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PROFESSOR: Now,
to start out today
00:00:25.240 --> 00:00:27.850
we're going to finish up
what we did last time.
00:00:27.850 --> 00:00:30.950
Which has to do with
partial fractions.
00:00:30.950 --> 00:00:33.160
I told you how to
do partial fractions
00:00:33.160 --> 00:00:35.080
in several special
cases and everybody
00:00:35.080 --> 00:00:37.920
was trying to figure out
what the general picture was.
00:00:37.920 --> 00:00:39.180
But I'd like to lay that out.
00:00:39.180 --> 00:00:41.730
I'll still only do
it for an example.
00:00:41.730 --> 00:00:43.810
But it will be somehow
a bigger example
00:00:43.810 --> 00:00:53.310
so that you can see what
the general pattern is.
00:00:53.310 --> 00:01:04.280
Partial fractions, remember,
is a method for breaking up
00:01:04.280 --> 00:01:06.730
so-called rational functions.
00:01:06.730 --> 00:01:09.510
Which are ratios of polynomials.
00:01:09.510 --> 00:01:13.110
And it shows you that you
can always integrate them.
00:01:13.110 --> 00:01:14.920
That's really the theme here.
00:01:14.920 --> 00:01:21.190
And this is what's reassuring
is that it always works.
00:01:21.190 --> 00:01:23.670
That's really the bottom line.
00:01:23.670 --> 00:01:26.550
And that's good
because there are
00:01:26.550 --> 00:01:34.100
a lot of integrals that don't
have formulas and these do.
00:01:34.100 --> 00:01:35.560
It always works.
00:01:35.560 --> 00:01:43.340
But, maybe with lots of help.
00:01:43.340 --> 00:01:46.257
So maybe slowly.
00:01:46.257 --> 00:01:47.840
Now, there's a little
bit of bad news,
00:01:47.840 --> 00:01:50.820
and I have to be totally
honest and tell you
00:01:50.820 --> 00:01:52.230
what all the bad news is.
00:01:52.230 --> 00:01:54.930
Along with the good news.
00:01:54.930 --> 00:02:00.070
The first step, which maybe
I should be calling Step 0,
00:02:00.070 --> 00:02:08.710
I had a Step 1, 2 and 3
last time, is long division.
00:02:08.710 --> 00:02:11.790
That's the step where you
take your polynomial divided
00:02:11.790 --> 00:02:17.440
by your other polynomial,
and you find the quotient
00:02:17.440 --> 00:02:22.470
plus some remainder.
00:02:22.470 --> 00:02:24.370
And you do that
by long division.
00:02:24.370 --> 00:02:27.520
And the quotient is easy
to take the antiderivative
00:02:27.520 --> 00:02:30.010
of because it's
just a polynomial.
00:02:30.010 --> 00:02:32.380
And the key extra
property here is
00:02:32.380 --> 00:02:35.680
that the degree of the numerator
now over here, this remainder,
00:02:35.680 --> 00:02:40.510
is strictly less than the
degree of the denominator.
00:02:40.510 --> 00:02:44.140
So that you can
do the next step.
00:02:44.140 --> 00:02:48.900
Now, the next step which I
called Step 1 last time, that's
00:02:48.900 --> 00:02:52.290
great imagination, it's
right after Step 0, Step 1
00:02:52.290 --> 00:02:54.930
was to factor the denominator.
00:02:54.930 --> 00:03:00.630
And I'm going to illustrate by
example what the setup is here.
00:03:00.630 --> 00:03:09.120
I don't know maybe,
we'll do this.
00:03:09.120 --> 00:03:12.920
Some polynomial here,
maybe cube this one.
00:03:12.920 --> 00:03:21.440
So here I've factored
the denominator.
00:03:21.440 --> 00:03:24.840
That's what I called
Step 1 last time.
00:03:24.840 --> 00:03:27.650
Now, here's the first
piece of bad news.
00:03:27.650 --> 00:03:31.620
In reality, if somebody
gave you a multiplied
00:03:31.620 --> 00:03:35.530
out degree-whatever
polynomial here,
00:03:35.530 --> 00:03:40.710
you would be very hard
pressed to factor it.
00:03:40.710 --> 00:03:44.132
A lot of them are extremely
difficult to factor.
00:03:44.132 --> 00:03:46.590
And so that's something you
would have to give to a machine
00:03:46.590 --> 00:03:47.940
to do.
00:03:47.940 --> 00:03:50.880
And it's just basically
a hard problem.
00:03:50.880 --> 00:03:54.050
So obviously, we're only
going to give you ones
00:03:54.050 --> 00:03:55.480
that you can do by hand.
00:03:55.480 --> 00:03:58.060
So very low degree examples.
00:03:58.060 --> 00:03:59.320
And that's just the way it is.
00:03:59.320 --> 00:04:03.600
So this is really a hard step
in disguise, in real life.
00:04:03.600 --> 00:04:06.000
Anyway, we're just going
to take it as given.
00:04:06.000 --> 00:04:07.820
And we have this
numerator, which
00:04:07.820 --> 00:04:10.720
is of degree less
than the denominator.
00:04:10.720 --> 00:04:14.830
So let's count up what
its degree has to be.
00:04:14.830 --> 00:04:18.640
This is 4 + 2 + 6.
00:04:18.640 --> 00:04:22.440
So this is degree 4 + 2 + 6.
00:04:22.440 --> 00:04:24.190
I added that up because
this is degree 4,
00:04:24.190 --> 00:04:28.600
this is degree 2 and
(x^2)^3 is the 6th power.
00:04:28.600 --> 00:04:32.330
So all together it's
this, which is 12.
00:04:32.330 --> 00:04:39.822
And so this thing
is of degree <= 11.
00:04:39.822 --> 00:04:41.280
All the way up to
degree 11, that's
00:04:41.280 --> 00:04:44.240
the possibilities for
the numerator here.
00:04:44.240 --> 00:04:49.810
Now, the extra information that
I want to impart right now,
00:04:49.810 --> 00:04:56.450
is just this setup.
00:04:56.450 --> 00:04:58.990
Which I called Step 2 last time.
00:04:58.990 --> 00:05:05.260
And the setup is this.
00:05:05.260 --> 00:05:07.590
Now, it's going to take
us a while to do this.
00:05:07.590 --> 00:05:10.810
We have this factor here.
00:05:10.810 --> 00:05:12.450
We have another factor.
00:05:12.450 --> 00:05:14.620
We have another term,
with the square.
00:05:14.620 --> 00:05:18.220
We have another
term with the cube.
00:05:18.220 --> 00:05:22.581
We have another term
with the fourth power.
00:05:22.581 --> 00:05:24.330
So this is what's going
to happen whenever
00:05:24.330 --> 00:05:25.530
you have linear factors.
00:05:25.530 --> 00:05:28.560
You'll have a collection
of terms like this.
00:05:28.560 --> 00:05:31.030
So you have four
constants to take care of.
00:05:31.030 --> 00:05:34.550
Now, with a quadratic
in the denominator,
00:05:34.550 --> 00:05:36.840
you need a linear
function in the numerator.
00:05:36.840 --> 00:05:41.230
So that's, if you like,
B_0 x + C_0 divided
00:05:41.230 --> 00:05:49.370
by this quadratic term here.
00:05:49.370 --> 00:05:52.500
And what I didn't
show you last time
00:05:52.500 --> 00:05:59.720
was how you deal with higher
powers of quadratic terms.
00:05:59.720 --> 00:06:04.090
So when you have a quadratic
term, what's going to happen
00:06:04.090 --> 00:06:07.370
is you're going to take
that first factor here.
00:06:07.370 --> 00:06:11.870
Just the way you
did in this case.
00:06:11.870 --> 00:06:15.500
But then you're going to
have to do the same thing
00:06:15.500 --> 00:06:24.120
with the next power.
00:06:24.120 --> 00:06:27.880
Now notice, just as in
the case of this top row,
00:06:27.880 --> 00:06:29.830
I have just a constant here.
00:06:29.830 --> 00:06:33.030
And even though I increased
the degree of the denominator
00:06:33.030 --> 00:06:34.540
I'm not increasing
the numerator.
