23 --> 00:25:25
So this is u v'.
428
00:25:28 --> 00:25:30
So this thing is u v'.
429
00:25:31 --> 00:25:33
And then we'll just
see what happens.
430
00:25:33 --> 00:25:38
Now, the game that we get is
that when I differentiate the
431
00:25:38 --> 00:25:42
logarithm squared, I'm going
to to get something simpler.
432
00:25:42 --> 00:25:47
It's not going to win us
the whole battle, but
433
00:25:47 --> 00:25:49
it will get us started.
434
00:25:49 --> 00:25:50
So here we get u'.
435
00:25:51 --> 00:25:56
And that's 2 ln x ( 1 / x).
436
00:25:56 --> 00:26:00
Applying the chain rule.
437
00:26:00 --> 00:26:06
And so the formula is that this
is x (ln x)^2, minus the
438
00:26:06 --> 00:26:12
integral of, well it's u'
v, right, that's what I
439
00:26:12 --> 00:26:13
have to put over here.
440
00:26:13 --> 00:26:22
So u' = 2 ln x ( 1
/ x), and v = x.
441
00:26:22 --> 00:26:25
And so now, you notice
something interesting
442
00:26:25 --> 00:26:25
happening here.
443
00:26:25 --> 00:26:28
So let me just demarcate
this a little bit.
444
00:26:28 --> 00:26:34
And let you see what it
is that I'm doing here.
445
00:26:34 --> 00:26:36
So notice, this is
the same integral.
446
00:26:36 --> 00:26:38
So here we have x (ln x) ^2.
447
00:26:38 --> 00:26:41
We've already solve that part.
448
00:26:41 --> 00:26:43
But now know notice that the
1 / x and the x cancel.
449
00:26:43 --> 00:26:46
So we're back to
the previous case.
450
00:26:46 --> 00:26:49
We didn't win all the way,
but actually we reduced
451
00:26:49 --> 00:26:51
ourselves to this integral.
452
00:26:51 --> 00:26:56
To the integral of ln x,
which we already know.
453
00:26:56 --> 00:26:58
So here, I can copy that down.
454
00:26:58 --> 00:27:04
That's - 2 (x ln x -
x), and then I have to
455
00:27:04 --> 00:27:05
throw in a constant, c.
456
00:27:05 --> 00:27:07
And that's the end of
the problem here.
457
00:27:07 --> 00:27:10
That's it.
458
00:27:10 --> 00:27:26
So this piece, I got
from Example 1.
459
00:27:26 --> 00:27:35
Now, this illustrates a
principle which is a little bit
460
00:27:35 --> 00:27:40
more complicated than just the
one of integration by parts.
461
00:27:40 --> 00:27:44
Which is a sort of a general
principle which I'll call my
462
00:27:44 --> 00:27:48
Example 3, which is
something which is called a
463
00:27:48 --> 00:27:56
reduction formula.
464
00:27:56 --> 00:28:00
A reduction formula is a case
where we apply some rule and
465
00:28:00 --> 00:28:03
we figure out one of these
integrals in terms
466
00:28:03 --> 00:28:05
of something else.
467
00:28:05 --> 00:28:07
Which is a little bit simpler.
468
00:28:07 --> 00:28:10
And eventually we'll get down
to the end, but it may take us
469
00:28:10 --> 00:28:12
n steps from the beginning.
470
00:28:12 --> 00:28:18
So the example is
l(n x^ n) dx. .
471
00:28:18 --> 00:28:21
And the claim is that if I do
what I did in Example 2, to
472
00:28:21 --> 00:28:26
this case, I'll get a simpler
one which will involve
473
00:28:26 --> 00:28:28
the n - 1st power.
474
00:28:28 --> 00:28:30
And that way I can get
all the way back down
475
00:28:30 --> 00:28:32
to the final answer.
476
00:28:32 --> 00:28:34
So here's what happens.
477
00:28:34 --> 00:28:36
We take u as ln x^ n.
478
00:28:37 --> 00:28:40
This is the same discussion
as before, v = x.
479
00:28:40 --> 00:28:47
And then u' is n l(n
x) ^ n - 1( 1 / x).
480
00:28:47 --> 00:28:50
And v' is 1.
481
00:28:50 --> 00:28:52
And so the setup is similar.
482
00:28:52 --> 00:28:59
We have here x ( ln x)^
n minus the integral.