00:06:34.540 --> 00:06:35.970
It's staying just a constant.
00:06:35.970 --> 00:06:38.230
It's not linear up here.
00:06:38.230 --> 00:06:39.850
It's better than that.
00:06:39.850 --> 00:06:41.990
It's just a constant.
00:06:41.990 --> 00:06:44.130
And here it stayed a constant.
00:06:44.130 --> 00:06:45.630
And here it stayed a constant.
00:06:45.630 --> 00:06:48.070
Similarly here, even
though I'm increasing
00:06:48.070 --> 00:06:49.620
the degree of the
denominator, I'm
00:06:49.620 --> 00:06:52.810
leaving the numerator, the
form of the numerator, alone.
00:06:52.810 --> 00:06:55.150
It's just a linear factor
and a linear factor.
00:06:55.150 --> 00:07:05.350
So that's the key
to this pattern.
00:07:05.350 --> 00:07:09.850
I don't have quite as
jazzy a song on mine.
00:07:09.850 --> 00:07:13.410
So this is so long that it
runs off the blackboard here.
00:07:13.410 --> 00:07:15.820
So let's continue
it on the next.
00:07:15.820 --> 00:07:20.200
We've got this B_2
x + C_2-- sorry,
00:07:20.200 --> 00:07:23.150
(B_3 x + C_3) / (x^2 + 4)^3.
00:07:26.590 --> 00:07:38.590
I guess I have room
for it over here.
00:07:38.590 --> 00:07:41.120
I'm going to talk about
this in just a second.
00:07:41.120 --> 00:07:43.590
Alright, so here's the pattern.
00:07:43.590 --> 00:07:51.270
Now, let me just do a count
of the number of unknowns
00:07:51.270 --> 00:07:52.344
we have here.
00:07:52.344 --> 00:07:54.010
The number of unknowns
that we have here
00:07:54.010 --> 00:07:58.750
is 1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11, 12.
00:07:58.750 --> 00:08:00.930
That 12 is no coincidence.
00:08:00.930 --> 00:08:03.815
That's the degree
of the polynomial.
00:08:03.815 --> 00:08:05.690
And it's the number of
unknowns that we have.
00:08:05.690 --> 00:08:08.060
And it's the number
of degrees of freedom
00:08:08.060 --> 00:08:11.215
in a polynomial of degree 11.
00:08:11.215 --> 00:08:13.090
If you have all these
free coefficients here,
00:08:13.090 --> 00:08:17.560
you have the coefficient x^0,
x^1, all the way up to x^ 11.
00:08:17.560 --> 00:08:23.100
And 0 through 11 is 12
different coefficients.
00:08:23.100 --> 00:08:26.380
And so this is a very
complicated system
00:08:26.380 --> 00:08:28.150
of equations for unknowns.
00:08:28.150 --> 00:08:33.259
This is twelve equations
for twelve unknowns.
00:08:33.259 --> 00:08:34.800
So I'll get rid of
this for a second.
00:08:34.800 --> 00:08:41.090
So we have twelve
equations, twelve unknowns.
00:08:41.090 --> 00:08:43.830
So that's the other bad news.
00:08:43.830 --> 00:08:46.020
Machines handle this very
well, but human beings
00:08:46.020 --> 00:08:47.630
have a little trouble with 12.
00:08:47.630 --> 00:08:51.570
Now, the cover-up
method works very neatly
00:08:51.570 --> 00:08:53.730
and picks out this term here.
00:08:53.730 --> 00:08:54.470
But that's it.
00:08:54.470 --> 00:08:56.550
So it reduces it to an 11 by 11.
00:08:56.550 --> 00:09:00.170
You'll be able to
evaluate this in no time.
00:09:00.170 --> 00:09:00.890
But that's it.
00:09:00.890 --> 00:09:04.240
That's the only simplification
of your previous method.
00:09:04.240 --> 00:09:06.250
We don't have a method for this.
00:09:06.250 --> 00:09:08.510
So I'm just showing what
the whole method looks
00:09:08.510 --> 00:09:10.343
like but really you'd
have to have a machine
00:09:10.343 --> 00:09:14.760
to implement this once it
gets to be any size at all.
00:09:14.760 --> 00:09:15.490
Yeah, question.
00:09:15.490 --> 00:09:18.060
STUDENT: [INAUDIBLE]
00:09:18.060 --> 00:09:22.510
PROFESSOR: It's
one big equation,
00:09:22.510 --> 00:09:24.920
but it's a polynomial equation.
00:09:24.920 --> 00:09:32.530
So there's an equation, there's
this function R(x) = a_11 x^11
00:09:32.530 --> 00:09:37.610
+ a_10 x^10...
00:09:37.610 --> 00:09:41.350
and these things are known.
00:09:41.350 --> 00:09:43.495
This is a known expression here.
00:09:43.495 --> 00:09:46.620
And then when you cross-multiply
on the other side,
00:09:46.620 --> 00:09:51.330
what you have is,
well, it's A_1 times--
00:09:51.330 --> 00:09:54.400
If you cancel this
denominator with that,
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
00:10:05.330 --> 00:10:08.260
the term for A_2, etc.
00:10:08.260 --> 00:10:10.350
It's a monster equation.
00:10:10.350 --> 00:10:12.620
And then to separate it out
into separate equations,
00:10:12.620 --> 00:10:19.225
you take the coefficient
on x^11, x^10, ...
00:10:19.225 --> 00:10:21.670
all the way down to x^0.
00:10:21.670 --> 00:10:27.280
And all told, that means there
are a total of 12 equations.
00:10:27.280 --> 00:10:31.030
11 through 0 is 12 equations.
00:10:31.030 --> 00:10:34.727
Yeah, another question.
00:10:34.727 --> 00:10:35.560
STUDENT: [INAUDIBLE]
00:10:35.560 --> 00:10:37.777
PROFESSOR: Should I
write down rest of this?
00:10:37.777 --> 00:10:38.610
STUDENT: [INAUDIBLE]
00:10:38.610 --> 00:10:40.820
PROFESSOR: Should you
write down all this stuff?
00:10:40.820 --> 00:10:43.950
Well, that's a good question.
00:10:43.950 --> 00:10:46.070
So you notice I
didn't write it down.
00:10:46.070 --> 00:10:47.300
Why didn't I write it down?
00:10:47.300 --> 00:10:50.200
Because it's incredibly long.
00:10:50.200 --> 00:10:54.104
In fact, you probably-- So
how many pages of writing
00:10:54.104 --> 00:10:54.770
would this take?
00:10:54.770 --> 00:10:56.210
This is about a page of writing.
00:10:56.210 --> 00:10:58.950
So just think of you're
a machine, how much time
00:10:58.950 --> 00:11:01.660
you want to spend on this.
00:11:01.660 --> 00:11:05.430
So the answer is that
you have to be realistic.
00:11:05.430 --> 00:11:07.420
You're a human
being, not a machine.
00:11:07.420 --> 00:11:10.070
And so there's certain things
that you can write down
00:11:10.070 --> 00:11:12.510
and other things you should
not attempt to write down.
00:11:12.510 --> 00:11:17.770
So do not do this at home.
00:11:17.770 --> 00:11:21.180
So that's the first
down-side to this method.
00:11:21.180 --> 00:11:24.350
It gets more and more
complicated as time goes on.
00:11:24.350 --> 00:11:27.280
The second down-side, I
want to point out to you,
00:11:27.280 --> 00:11:35.100
is what's happening
with the pieces.
00:11:35.100 --> 00:11:42.830
So the pieces still
need to be integrated.
00:11:42.830 --> 00:11:48.130
We simplified this problem,
but we didn't get rid of it.
00:11:48.130 --> 00:11:50.890
We still have the problem
of integrating the pieces.
00:11:50.890 --> 00:11:52.740
Now, some of the
pieces are very easy.
00:11:52.740 --> 00:11:55.540
This top row here, the
antiderivatives of these,
00:11:55.540 --> 00:11:59.150
you can just write down.
00:11:59.150 --> 00:12:01.390
By advanced guessing.
00:12:01.390 --> 00:12:04.300
I'm going to skip over to the
most complicated one over here.
00:12:04.300 --> 00:12:06.220
For one second here.
00:12:06.220 --> 00:12:09.240
And what is it that you'd have
to deal with for that one.
00:12:09.240 --> 00:12:11.810
You'd have to deal
with, for example,
00:12:11.810 --> 00:12:21.660
so e.g., for example, I
need to deal with this guy.
00:12:21.660 --> 00:12:26.590
I've got to get this
antiderivative here.
00:12:26.590 --> 00:12:28.972
Now, this one you're
supposed to be able to know.
00:12:28.972 --> 00:12:30.430
So this is why I'm
mentioning this.
00:12:30.430 --> 00:12:33.270
Because this kind of
ingredient is something
00:12:33.270 --> 00:12:34.860
you already covered.
00:12:34.860 --> 00:12:35.740
And what is it?
00:12:35.740 --> 00:12:39.060
Well, you do this one
by advanced guessing,
00:12:39.060 --> 00:12:42.000
although you learned it as
the method of substitution.
00:12:42.000 --> 00:12:47.900
You realize that it's going to
be of the form (x^2 + 4)^(-2),
00:12:47.900 --> 00:12:49.400
roughly speaking.
00:12:49.400 --> 00:12:51.170
And now we're going to fix that.
00:12:51.170 --> 00:12:53.950
Because if you differentiate
it you get 2x times the -2,
00:12:53.950 --> 00:12:56.410
that's -4 times x times this.
00:12:56.410 --> 00:12:58.370
There's an x in
the numerator here.
00:12:58.370 --> 00:13:02.600
So it's -1/4 of that
will fix the factor.
00:13:02.600 --> 00:13:06.550
And here's the
answer for that one.
00:13:06.550 --> 00:13:10.560
So that's one you can do.
00:13:10.560 --> 00:13:19.020
The second piece is this guy.
00:13:19.020 --> 00:13:20.480
This is the other piece.
00:13:20.480 --> 00:13:25.680
Now, this was the piece
that came from B_3.
00:13:25.680 --> 00:13:27.170
This is the one
that came from B_3.
00:13:27.170 --> 00:13:30.430
And this is the one
that's coming from C_3.
00:13:30.430 --> 00:13:32.360
This is coming from C_3.
00:13:32.360 --> 00:13:35.000
We still need to get
this one out there.
00:13:35.000 --> 00:13:37.630
So C_3 times that will
be the correct answer,
00:13:37.630 --> 00:13:40.960
once we've found these numbers.
00:13:40.960 --> 00:13:44.160
So how do we do this?
00:13:44.160 --> 00:13:45.490
How's this one integrated?
00:13:45.490 --> 00:13:49.690
STUDENT: Trig substitution?
00:13:49.690 --> 00:13:51.640
PROFESSOR: Trig substitution.
00:13:51.640 --> 00:13:57.630
So the trig substitution
here is x = 2 tan u.
00:13:57.630 --> 00:14:00.800
Or 2 tan theta.
00:14:00.800 --> 00:14:03.860
And when you do that, there are
a couple of simplifications.
00:14:03.860 --> 00:14:06.450
Well, I wouldn't call
this a simplification.
00:14:06.450 --> 00:14:14.830
This is just the differentiation
formula. dx = 2 sec^2 u du.
00:14:14.830 --> 00:14:19.030
And then you have to plug in,
and you're using the fact that
00:14:19.030 --> 00:14:22.960
when you plug in the
tan^2, 4 tan ^2 + 4,
00:14:22.960 --> 00:14:24.370
you'll get a secant squared.
00:14:24.370 --> 00:14:32.240
So altogether, this
thing is, 2 sec^2 u du.
00:14:32.240 --> 00:14:40.300
And then there's a (4 sec^2
u)^3, in the denominator.
00:14:40.300 --> 00:14:44.340
So that's what happens when
you change variables here.
00:14:44.340 --> 00:14:46.790
And now look, this
keeps on going.
00:14:46.790 --> 00:14:49.120
This is not the
end of the problem.
00:14:49.120 --> 00:14:50.790
Because what does
that simplify to?
00:14:50.790 --> 00:14:57.860
That is, let's see, it's
2/64, the integral of sec^6
00:14:57.860 --> 00:14:58.620
and sec^2.
00:14:58.620 --> 00:15:00.090
That's the same as cos^4.
00:15:04.300 --> 00:15:06.370
And now, you did a
trig substitution
00:15:06.370 --> 00:15:11.140
but you still have
a trig integral.
00:15:11.140 --> 00:15:15.540
The trig integral now,
there's a method for this.
00:15:15.540 --> 00:15:18.730
The method for this is
when it's an even power,
00:15:18.730 --> 00:15:22.280
you have to use the
double angle formula.
00:15:22.280 --> 00:15:31.890
So that's this guy here.
00:15:31.890 --> 00:15:33.470
And you're still not done.
00:15:33.470 --> 00:15:35.040
You have to square
this thing out.
00:15:35.040 --> 00:15:37.120
And then you'll still
get a cos^2 (2u).
00:15:37.120 --> 00:15:38.160
And it keeps on going.
00:15:38.160 --> 00:15:41.737
So this thing goes
on for a long time.
00:15:41.737 --> 00:15:43.320
But I'm not even
going to finish this,
00:15:43.320 --> 00:15:44.780
but I just want to show you.
00:15:44.780 --> 00:15:46.450
The point is, we're
not showing you how
00:15:46.450 --> 00:15:48.270
to do any complicated problem.
00:15:48.270 --> 00:15:50.550
We're just showing you all
the little ingredients.
00:15:50.550 --> 00:15:52.050
And you have to
string them together
00:15:52.050 --> 00:15:56.170
a long, long, long process to
get to the final answer of one
00:15:56.170 --> 00:15:57.680
of these questions.
00:15:57.680 --> 00:16:07.300
So it always works,
but maybe slowly.
00:16:07.300 --> 00:16:13.440
By the way, there's even another
horrible thing that happens.
00:16:13.440 --> 00:16:22.669
Which is, if you handle this
guy here, what's the technique.
00:16:22.669 --> 00:16:24.460
This is another technique
that you learned,
00:16:24.460 --> 00:16:28.770
supposedly within
the last few days.
00:16:28.770 --> 00:16:30.810
Completing the square.
00:16:30.810 --> 00:16:39.020
So this, it turns out, you
have to rewrite it this way.
00:16:39.020 --> 00:16:42.530
And then the evaluation is going
to be expressed in terms of,
00:16:42.530 --> 00:16:44.440
I'm going to jump to the end.
00:16:44.440 --> 00:16:49.310
It's going to turn out to be
expressed in terms of this.
00:16:49.310 --> 00:16:53.940
That's what will eventually
show up in the formula.
00:16:53.940 --> 00:16:56.370
And not only that,
but if you deal
00:16:56.370 --> 00:16:59.890
with ones involving
x as well, you'll
00:16:59.890 --> 00:17:07.420
also need to deal with something
like log of this denominator
00:17:07.420 --> 00:17:09.560
here.
00:17:09.560 --> 00:17:13.120
So all of these things
will be involved.
00:17:13.120 --> 00:17:16.700
So now, the last message that
I have for you is just this.
00:17:16.700 --> 00:17:18.150
This thing is very complicated.
00:17:18.150 --> 00:17:20.150
We're certainly never
going to ask you to do it.
00:17:20.150 --> 00:17:23.160
But you should just be aware
that this level of complexity,
00:17:23.160 --> 00:17:26.690
we are absolutely stuck
with in this problem.
00:17:26.690 --> 00:17:29.580
And the reason why
we're stuck with it
00:17:29.580 --> 00:17:36.160
is that this is what the
formulas look like in the end.
00:17:36.160 --> 00:17:39.130
If the answers look
like this, the formulas
00:17:39.130 --> 00:17:41.045
have to be this complicated.
00:17:41.045 --> 00:17:43.170
If you differentiate this,
you get your polynomial,
00:17:43.170 --> 00:17:44.254
your ratio of polynomials.
00:17:44.254 --> 00:17:46.794
If you differentiate this, you
get some ratio of polynomials.
00:17:46.794 --> 00:17:48.480
These are the
things that come up
00:17:48.480 --> 00:17:51.710
when you take antiderivatives
of those rational functions.