483
00:28:59 --> 00:29:05
And there's n times, it turns
out to be (ln x)^ n - 1.
484
00:29:05 --> 00:29:26
And then there's a 1 / x
and an x, which cancel.
485
00:29:26 --> 00:29:32
So I'm going to explain this
also abstractly a little bit
486
00:29:32 --> 00:29:35
just to show you what's
happening here.
487
00:29:35 --> 00:29:44
If you use the notation Fn (x)
is the integral of (ln x)^n dx,
488
00:29:44 --> 00:29:46
and we're going to forget
the constant here.
489
00:29:46 --> 00:29:52
Then the relationship that we
have here is that Fn (x) = n
490
00:29:52 --> 00:29:56
ln, I'm sorry, x (ln x)^ n.
491
00:29:56 --> 00:29:59
That's the first
term over here.
492
00:29:59 --> 00:30:03
Minus n times the
preceding one.
493
00:30:03 --> 00:30:07
This one here.
494
00:30:07 --> 00:30:11
And the idea is that
eventually we can get down.
495
00:30:11 --> 00:30:14
If we start with the nth one,
we have a formula that
496
00:30:14 --> 00:30:17
includes, so the reduction
is to the n - 1st.
497
00:30:17 --> 00:30:21
Then we can reduce to
the n - 2nd, and so on.
498
00:30:21 --> 00:30:23
Until we reduce to
the 1, the first 1.
499
00:30:23 --> 00:30:29
And then in fact we can even
go down to the 0th one.
500
00:30:29 --> 00:30:32
So this is the idea of
a reduction formula.
501
00:30:32 --> 00:30:36
And let me illustrate it
exactly in the context
502
00:30:36 --> 00:30:38
of Examples 1 and 2.
503
00:30:38 --> 00:30:44
So the first step would be
to evaluate the first one.
504
00:30:44 --> 00:30:48
Which is, if you
like, (ln x)^ 0 dx.
505
00:30:48 --> 00:30:52
That's very easy, that's x.
506
00:30:52 --> 00:31:01
And then F1 ( x) =
x ln x - F0 (x).
507
00:31:01 --> 00:31:03
Now, that's applying this rule.
508
00:31:03 --> 00:31:06
So let me just put
it in a box here.
509
00:31:06 --> 00:31:09
This is the method
of induction.
510
00:31:09 --> 00:31:13
Here's the rule.
511
00:31:13 --> 00:31:21
And I'm applying it for n = 1.
512
00:31:21 --> 00:31:23
I plugged in n = 1 here.
513
00:31:23 --> 00:31:32
So here, I have x ln
x ^ 1 - 1 ( F0 ( x).
514
00:31:32 --> 00:31:39
And that's what I put right
here, on the right-hand side.
515
00:31:39 --> 00:31:41
And that's going to generate
for me the formula that I
516
00:31:41 --> 00:31:44
want, which is x ln x - x.
517
00:31:44 --> 00:31:49
That's the answer to
this problem over here.
518
00:31:49 --> 00:31:51
This was Example 1.
519
00:31:51 --> 00:31:53
Notice I dropped the
constants because I can
520
00:31:53 --> 00:31:54
add them in at the end.
521
00:31:54 --> 00:31:57
So I'll put in
parentheses here, (+ c).
522
00:31:57 --> 00:32:01
That's what would happen at
the end of the problem.
523
00:32:01 --> 00:32:08
The next step, so that
was Example 1, and now
524
00:32:08 --> 00:32:12
Example 2 works more
or less the same way.
525
00:32:12 --> 00:32:14
I'm just summarizing what
I did on that blackboard
526
00:32:14 --> 00:32:16
right up here.
527
00:32:16 --> 00:32:21
The same thing, but in much
more compact notation.
528
00:32:21 --> 00:32:25
If I take F2 ( x), that's
going to be equal to x
529
00:32:25 --> 00:32:31
(ln x)^2 - 2 F1 ( x).
530
00:32:31 --> 00:32:41
Again, this is box for n = 2.
531
00:32:41 --> 00:32:46
And if I plug it in, what I'm
getting here is x (ln x) ^2
532
00:32:46 --> 00:32:49
- twice this stuff here.
533
00:32:49 --> 00:32:55
Which is right
here. x ln x - x.
534
00:32:55 --> 00:32:58
If you like, + c.
535
00:32:58 --> 00:33:07
So I'll leave the c off.
536
00:33:07 --> 00:33:12
So this is how reduction
formulas work in general.
537
00:33:12 --> 00:33:22
I'm going to give you one more
example of a reduction formula.
538
00:33:22 --> 00:33:30
So I guess we have to
call this Example 4.
539
00:33:30 --> 00:33:34
Let's be fancy, let's
make it the sine.
540
00:33:34 --> 00:33:35
No no, no, let's
be fancier still.
541
00:33:35 --> 00:33:38
Let's make it e^ x.
542
00:33:38 --> 00:33:48
So this would also work
for cosine x and sine x.
543
00:33:48 --> 00:33:50
The same sort of thing.
544
00:33:50 --> 00:33:53
And I should mention that
on your homework, you have
545
00:33:53 --> 00:33:54
to do it for cosine of x.
546
00:33:54 --> 00:33:56
I decided to change my mind
on the spur of the moment.
547
00:33:56 --> 00:33:58
I'm not going to do it for
cosine because you have
548
00:33:58 --> 00:34:00
to work it out on your
homework for cosine.
549
00:34:00 --> 00:34:03
In a later homework you'll
even do this case.
550
00:34:03 --> 00:34:05
So it's fine.
551
00:34:05 --> 00:34:07
You need the practice.
552
00:34:07 --> 00:34:10
OK, so how am I going
to do it this time.
553
00:34:10 --> 00:34:13
This is again, a
reduction formula.
554
00:34:13 --> 00:34:19
And the trick here is to pick
u to be this function here.
555
00:34:19 --> 00:34:20
And the reason is
the following.
556
00:34:20 --> 00:34:23
So it's very important to pick
which function is the u and
557
00:34:23 --> 00:34:25
which function is the v.
558
00:34:25 --> 00:34:27
That's the only decision you
have to make if you're going to
559
00:34:27 --> 00:34:30
apply integration by parts.
560
00:34:30 --> 00:34:33
When I pick this function as
the u, the advantage that I
561
00:34:33 --> 00:34:38
have is that u' is simpler.
562
00:34:38 --> 00:34:39
How is it simpler?
563
00:34:39 --> 00:34:42
It's simpler because
it's one degree down.
564
00:34:42 --> 00:34:45
So that's making
progress for us.
565
00:34:45 --> 00:34:49
On the other hand, this
function here is going to
566
00:34:49 --> 00:34:52
be what I'll use for v.
567
00:34:52 --> 00:34:55
And if I differentiated that,
if I did it the other way
568
00:34:55 --> 00:34:57
around and I differentiated
that, I would just get the
569
00:34:57 --> 00:34:58
same level of complexity.
570
00:34:58 --> 00:35:01
Differentiating e^x just
gives you back e ^ x.
571
00:35:01 --> 00:35:02
So that's boring.
572
00:35:02 --> 00:35:05
It doesn't make any
progress in this process.
573
00:35:05 --> 00:35:11
And so I'm going to instead
let v = e ^ x and,
574
00:35:11 --> 00:35:12
sorry this is v '.
575
00:35:12 --> 00:35:14
Make it v ' = e ^ x.
576
00:35:14 --> 00:35:15
And then v = e ^x.
577
00:35:15 --> 00:35:17
At least it isn't any
worse when I went
578
00:35:17 --> 00:35:20
backwards like that.
579
00:35:20 --> 00:35:28
So now, I have u and v ', and
now I get (x ^ n)( e ^ x).
580
00:35:28 --> 00:35:30
This again is u, and this is v.
581
00:35:30 --> 00:35:34
So it happens that v = t v ' so
it's a little confusing here.
582
00:35:34 --> 00:35:37
But this is the one
we're calling v '.
583
00:35:37 --> 00:35:38
And here's v.
584
00:35:38 --> 00:35:43
And now minus the integral
and I have here nx ^ n - 1.
585
00:35:43 --> 00:35:45
And I have here e ^ x.
586
00:35:45 --> 00:35:52
So this is u ' and
this is v dx.
587
00:35:52 --> 00:35:55
So this recurrence is
a new recurrence.
588
00:35:55 --> 00:35:57
And let me summarize it here.
589
00:35:57 --> 00:36:00
It's saying that Gn ( x)
should be the integral
590
00:36:00 --> 00:36:05
of (x ^ n)( e ^ x) dx.