00:17:51.710 --> 00:17:56.080
So we're just stuck
with these guys.
00:17:56.080 --> 00:17:58.770
And so don't let it
get to you too much.
00:17:58.770 --> 00:17:59.770
I mean, it's not so bad.
00:17:59.770 --> 00:18:01.510
In fact, there are
computer programs
00:18:01.510 --> 00:18:03.510
that will do this for
you anytime you want.
00:18:03.510 --> 00:18:05.800
And you just have to be
not intimidated by them.
00:18:05.800 --> 00:18:10.260
They're like other functions.
00:18:10.260 --> 00:18:20.600
OK, that's it for the general
comments on partial fractions.
00:18:20.600 --> 00:18:24.215
Now we're going to change
subjects to our last technique.
00:18:24.215 --> 00:18:25.840
This is one more
technical thing to get
00:18:25.840 --> 00:18:27.540
you familiar with functions.
00:18:27.540 --> 00:18:32.260
And this technique is
called integration by parts.
00:18:32.260 --> 00:18:34.580
Please, just because
its name sort
00:18:34.580 --> 00:18:35.957
of sounds like
partial fractions,
00:18:35.957 --> 00:18:37.290
don't think it's the same thing.
00:18:37.290 --> 00:18:38.450
It's totally different.
00:18:38.450 --> 00:18:44.340
It's not the same.
00:18:44.340 --> 00:19:06.640
So this one is called
integration by parts.
00:19:06.640 --> 00:19:09.570
Now, unlike the previous case,
where I couldn't actually
00:19:09.570 --> 00:19:12.736
justify to you that the
linear algebra always works.
00:19:12.736 --> 00:19:14.860
I claimed it worked, but
I wasn't able to prove it.
00:19:14.860 --> 00:19:17.390
That's a complicated
theorem which I'm not
00:19:17.390 --> 00:19:19.560
able to do in this class.
00:19:19.560 --> 00:19:22.270
Here I can explain to
you what's going on
00:19:22.270 --> 00:19:24.200
with integration by parts.
00:19:24.200 --> 00:19:26.600
It's just the fundamental
theorem of calculus,
00:19:26.600 --> 00:19:30.430
if you like, coupled
with the product formula.
00:19:30.430 --> 00:19:33.740
Sort of unwound and
read in reverse.
00:19:33.740 --> 00:19:35.610
And here's how that works.
00:19:35.610 --> 00:19:38.480
If you take the product of two
functions and you differentiate
00:19:38.480 --> 00:19:41.910
them, then we know that the
product rule says that this is
00:19:41.910 --> 00:19:45.790
u'v + uv'.
00:19:45.790 --> 00:19:50.400
And now I'm just going to
rearrange in the following way.
00:19:50.400 --> 00:19:53.370
I'm going to solve for uv'.
00:19:53.370 --> 00:19:54.710
That is, this term here.
00:19:54.710 --> 00:19:56.370
So what is this term?
00:19:56.370 --> 00:19:59.990
It's this other term, (uv)'.
00:19:59.990 --> 00:20:04.520
Minus the other piece.
00:20:04.520 --> 00:20:08.360
So I just rewrote this equation.
00:20:08.360 --> 00:20:10.900
And now I'm going
to integrate it.
00:20:10.900 --> 00:20:11.860
So here's the formula.
00:20:11.860 --> 00:20:15.160
The integral of
the left-hand side
00:20:15.160 --> 00:20:17.200
is equal to the integral
of the right-hand side.
00:20:17.200 --> 00:20:18.670
Well when I integrate
a derivative,
00:20:18.670 --> 00:20:21.070
of I get back the
function itself.
00:20:21.070 --> 00:20:27.010
That's the fundamental theorem.
00:20:27.010 --> 00:20:27.600
So this is it.
00:20:27.600 --> 00:20:30.500
Sorry, I missed the
dx, which is important.
00:20:30.500 --> 00:20:32.460
I apologize.
00:20:32.460 --> 00:20:35.410
Let's put that in there.
00:20:35.410 --> 00:20:41.540
So this is the integration
by parts formula.
00:20:41.540 --> 00:20:46.760
I'm going to write it one more
time with the limits stuck in.
00:20:46.760 --> 00:21:02.170
It's also written this way, when
you have a definite integral.
00:21:02.170 --> 00:21:13.260
Just the same formula,
written twice.
00:21:13.260 --> 00:21:14.910
Alright, now I'm
going to show you
00:21:14.910 --> 00:21:24.360
how it works on a few examples.
00:21:24.360 --> 00:21:29.470
And I have to give you a
flavor for how it works.
00:21:29.470 --> 00:21:34.230
But it'll grow as we get
more and more experience.
00:21:34.230 --> 00:21:38.360
The first example
that I'm going to take
00:21:38.360 --> 00:21:43.040
is one that looks intractable
on the face of it.
00:21:43.040 --> 00:21:49.740
Which is the
integral of ln x dx.
00:21:49.740 --> 00:21:52.560
Now, it looks like there's sort
of nothing we can do with this.
00:21:52.560 --> 00:21:55.310
And we don't know
what the solution is.
00:21:55.310 --> 00:21:59.480
However, I claim that if
we fit it into this form,
00:21:59.480 --> 00:22:03.100
we can figure out what the
integral is relatively easily.
00:22:03.100 --> 00:22:07.160
By some little magic of
cancellation, it happens.
00:22:07.160 --> 00:22:08.960
The idea is the following.
00:22:08.960 --> 00:22:13.130
If I consider this
function to be u,
00:22:13.130 --> 00:22:15.680
then what's going to
appear on the other side
00:22:15.680 --> 00:22:19.540
in the integrated form
is the function u', which
00:22:19.540 --> 00:22:22.680
is-- so, if you like, u = ln x.
00:22:22.680 --> 00:22:25.620
So u' = 1 / x.
00:22:25.620 --> 00:22:28.680
Now, 1 / x is a more
manageable function than ln x.
00:22:28.680 --> 00:22:31.370
What we're using is that when
we differentiate the function,
00:22:31.370 --> 00:22:33.100
it's getting nicer.
00:22:33.100 --> 00:22:36.830
It's getting more
tractable for us.
00:22:36.830 --> 00:22:38.860
In order for this to
fit into this pattern,
00:22:38.860 --> 00:22:45.410
however, I need a function
v. So what in the world
00:22:45.410 --> 00:22:48.220
am I going to put here for v?
00:22:48.220 --> 00:22:51.920
The answer is, well, dx is
almost the right answer.
00:22:51.920 --> 00:22:53.860
The answer turns out to be x.
00:22:53.860 --> 00:23:01.260
And the reason is that
that makes v' = 1.
00:23:01.260 --> 00:23:02.630
It makes v' = 1.
00:23:02.630 --> 00:23:05.560
So that means that this
is u, but it's also uv'.
00:23:05.560 --> 00:23:11.240
Which was what I had
on the left-hand side.
00:23:11.240 --> 00:23:13.470
So it's both u and uv'.
00:23:13.470 --> 00:23:14.710
So this is the setup.
00:23:14.710 --> 00:23:19.300
And now all I'm going to do is
read off what the formula says.
00:23:19.300 --> 00:23:24.190
What it says is, this is
equal to u times v. So u
00:23:24.190 --> 00:23:25.270
is this and v is that.
00:23:25.270 --> 00:23:32.440
So it's x ln x minus, so
that again, this is uv.
00:23:32.440 --> 00:23:37.170
Except in the other order, vu.
00:23:37.170 --> 00:23:40.510
And then I'm integrating, and
what do I have to integrate?
00:23:40.510 --> 00:23:42.540
u'v. So look up there.
00:23:42.540 --> 00:23:47.760
u'v with a minus sign here.
u' = 1 / x, and v = x.
00:23:47.760 --> 00:23:50.890
So it's 1 / x, that's u'.
00:23:50.890 --> 00:23:56.070
And here is x, that's v, dx.
00:23:56.070 --> 00:23:58.270
Now, that one is
easy to integrate.
00:23:58.270 --> 00:24:00.870
Because (1/x) x = 1.
00:24:00.870 --> 00:24:07.630
And the integral of 1 dx
is x, plus c, if you like.
00:24:07.630 --> 00:24:10.510
So the antiderivative of 1 is x.
00:24:10.510 --> 00:24:11.610
And so here's our answer.
00:24:11.610 --> 00:24:34.480
Our answer is that
this is x ln x - x + c.
00:24:34.480 --> 00:24:37.610
I'm going to do two
more slightly more
00:24:37.610 --> 00:24:39.380
complicated examples.
00:24:39.380 --> 00:24:42.590
And then really,
the main thing is
00:24:42.590 --> 00:24:44.580
to get yourself
used to this method.
00:24:44.580 --> 00:24:47.930
And there's no one
way of doing that.
00:24:47.930 --> 00:24:49.970
Just practice makes perfect.
00:24:49.970 --> 00:24:53.070
And so we'll just do
a few more examples.
00:24:53.070 --> 00:24:55.870
And illustrate them.
00:24:55.870 --> 00:24:59.960
The second example that I'm
going to use is the integral
00:24:59.960 --> 00:25:03.590
of (ln x)^2 dx.
00:25:03.590 --> 00:25:08.510
And this is just slightly
more recalcitrant.
00:25:08.510 --> 00:25:13.270
Namely, I'm going to
let u be (ln x)^2.
00:25:17.740 --> 00:25:20.440
And again, v = x.
00:25:20.440 --> 00:25:21.890
So that matches up here.
00:25:21.890 --> 00:25:23.730
That is, v' = 1.
00:25:23.730 --> 00:25:28.390
So this is uv'.
00:25:28.390 --> 00:25:31.440
So this thing is uv'.
00:25:31.440 --> 00:25:33.510
And then we'll just
see what happens.
00:25:33.510 --> 00:25:38.060
Now, the game that we get
is that when I differentiate
00:25:38.060 --> 00:25:42.870
the logarithm squared, I'm going
to to get something simpler.
00:25:42.870 --> 00:25:46.850
It's not going to win
us the whole battle,
00:25:46.850 --> 00:25:49.860
but it will get us started.
00:25:49.860 --> 00:25:51.770
So here we get u'.
00:25:51.770 --> 00:25:56.650
And that's 2 ln x times 1/x.
00:25:56.650 --> 00:26:00.470
Applying the chain rule.
00:26:00.470 --> 00:26:06.020
And so the formula is
that this is x (ln x)^2,
00:26:06.020 --> 00:26:11.710
minus the integral of,
well it's u'v, right,
00:26:11.710 --> 00:26:13.210
that's what I have
to put over here.
00:26:13.210 --> 00:26:22.190
So u' = 2 ln x 1/x and v = x.
00:26:22.190 --> 00:26:25.880
And so now, you notice something
interesting happening here.
00:26:25.880 --> 00:26:28.960
So let me just demarcate
this a little bit.
00:26:28.960 --> 00:26:34.680
And let you see what it
is that I'm doing here.
00:26:34.680 --> 00:26:36.820
So notice, this is
the same integral.
00:26:36.820 --> 00:26:38.650
So here we have x (ln x)^2.
00:26:38.650 --> 00:26:41.050
We've already solved that part.
00:26:41.050 --> 00:26:43.870
But now know notice that
the 1/x and the x cancel.
00:26:43.870 --> 00:26:46.890
So we're back to
the previous case.
00:26:46.890 --> 00:26:49.570
We didn't win all the way, but
actually we reduced ourselves
00:26:49.570 --> 00:26:51.350
to this integral.
00:26:51.350 --> 00:26:56.630
To the integral of ln x,
which we already know.
00:26:56.630 --> 00:26:58.700
So here, I can copy that down.
00:26:58.700 --> 00:27:04.450
That's - -2(x ln x - x),
and then I have to throw
00:27:04.450 --> 00:27:05.460
in a constant, c.
00:27:05.460 --> 00:27:07.270
And that's the end
of the problem here.
00:27:07.270 --> 00:27:10.260
That's it.
00:27:10.260 --> 00:27:26.400
So this piece, I
got from Example 1.
00:27:26.400 --> 00:27:34.350
Now, this illustrates
a principle
00:27:34.350 --> 00:27:36.200
which is a little
bit more complicated
00:27:36.200 --> 00:27:40.390
than just the one of
integration by parts.
00:27:40.390 --> 00:27:43.630
Which is a sort of a
general principle which
00:27:43.630 --> 00:27:48.150
I'll call my Example 3,
which is something which
00:27:48.150 --> 00:27:56.520
is called a reduction formula.
00:27:56.520 --> 00:28:00.540
A reduction formula is a
case where we apply some rule
00:28:00.540 --> 00:28:03.220
and we figure out one
of these integrals
00:28:03.220 --> 00:28:05.440
in terms of something else.
00:28:05.440 --> 00:28:07.320
Which is a little bit simpler.
00:28:07.320 --> 00:28:09.070
And eventually we'll
get down to the end,
00:28:09.070 --> 00:28:12.810
but it may take us n
steps from the beginning.
00:28:12.810 --> 00:28:17.541
So the example is (ln x)^n dx.
00:28:17.541 --> 00:28:18.040
.
00:28:18.040 --> 00:28:21.710
And the claim is that if I
do what I did in Example 2,
00:28:21.710 --> 00:28:26.500
to this case, I'll get a
simpler one which will involve
00:28:26.500 --> 00:28:28.224
the (n-1)st power.
00:28:28.224 --> 00:28:29.640
And that way I can
get all the way
00:28:29.640 --> 00:28:32.440
back down to the final answer.
00:28:32.440 --> 00:28:34.160
So here's what happens.
00:28:34.160 --> 00:28:37.090
We take u as (ln x)^n.
00:28:37.090 --> 00:28:40.980
This is the same discussion
as before, v = x.
00:28:40.980 --> 00:28:44.850
And then u' is n n
(ln x)^(n-1) 1/x.
00:28:47.440 --> 00:28:50.020
And v' is 1.
00:28:50.020 --> 00:28:52.800
And so the setup is similar.
00:28:52.800 --> 00:28:59.320
We have here x (ln x)^n
minus the integral.
00:28:59.320 --> 00:29:05.020
And there's n times, it
turns out to be (ln x)^(n-1).
00:29:05.020 --> 00:29:26.410
And then there's a 1/x
and an x, which cancel.
00:29:26.410 --> 00:29:31.010
So I'm going to explain
this also abstractly
00:29:31.010 --> 00:29:35.450
a little bit just to show
you what's happening here.
00:29:35.450 --> 00:29:44.360
If you use the notation F_n(x)
is the integral of (ln x)^n dx,
00:29:44.360 --> 00:29:46.780
and we're going to
forget the constant here.
00:29:46.780 --> 00:29:51.710
Then the relationship that we
have here is that F_n(x) is
00:29:51.710 --> 00:29:56.590
equal to n ln-- I'm
sorry, x (ln x)^n.
00:29:56.590 --> 00:29:59.490
That's the first term over here.
00:29:59.490 --> 00:30:03.650
Minus n times the preceding one.
00:30:03.650 --> 00:30:07.670
This one here.
00:30:07.670 --> 00:30:11.060
And the idea is that
eventually we can get down.
00:30:11.060 --> 00:30:15.010
If we start with the nth one, we
have a formula that includes--
00:30:15.010 --> 00:30:17.440
So the reduction is
to the n (n-1)st.
00:30:17.440 --> 00:30:21.280
Then we can reduce to
the (n-2)nd and so on.
00:30:21.280 --> 00:30:23.610
Until we reduce to
the 1, the first one.
00:30:23.610 --> 00:30:29.390
And then in fact we can
even go down to the 0th one.
00:30:29.390 --> 00:30:32.510
So this is the idea of
a reduction formula.
00:30:32.510 --> 00:30:37.320
And let me illustrate it exactly
in the context of Examples 1
00:30:37.320 --> 00:30:38.870
and 2.
00:30:38.870 --> 00:30:44.670
So the first step would be
to evaluate the first one.
00:30:44.670 --> 00:30:48.190
Which is, if you
like, (ln x)^0 dx.
00:30:48.190 --> 00:30:52.370
That's very easy, that's x.
00:30:52.370 --> 00:31:01.000
And then F_1(x) =
x ln x - F_0(x).