591
00:36:05 --> 00:36:06
Again, I'm dropping the c.
592
00:36:06 --> 00:36:17
And then the reduction formula
is that Gn (x) = this
593
00:36:17 --> 00:36:25
expression here, (x ^ n)(
e ^ x) - n Gn - 1 (x).
594
00:36:25 --> 00:36:32
So here's our
reduction formula.
595
00:36:32 --> 00:36:38
And to illustrate this, if I
take G0 (x), if you think
596
00:36:38 --> 00:36:40
about it for a second that's
just, there's nothing here.
597
00:36:40 --> 00:36:44
The antiderivative of e ^ x,
that's going to be e ^ x,
598
00:36:44 --> 00:36:48
that getting started at
the real basement here.
599
00:36:48 --> 00:36:52
Again, as always, 0 is
my favorite number.
600
00:36:52 --> 00:36:52
Not 1.
601
00:36:52 --> 00:36:55
I always start with the
easiest one, if possible.
602
00:36:55 --> 00:37:00
And now G1, applying this
formula, is going to be
603
00:37:00 --> 00:37:06
equal to x e ^ x - G0 ( x).
604
00:37:06 --> 00:37:11
Which is just right, because
n is 1 and n - 1 is 0.
605
00:37:11 --> 00:37:17
And so that's just
x e ^ x - e^ x.
606
00:37:17 --> 00:37:20
So this is a very, very
fancy way of saying
607
00:37:20 --> 00:37:22
the following fact.
608
00:37:22 --> 00:37:32
I'll put it over on
this other board.
609
00:37:32 --> 00:37:44
Which is that the integral of
x e^ x dx = x e^ x - x + c.
610
00:37:44 --> 00:37:45
Yeah, question.
611
00:37:45 --> 00:37:50
STUDENT: [INAUDIBLE]
612
00:37:50 --> 00:37:53
PROFESSOR: The question
is, why is this true.
613
00:37:53 --> 00:37:54
Why is this statement true.
614
00:37:54 --> 00:37:56
Why is G 0 = e^ x.
615
00:37:56 --> 00:37:58
I did that in my head.
616
00:37:58 --> 00:38:02
What I did was, I first wrote
down the formula for G0.
617
00:38:02 --> 00:38:11
Which was G0 is equal to
the integral of e^ x dx.
618
00:38:11 --> 00:38:12
Because there's an x to
the 0 power in there,
619
00:38:12 --> 00:38:15
which is just 1.
620
00:38:15 --> 00:38:17
And then I know the
antiderivative of e ^ x.
621
00:38:17 --> 00:38:23
It's e ^x.
622
00:38:23 --> 00:38:31
STUDENT: [INAUDIBLE]
623
00:38:31 --> 00:38:33
PROFESSOR: How do you know
when this method will work?
624
00:38:33 --> 00:38:37
The answer is only
by experience.
625
00:38:37 --> 00:38:40
You must get practice
doing this.
626
00:38:40 --> 00:38:42
If you look in your
textbook, you'll see
627
00:38:42 --> 00:38:44
hints as to what to do.
628
00:38:44 --> 00:38:47
The other hint that I want to
say is that if you find that
629
00:38:47 --> 00:38:50
you have one factor in your
expression which when you
630
00:38:50 --> 00:38:52
differentiate it,
it gets easier.
631
00:38:52 --> 00:38:56
And when you antidifferentiate
the other half, it doesn't get
632
00:38:56 --> 00:38:58
any worse, then that's
when this method has
633
00:38:58 --> 00:39:01
a chance of helping.
634
00:39:01 --> 00:39:04
And there is there's
no general thing.
635
00:39:04 --> 00:39:08
The thing is, though, if you it
with x^ n (e^ x), x ^ n cosine
636
00:39:08 --> 00:39:11
x, especially on sine x, those
are examples where it works.
637
00:39:11 --> 00:39:15
This power of the ln.
638
00:39:15 --> 00:39:19
I'll give you er one
more example here.
639
00:39:19 --> 00:39:26
So this was G1 ( x), right.
640
00:39:26 --> 00:39:28
I'll give you one more
example in a second.
641
00:39:28 --> 00:39:29
Yeah.
642
00:39:29 --> 00:39:33
STUDENT: [INAUDIBLE]
643
00:39:33 --> 00:39:35
PROFESSOR: Thank you.
644
00:39:35 --> 00:39:38
There's a mistake here.