00:31:01.000 --> 00:31:03.240
Now, that's applying this rule.
00:31:03.240 --> 00:31:06.830
So let me just put
it in a box here.
00:31:06.830 --> 00:31:09.380
This is the method of induction.
00:31:09.380 --> 00:31:13.510
Here's the rule.
00:31:13.510 --> 00:31:21.930
And I'm applying it for n = 1.
00:31:21.930 --> 00:31:23.810
I plugged in n = 1 here.
00:31:23.810 --> 00:31:26.720
So here, I have x
(ln x)^1 - 1*F_0(x).
00:31:32.430 --> 00:31:39.240
And that's what I put right
here, on the right-hand side.
00:31:39.240 --> 00:31:42.440
And that's going to generate
for me the formula that I want,
00:31:42.440 --> 00:31:44.920
which is x ln x - x.
00:31:44.920 --> 00:31:49.160
That's the answer to
this problem over here.
00:31:49.160 --> 00:31:51.230
This was Example 1.
00:31:51.230 --> 00:31:52.980
Notice I dropped the
constants because I
00:31:52.980 --> 00:31:54.880
can add them in at the end.
00:31:54.880 --> 00:31:57.590
So I'll put in
parentheses here, plus c.
00:31:57.590 --> 00:32:01.850
That's what would happen
at the end of the problem.
00:32:01.850 --> 00:32:10.320
The next step, so that was
Example 1, and now Example 2
00:32:10.320 --> 00:32:12.190
works more or less the same way.
00:32:12.190 --> 00:32:14.590
I'm just summarizing what
I did on that blackboard
00:32:14.590 --> 00:32:16.640
right up here.
00:32:16.640 --> 00:32:21.030
The same thing, but in
much more compact notation.
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 -
00:32:29.950 --> 00:32:31.820
2 F_1(x).
00:32:31.820 --> 00:32:41.550
Again, this is box for n = 2.
00:32:41.550 --> 00:32:46.730
And if I plug it in, what I'm
getting here is x (ln x)^2
00:32:46.730 --> 00:32:49.570
minus twice this stuff here.
00:32:49.570 --> 00:32:55.780
Which is right here. x ln x - x.
00:32:55.780 --> 00:32:58.580
If you like, plus c.
00:32:58.580 --> 00:33:07.360
So I'll leave the c off.
00:33:07.360 --> 00:33:12.170
So this is how reduction
formulas work in general.
00:33:12.170 --> 00:33:22.620
I'm going to give you one more
example of a reduction formula.
00:33:22.620 --> 00:33:30.560
So I guess we have to
call this Example 4.
00:33:30.560 --> 00:33:34.050
Let's be fancy, let's
make it the sine.
00:33:34.050 --> 00:33:35.950
No no, no, let's
be fancier still.
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.
00:33:48.790 --> 00:33:50.110
The same sort of thing.
00:33:50.110 --> 00:33:52.840
And I should mention
that on your homework,
00:33:52.840 --> 00:33:54.300
you have to do it for cos x.
00:33:54.300 --> 00:33:56.550
I decided to change my mind
on the spur of the moment.
00:33:56.550 --> 00:33:57.924
I'm not going to
do it for cosine
00:33:57.924 --> 00:34:00.530
because you have to work it out
on your homework for cosine.
00:34:00.530 --> 00:34:03.100
In a later homework
you'll even do this case.
00:34:03.100 --> 00:34:05.190
So it's fine.
00:34:05.190 --> 00:34:07.400
You need the practice.
00:34:07.400 --> 00:34:10.240
OK, so how am I going
to do it this time.
00:34:10.240 --> 00:34:13.970
This is again, a
reduction formula.
00:34:13.970 --> 00:34:19.420
And the trick here is to pick
u to be this function here.
00:34:19.420 --> 00:34:20.780
And the reason is the following.
00:34:20.780 --> 00:34:23.029
So it's very important to
pick which function is the u
00:34:23.029 --> 00:34:26.450
and which function is the v.
That's the only decision you
00:34:26.450 --> 00:34:30.020
have to make if you're going
to apply integration by parts.
00:34:30.020 --> 00:34:34.420
When I pick this function as
the u, the advantage that I have
00:34:34.420 --> 00:34:38.150
is that u' is simpler.
00:34:38.150 --> 00:34:39.630
How is it simpler?
00:34:39.630 --> 00:34:42.820
It's simpler because
it's one degree down.
00:34:42.820 --> 00:34:45.420
So that's making
progress for us.
00:34:45.420 --> 00:34:48.820
On the other hand,
this function here
00:34:48.820 --> 00:34:52.530
is going to be what
I'll use for v.
00:34:52.530 --> 00:34:55.279
And if I differentiated that,
if I did it the other way around
00:34:55.279 --> 00:34:57.070
and I differentiated
that, I would just get
00:34:57.070 --> 00:34:58.900
the same level of complexity.
00:34:58.900 --> 00:35:01.120
Differentiating e^x
just gives you back e^x.
00:35:01.120 --> 00:35:02.000
So that's boring.
00:35:02.000 --> 00:35:05.750
It doesn't make any
progress in this process.
00:35:05.750 --> 00:35:11.460
And so I'm going to instead
let v = e^x and-- Sorry,
00:35:11.460 --> 00:35:12.830
this is v'.
00:35:12.830 --> 00:35:14.380
Make it v' = e^x.
00:35:14.380 --> 00:35:15.950
And then v = e^x.
00:35:15.950 --> 00:35:20.640
At least it isn't any worse
when I went backwards like that.
00:35:20.640 --> 00:35:28.150
So now, I have u and v',
and now I get x^n e^x.
00:35:28.150 --> 00:35:31.490
This again is u, and this is v.
So it happens that v is equal
00:35:31.490 --> 00:35:34.200
to v ' so it's a
little confusing here.
00:35:34.200 --> 00:35:37.640
But this is the one
we're calling v'.
00:35:37.640 --> 00:35:41.510
And here's v. And now minus
the integral and I have here
00:35:41.510 --> 00:35:43.760
nx^(n-1).
00:35:43.760 --> 00:35:45.120
And I have here e^x.
00:35:45.120 --> 00:35:52.060
So this is u' and this is v dx.
00:35:52.060 --> 00:35:55.180
So this recurrence
is a new recurrence.
00:35:55.180 --> 00:35:57.050
And let me summarize it here.
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.
00:36:05.210 --> 00:36:06.810
Again, I'm dropping the c.
00:36:06.810 --> 00:36:17.060
And then the reduction formula
is that G_n(x) is equal to this
00:36:17.060 --> 00:36:25.300
expression here: x^n
e^x - n*G_(n-1)(x).
00:36:25.300 --> 00:36:32.830
So here's our reduction formula.
00:36:32.830 --> 00:36:37.912
And to illustrate
this, if I take G_0(x),
00:36:37.912 --> 00:36:39.870
if you think about it
for a second that's just,
00:36:39.870 --> 00:36:40.744
there's nothing here.
00:36:40.744 --> 00:36:44.680
The antiderivative of e^x,
that's going to be e^x,
00:36:44.680 --> 00:36:48.220
that's getting started at
the real basement here.
00:36:48.220 --> 00:36:52.000
Again, as always, 0
is my favorite number.
00:36:52.000 --> 00:36:52.820
Not 1.
00:36:52.820 --> 00:36:55.850
I always start with the
easiest one, if possible.
00:36:55.850 --> 00:37:00.150
And now G_1, applying
this formula,
00:37:00.150 --> 00:37:06.830
is going to be equal
to x e^x - G_0(x).
00:37:06.830 --> 00:37:11.180
Which is just-- Right,
because n is 1 and n - 1 is 0.
00:37:11.180 --> 00:37:13.970
And so that's just
^ x e^x - e^x.
00:37:17.220 --> 00:37:19.770
So this is a very,
very fancy way
00:37:19.770 --> 00:37:22.620
of saying the following fact.
00:37:22.620 --> 00:37:32.210
I'll put it over on
this other board.
00:37:32.210 --> 00:37:38.270
Which is that the integral of
x e^x dx is equal to x e^x -
00:37:38.270 --> 00:37:44.600
x + c.
00:37:44.600 --> 00:37:45.270
Yeah, question.
00:37:45.270 --> 00:37:50.950
STUDENT: [INAUDIBLE]
00:37:50.950 --> 00:37:53.020
PROFESSOR: The question
is, why is this true.
00:37:53.020 --> 00:37:54.830
Why is this statement true.
00:37:54.830 --> 00:37:56.420
Why is G_0 equal to e^x.
00:37:56.420 --> 00:37:58.410
I did that in my head.
00:37:58.410 --> 00:38:02.910
What I did was, I first wrote
down the formula for G_0.
00:38:02.910 --> 00:38:08.100
Which was G_0 is equal to
the integral of e^x dx.
00:38:11.076 --> 00:38:12.950
Because there's an x to
the 0 power in there,
00:38:12.950 --> 00:38:15.010
which is just 1.
00:38:15.010 --> 00:38:17.780
And then I know the
antiderivative of e^x.
00:38:17.780 --> 00:38:23.230
It's e^x.
00:38:23.230 --> 00:38:30.780
STUDENT: [INAUDIBLE]
00:38:30.780 --> 00:38:33.030
PROFESSOR: How do you know
when this method will work?
00:38:33.030 --> 00:38:37.370
The answer is only
by experience.
00:38:37.370 --> 00:38:40.091
You must get
practice doing this.
00:38:40.091 --> 00:38:41.590
If you look in your
textbook, you'll
00:38:41.590 --> 00:38:44.430
see hints as to what to do.
00:38:44.430 --> 00:38:46.050
The other hint
that I want to say
00:38:46.050 --> 00:38:48.180
is that if you
find that you have
00:38:48.180 --> 00:38:51.000
one factor in your expression
which when you differentiate
00:38:51.000 --> 00:38:52.590
it, it gets easier.
00:38:52.590 --> 00:38:55.140
And when you antidifferentiate
the other half,
00:38:55.140 --> 00:38:57.780
it doesn't get any
worse, then that's
00:38:57.780 --> 00:39:01.790
when this method has
a chance of helping.
00:39:01.790 --> 00:39:04.430
And there is-- there's
no general thing.
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,
00:39:09.330 --> 00:39:11.970
x^n sin x, those are
examples where it works.
00:39:11.970 --> 00:39:15.600
This power of the log.
00:39:15.600 --> 00:39:19.150
I'll give you one
more example here.
00:39:19.150 --> 00:39:26.519
So this was G_1(x), right.
00:39:26.519 --> 00:39:28.310
I'll give you one more
example in a second.
00:39:28.310 --> 00:39:29.330
Yeah.
00:39:29.330 --> 00:39:33.220
STUDENT: [INAUDIBLE]
00:39:33.220 --> 00:39:35.680
PROFESSOR: Thank you.
00:39:35.680 --> 00:39:38.490
There's a mistake here.
00:39:38.490 --> 00:39:39.240
That's bad.
00:39:39.240 --> 00:39:45.910
I was thinking in the back of my
head of the following formula.
00:39:45.910 --> 00:39:51.159
Which is another one
which we've just done.
00:39:51.159 --> 00:39:53.450
So these are the types of
formulas that you can get out
00:39:53.450 --> 00:39:57.620
of integration by parts.
00:39:57.620 --> 00:40:00.650
There's also another way of
getting these, which I'm not
00:40:00.650 --> 00:40:02.240
going to say anything about.
00:40:02.240 --> 00:40:04.282
Which is called
advance guessing.
00:40:04.282 --> 00:40:06.740
You guess in advance what the
form is, you differentiate it
00:40:06.740 --> 00:40:08.160
and you check.
00:40:08.160 --> 00:40:14.250
That does work too, with
many of these cases.
00:40:14.250 --> 00:40:21.580
I want to give you
an illustration.
00:40:21.580 --> 00:40:30.760
Just because, you know, these
formulas are somewhat dry.
00:40:30.760 --> 00:40:34.570
So I want to give you just
at least one application.
00:40:34.570 --> 00:40:42.230
We're almost done with the
idea of these formulas.
00:40:42.230 --> 00:40:44.880
And we're going to
get back now to being
00:40:44.880 --> 00:40:47.990
able to handle lots more
integrals than we could before.
00:40:47.990 --> 00:40:49.810
And what's satisfying
is that now we
00:40:49.810 --> 00:40:53.830
can get numbers out instead
of being stuck and hamstrung
00:40:53.830 --> 00:40:55.120
with only a few techniques.
00:40:55.120 --> 00:40:57.680
Now we have all of the
techniques of integration
00:40:57.680 --> 00:40:59.250
that anybody has.
00:40:59.250 --> 00:41:01.820
And so we can do
pretty much anything
00:41:01.820 --> 00:41:04.430
we want that's possible to do.
00:41:04.430 --> 00:41:14.250
So here's, if you like,
an application that
00:41:14.250 --> 00:41:18.890
illustrates how integration
by parts can be helpful.
00:41:18.890 --> 00:41:27.030
And we're going to find the
volume of an exponential wine
00:41:27.030 --> 00:41:34.290
glass here.
00:41:34.290 --> 00:41:38.350
Again, don't try
this at home, but.
00:41:38.350 --> 00:41:40.500
So let's see.
00:41:40.500 --> 00:41:44.660
It's going to be this
beautiful guy here.
00:41:44.660 --> 00:41:46.930
I think.
00:41:46.930 --> 00:41:49.060
OK, so what's it going to be.
00:41:49.060 --> 00:41:52.780
This graph is going
to be y = e^x.
00:41:52.780 --> 00:42:04.030
Then we're going to rotate
it around the y-axis.
00:42:04.030 --> 00:42:10.290
And this level here
is the height y = 1.
00:42:10.290 --> 00:42:12.990
And the top, let's
say, is y = e.
00:42:12.990 --> 00:42:22.160
So that the horizontal
here, coming down, is x = 1.
00:42:22.160 --> 00:42:35.050
Now, there are two ways
to set up this problem.
00:42:35.050 --> 00:42:40.050
And so there are two methods.
00:42:40.050 --> 00:42:44.110
And this is also a good
review because, of course,
00:42:44.110 --> 00:42:46.330
we did this in the last unit.
00:42:46.330 --> 00:42:58.480
The two methods are horizontal
and vertical slices.
00:42:58.480 --> 00:43:00.660
Those are the two
ways we can do this.
00:43:00.660 --> 00:43:03.710
Now, if we do it with--
So let's start out
00:43:03.710 --> 00:43:09.370
with the horizontal ones.
00:43:09.370 --> 00:43:12.370
That's this shape here.
00:43:12.370 --> 00:43:15.370
And we're going like that.
00:43:15.370 --> 00:43:19.900
And the horizontal slices
mean that this little bit here
00:43:19.900 --> 00:43:22.842
is of thickness dy.
00:43:22.842 --> 00:43:24.550
And then we're going
to wrap that around.
00:43:24.550 --> 00:43:30.810
So this is going
to become a disk.
00:43:30.810 --> 00:43:34.070
This is the method of disks.
00:43:34.070 --> 00:43:35.830
And what's this distance here?
00:43:35.830 --> 00:43:37.770
Well, this place is x.
00:43:37.770 --> 00:43:40.680
And so the disk has area pi x^2.
00:43:40.680 --> 00:43:42.840
And we're going to
add up the thickness
00:43:42.840 --> 00:43:45.730
of the disks and we're going
to integrate from 1 to e.
00:43:45.730 --> 00:43:51.930
So here's our volume.
00:43:51.930 --> 00:43:54.510
And now we have one last
little item of business
00:43:54.510 --> 00:43:56.230
before we can evaluate
this integral.
00:43:56.230 --> 00:43:58.480
And that is that we need to
know the relationship here
00:43:58.480 --> 00:44:01.360
on the curve, that y = e^x.
00:44:01.360 --> 00:44:07.490
So that means x = ln y.
00:44:07.490 --> 00:44:09.180
And in order to
evaluate this integral,
00:44:09.180 --> 00:44:13.050
we have to evaluate x
correctly as a function of y.
00:44:13.050 --> 00:44:26.200
So that's the integral from 1
to e of (ln y)^2, times pi, dy.