645
00:39:38 --> 00:39:39
That's bad.
646
00:39:39 --> 00:39:40
I was thinking in the
back of my head of the
647
00:39:40 --> 00:39:45
following formula.
648
00:39:45 --> 00:39:51
Which is another one
which we've just done.
649
00:39:51 --> 00:39:53
So these are the types of
formulas that you can get out
650
00:39:53 --> 00:39:57
of integration by parts.
651
00:39:57 --> 00:40:00
There's also another way of
getting these, which I'm not
652
00:40:00 --> 00:40:02
going to say anything about.
653
00:40:02 --> 00:40:04
Which is called
advance guessing.
654
00:40:04 --> 00:40:06
You guess in advance what the
form is, you differentiate
655
00:40:06 --> 00:40:08
it and you check.
656
00:40:08 --> 00:40:14
That does work too, with
many of these cases.
657
00:40:14 --> 00:40:21
I want to give you
an illustration.
658
00:40:21 --> 00:40:30
Just because you these
formulas are somewhat dry.
659
00:40:30 --> 00:40:34
So I want to give you just
at least one application.
660
00:40:34 --> 00:40:42
We're almost done with the
idea of these formulas.
661
00:40:42 --> 00:40:46
And we're going to get back now
to being able to handle lots
662
00:40:46 --> 00:40:47
more integrals than
we could before.
663
00:40:47 --> 00:40:50
And what's satisfying is that
now we can get numbers out
664
00:40:50 --> 00:40:54
instead of being stuck and
hamstrung with only
665
00:40:54 --> 00:40:55
a few techniques.
666
00:40:55 --> 00:40:57
Now we have all of the
techniques of integration
667
00:40:57 --> 00:40:59
that anybody has.
668
00:40:59 --> 00:41:02
And so we can do pretty
much anything we want
669
00:41:02 --> 00:41:04
that's possible to do.
670
00:41:04 --> 00:41:14
So here's, if you like, an
application that illustrates
671
00:41:14 --> 00:41:18
how integration by
parts can be helpful.
672
00:41:18 --> 00:41:26
And we're going to find the
volume of an exponential
673
00:41:26 --> 00:41:34
wine glass here.
674
00:41:34 --> 00:41:38
Again, don't try
this at home, but.
675
00:41:38 --> 00:41:40
So let's see.
676
00:41:40 --> 00:41:44
It's going to be this
beautiful guy here.
677
00:41:44 --> 00:41:46
I think.
678
00:41:46 --> 00:41:49
OK, so what's it going to be.
679
00:41:49 --> 00:41:52
This graph is going
to be y = e^ x.
680
00:41:52 --> 00:42:04
Then we're going to rotate
it around the y axis.
681
00:42:04 --> 00:42:10
And this level here
is the height y = 1.
682
00:42:10 --> 00:42:12
And the top, let's
say, is y = e.
683
00:42:12 --> 00:42:22
So that the horizontal here,
coming down, is x = 1.
684
00:42:22 --> 00:42:35
Now, there are two ways
to set up this problem.
685
00:42:35 --> 00:42:40
And so there are two methods.
686
00:42:40 --> 00:42:44
And this is also a good review
because, of course, we did
687
00:42:44 --> 00:42:46
this in the last unit.
688
00:42:46 --> 00:42:58
The two methods are horizontal
and vertical slices.
689
00:42:58 --> 00:43:00
Those are the two
ways we can do this.
690
00:43:00 --> 00:43:03
Now, if we do it with,
so let's start out with
691
00:43:03 --> 00:43:09
the horizontal ones.
692
00:43:09 --> 00:43:12
That's this shape here.
693
00:43:12 --> 00:43:15
And we're going like that.
694
00:43:15 --> 00:43:19
And the horizontal slices
mean that this little bit
695
00:43:19 --> 00:43:22
here is a thickness dy.
696
00:43:22 --> 00:43:24
And then we're going
to wrap that around.
697
00:43:24 --> 00:43:30
So this is going
to become a disk.
698
00:43:30 --> 00:43:34
This is the method of disks.
699
00:43:34 --> 00:43:35
And what's this distance here?
700
00:43:35 --> 00:43:37
Well, this place is x.
701
00:43:37 --> 00:43:40
And so the disk
has area pi x ^2.
702
00:43:40 --> 00:43:43
And we're going to add
up the thickness of the
703
00:43:43 --> 00:43:45
disks and we're going to
integrate from 1 to e.