00:44:26.200 --> 00:44:27.860
So now you see that
this is an integral
00:44:27.860 --> 00:44:30.140
that we did calculate already.
00:44:30.140 --> 00:44:34.280
And in fact, it's
sitting right here.
00:44:34.280 --> 00:44:37.030
Except with the variable x
instead of the variable y.
00:44:37.030 --> 00:44:44.830
So the answer, which we already
had, is this F_2(y) here.
00:44:44.830 --> 00:44:47.820
So maybe I'll write it that way.
00:44:47.820 --> 00:44:52.010
So this is F_2(y)
between 1 and e.
00:44:52.010 --> 00:45:00.040
And now let's figure
out what it is.
00:45:00.040 --> 00:45:02.060
It's written over there.
00:45:02.060 --> 00:45:15.100
It's y (ln y)^2 - 2(y ln y - y).
00:45:15.100 --> 00:45:24.460
The whole thing
evaluated at 1, e.
00:45:24.460 --> 00:45:29.130
And that is, if I plug
in e here, I get e.
00:45:29.130 --> 00:45:32.150
Except there's a factor
of pi there, sorry.
00:45:32.150 --> 00:45:36.360
Missed the pi factor.
00:45:36.360 --> 00:45:38.780
So there's an e here.
00:45:38.780 --> 00:45:43.050
And then I subtract off,
well, at 1 this is e - e.
00:45:43.050 --> 00:45:44.330
So it cancels.
00:45:44.330 --> 00:45:45.440
There's nothing left.
00:45:45.440 --> 00:45:50.200
And then at 1, I
get ln 1 is 0, ln 1
00:45:50.200 --> 00:45:53.500
is 0, there's only one
term left, which is 2.
00:45:53.500 --> 00:45:55.720
So it's -2.
00:45:55.720 --> 00:46:03.790
That's the answer.
00:46:03.790 --> 00:46:10.830
Now we get to compare
that with what happens
00:46:10.830 --> 00:46:15.950
if we do it the other way.
00:46:15.950 --> 00:46:19.870
So what's the vertical?
00:46:19.870 --> 00:46:31.830
So by vertical
slicing, we get shells.
00:46:31.830 --> 00:46:38.530
And that starts-- That's
in the x variable.
00:46:38.530 --> 00:46:43.480
It starts at 0 and
ends at 1 and it's dx.
00:46:43.480 --> 00:46:46.300
And what are the shells?
00:46:46.300 --> 00:46:51.750
Well, the shells are, if I
can draw the picture again,
00:46:51.750 --> 00:46:55.120
they start-- the top value is e.
00:46:55.120 --> 00:47:02.530
And the bottom value is, I need
a little bit of room for this.
00:47:02.530 --> 00:47:06.810
The bottom value is y.
00:47:06.810 --> 00:47:12.670
And then we have 2 pi
x is the circumference,
00:47:12.670 --> 00:47:15.970
as we sweep it around dx.
00:47:15.970 --> 00:47:18.260
So here's our new volume.
00:47:18.260 --> 00:47:23.600
Expressed in this different way.
00:47:23.600 --> 00:47:26.220
So now I'm going to
plug in what this is.
00:47:26.220 --> 00:47:30.380
It's the integral from
0 to 1 of e minus e^x,
00:47:30.380 --> 00:47:36.530
that's the formula
for y, 2 pi x dx.
00:47:36.530 --> 00:47:39.670
And what you see is that
you get the integral
00:47:39.670 --> 00:47:45.150
from 0 to 1 of 2 pi e x dx.
00:47:45.150 --> 00:47:46.540
That's easy, right?
00:47:46.540 --> 00:47:51.940
That's just 2 pi e times 1/2.
00:47:51.940 --> 00:47:54.820
This one is just the
area of a triangle.
00:47:54.820 --> 00:47:56.890
If I factor out the 2 pi e.
00:47:56.890 --> 00:48:03.230
And then the other piece is
the integral of 2 pi x e^x dx.
00:48:03.230 --> 00:48:08.610
From 0 to 1.
00:48:08.610 --> 00:48:12.480
STUDENT: [INAUDIBLE] PROFESSOR:
Are you asking me whether I
00:48:12.480 --> 00:48:14.370
need an x^2 here?
00:48:14.370 --> 00:48:15.960
I just evaluated the integral.
00:48:15.960 --> 00:48:17.290
I just did it geometrically.
00:48:17.290 --> 00:48:19.570
I said, this is the
area of a triangle.
00:48:19.570 --> 00:48:21.850
I didn't antidifferentiate
and evaluate it,
00:48:21.850 --> 00:48:23.880
I just told you the number.
00:48:23.880 --> 00:48:27.580
Because it's a
definite integral.
00:48:27.580 --> 00:48:31.650
So now, this one here, I
can read off from right up
00:48:31.650 --> 00:48:33.980
here, above it.
00:48:33.980 --> 00:48:37.050
This is G_1.
00:48:37.050 --> 00:48:42.060
So this is equal to,
let's check it out here.
00:48:42.060 --> 00:48:49.170
So this is pi e, right,
minus 2 pi G_1(x),
00:48:49.170 --> 00:48:52.280
evaluated at 0 and 1.
00:48:52.280 --> 00:48:54.990
So let's make sure that it's
the same as what we had before.
00:48:54.990 --> 00:48:59.860
It's pi e minus 2 pi
times-- here's G_1.
00:48:59.860 --> 00:49:03.230
So it's x e^x - e^x.
00:49:03.230 --> 00:49:05.720
So at x = 1, that cancels.
00:49:05.720 --> 00:49:08.260
But at the bottom end, it's e^0.
00:49:08.260 --> 00:49:12.040
So it's -1 here.
00:49:12.040 --> 00:49:13.130
Is that right?
00:49:13.130 --> 00:49:13.710
Yep.
00:49:13.710 --> 00:49:17.060
So it's pi e - 2.
00:49:17.060 --> 00:49:21.650
It's the same.
00:49:21.650 --> 00:49:22.150
Question.
00:49:22.150 --> 00:49:28.030
STUDENT: [INAUDIBLE]
00:49:28.030 --> 00:49:33.380
PROFESSOR: From here to
here, is that the question?
00:49:33.380 --> 00:49:39.710
STUDENT: [INAUDIBLE]
00:49:39.710 --> 00:49:43.730
PROFESSOR: So the step here
is just the distributive law.
00:49:43.730 --> 00:49:46.810
This is e 2 pi x,
that's this term.
00:49:46.810 --> 00:49:49.550
And the other terms, the
minus sign is outside.
00:49:49.550 --> 00:49:51.320
The 2 pi I factored out.
00:49:51.320 --> 00:49:56.980
And the x and the e^x stayed
inside the integral sign.
00:49:56.980 --> 00:49:59.140
Thank you.
00:49:59.140 --> 00:50:01.670
The correction is that
there was a missing minus
00:50:01.670 --> 00:50:03.780
sign, last time.
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.
00:50:13.100 --> 00:50:15.090
Evaluated at 0 and 1.
00:50:15.090 --> 00:50:18.350
And that's equal to +1.
00:50:18.350 --> 00:50:21.580
I was missing this minus sign.
00:50:21.580 --> 00:50:30.840
The place where it came in
was in this wineglass example.
00:50:30.840 --> 00:50:36.210
We had the integral of
2 pi x (e - e^x) dx.
00:50:39.340 --> 00:50:48.900
And that was 2 pi e integral
of x dx, from 0 to 1, -2 pi,
00:50:48.900 --> 00:50:52.810
integral from 0
to 1 of x e^x dx.
00:50:52.810 --> 00:50:58.400
And then I worked this
out and it was pi e.
00:50:58.400 --> 00:51:03.030
And then this one was -2 pi,
and what I wrote down was -1.
00:51:03.030 --> 00:51:05.410
But there should have been
an extra minus sign there.
00:51:05.410 --> 00:51:08.430
So it's this.
00:51:08.430 --> 00:51:11.930
The final answer was
correct, but this minus sign
00:51:11.930 --> 00:51:13.590
was missing.
00:51:13.590 --> 00:51:16.930
Right there.
00:51:16.930 --> 00:51:20.460
So just, right there.