704
00:43:45 --> 00:43:51
So here's our volume.
705
00:43:51 --> 00:43:54
And now we have one last little
item of business before we
706
00:43:54 --> 00:43:56
can evaluate this integral.
707
00:43:56 --> 00:43:58
And that is that we need to
know the relationship here on
708
00:43:58 --> 00:44:01
the curve, that y = e ^ x.
709
00:44:01 --> 00:44:07
So that means x = ln y.
710
00:44:07 --> 00:44:10
And in order to evaluate this
integral, we have to evaluate x
711
00:44:10 --> 00:44:13
correctly as a function of y.
712
00:44:13 --> 00:44:26
So that's the integral from 1
to e of (ln y)^2, times pi, dy.
713
00:44:26 --> 00:44:28
So now you see that this
is an integral that we
714
00:44:28 --> 00:44:30
did calculate already.
715
00:44:30 --> 00:44:34
And in fact, it's
sitting right here.
716
00:44:34 --> 00:44:37
Except with the variable x
instead of the variable y.
717
00:44:37 --> 00:44:44
So the answer, which we already
had, is this F2 ( y) here.
718
00:44:44 --> 00:44:47
So maybe I'll write
it that way.
719
00:44:47 --> 00:44:52
So this is F2 (y)
between 1 and e.
720
00:44:52 --> 00:45:00
And now let's figure
out what it is.
721
00:45:00 --> 00:45:02
It's written over there.
722
00:45:02 --> 00:45:15
It's y (ln y) ^2
- 2(y ln y - y).
723
00:45:15 --> 00:45:24
The whole thing
evaluated at 1e.
724
00:45:24 --> 00:45:29
And that is, if I plug
in e here, I get e.
725
00:45:29 --> 00:45:32
Except there's a factor
of pi there, sorry.
726
00:45:32 --> 00:45:36
Missed the pi factor.
727
00:45:36 --> 00:45:38
So there's an e here.
728
00:45:38 --> 00:45:43
And then I subtract off,
well, at 1 this is e - e.
729
00:45:43 --> 00:45:44
So it cancels.
730
00:45:44 --> 00:45:45
There's nothing left.
731
00:45:45 --> 00:45:50
And then at 1, I get ln 1 is
0, ln 1 is 0, there's only
732
00:45:50 --> 00:45:53
one term left, which is 2.
733
00:45:53 --> 00:45:55
So it's - 2.
734
00:45:55 --> 00:46:03
That's the answer.
735
00:46:03 --> 00:46:11
Now we get to compare that
with what happens if we
736
00:46:11 --> 00:46:15
do it the other way.
737
00:46:15 --> 00:46:19
So what's the vertical?
738
00:46:19 --> 00:46:31
So by vertical slicing,
we get shells.
739
00:46:31 --> 00:46:38
And that starts, that's
in the x variable.
740
00:46:38 --> 00:46:43
It starts at 0 and ends
at 1 and it's dx.
741
00:46:43 --> 00:46:46
And what are the shells?
742
00:46:46 --> 00:46:51
Well, the shells are, if I can
draw the picture again, they
743
00:46:51 --> 00:46:55
start, the top value is e.
744
00:46:55 --> 00:47:02
And the bottom value is, I need
a little bit of room for this.
745
00:47:02 --> 00:47:06
The bottom value is y.
746
00:47:06 --> 00:47:12
And then we have 2 pi x
is the circumference, as
747
00:47:12 --> 00:47:15
we sweep it around dx.
748
00:47:15 --> 00:47:18
So here's our new volume.
749
00:47:18 --> 00:47:23
Expressed in this
different way.
750
00:47:23 --> 00:47:26
So now I'm going to
plug in what this is.
751
00:47:26 --> 00:47:30
It's the integral from
0 to 1 of e - e ^ x.
752
00:47:30 --> 00:47:32
That's the formula for y.
753
00:47:32 --> 00:47:36
2 pi x dx.
754
00:47:36 --> 00:47:39
And what you see is that
you get the integral from
755
00:47:39 --> 00:47:45
0 to 1 of 2 pi e x dx.
756
00:47:45 --> 00:47:46
That's easy, right?
757
00:47:46 --> 00:47:51
That's just 2 pi e ( 1/2).
758
00:47:51 --> 00:47:54
This one is just the
area of a triangle.
759
00:47:54 --> 00:47:56
If I factor out the 2 pi e.
760
00:47:56 --> 00:48:03
And then the other piece is the
integral of 2 pi x e^ x dx.
761
00:48:03 --> 00:48:08
From 0 to 1.
762
00:48:08 --> 00:48:11
STUDENT: [INAUDIBLE]
763
00:48:11 --> 00:48:14
PROFESSOR: Are you asking me
whether I need an x ^2 here?
764
00:48:14 --> 00:48:15
I just evaluated the integral.
765
00:48:15 --> 00:48:17
I just did it geometrically.
766
00:48:17 --> 00:48:19
I said, this is the
area of a triangle.
767
00:48:19 --> 00:48:22
I didn't antidifferentiate
and evaluate it, I just
768
00:48:22 --> 00:48:23
told you the number.
769
00:48:23 --> 00:48:27
Because it's a
definite integral.
770
00:48:27 --> 00:48:33
So now, this one here, I can
read off from right up here.
771
00:48:33 --> 00:48:37
Above it, this is G1.
772
00:48:37 --> 00:48:42
So this is equal to,
let's check it out here.
773
00:48:42 --> 00:48:52
So this is pi e, right, - 2 pi
G1 ( x), evaluated at 0 and 1.
774
00:48:52 --> 00:48:54
So let's make sure that it's
the same as what we had before.
775
00:48:54 --> 00:48:59
It's pi e - 2 pi
times here's g1.
776
00:48:59 --> 00:49:03
So it's x e ^ x - e^ x.
777
00:49:03 --> 00:49:05
So at x = 1, that cancels.
778
00:49:05 --> 00:49:08
But at the bottom
end, it's e^ 0.
779
00:49:08 --> 00:49:12
So it's - 1 here.
780
00:49:12 --> 00:49:13
Is that right?
781
00:49:13 --> 00:49:13
Yep.
782
00:49:13 --> 00:49:17
So it's pi e - 2.
783
00:49:17 --> 00:49:21
It's the same.
784
00:49:21 --> 00:49:22
Question.
785
00:49:22 --> 00:49:28
STUDENT: [INAUDIBLE]
786
00:49:28 --> 00:49:33
PROFESSOR: From here to
here, is that the question?
787
00:49:33 --> 00:49:39
STUDENT: [INAUDIBLE]
788
00:49:39 --> 00:49:43
PROFESSOR: So the step here is
just the distributive law.
789
00:49:43 --> 00:49:46
This is e 2 pi x,
that's this term.
790
00:49:46 --> 00:49:49
And the other terms, the
minus sign is outside.
791
00:49:49 --> 00:49:51
The 2 pi I factored out.
792
00:49:51 --> 00:49:56
And the x and the e ^x stayed
inside the integral sign.
793
00:49:56 --> 00:49:59
Thank you.
794
00:49:59 --> 00:50:01
The correction is that
there was a missing
795
00:50:01 --> 00:50:03
minus sign, last time.
796
00:50:03 --> 00:50:13
When I integrated from 0 to 1 x
e^ x dx, I had a x e^ x - e^x.
797
00:50:13 --> 00:50:15
Evaluated at 0 and 1.
798
00:50:15 --> 00:50:18
And that's equal to + 1.
799
00:50:18 --> 00:50:21
I was missing this minus sign.
800
00:50:21 --> 00:50:30
The place where it came in was
in this wineglass example.
801
00:50:30 --> 00:50:39
We had the integral of
2 pi x e - e ^x dx.
802
00:50:39 --> 00:50:48
And that was 2 pi e integral of
x dx, from 0 to 1, - 2 pi,
803
00:50:48 --> 00:50:52
integral from 0 to
1 of x e^x dx.
804
00:50:52 --> 00:50:58
And then I worked this
out and it was pi e.
805
00:50:58 --> 00:51:03
And then this one was - 2 pi,
and what I wrote down was - 1.
806
00:51:03 --> 00:51:05
But there should have been
an extra minus sign there.
807
00:51:05 --> 00:51:08
So it's this.
808
00:51:08 --> 00:51:11
The final answer was
correct, but this minus
809
00:51:11 --> 00:51:13
sign was missing.
810
00:51:13 --> 00:51:16
Right there.
811
00:51:16 --> 00:51:20
So just, right there.
812
00:51:20 --> 00:51:23