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PROFESSOR: Today we're moving
on from theoretical things,
00:00:24.970 --> 00:00:30.570
from the mean value theorem,
to the introduction to what's
00:00:30.570 --> 00:00:33.120
going to occupy us for the
whole rest of the course, which
00:00:33.120 --> 00:00:34.850
is integration.
00:00:34.850 --> 00:00:38.850
So, in order to
introduce that subject,
00:00:38.850 --> 00:00:42.660
I need to introduce for
you a new notation, which
00:00:42.660 --> 00:00:52.640
is called differentials.
00:00:52.640 --> 00:00:55.640
I'm going to tell you
what a differential is,
00:00:55.640 --> 00:01:02.040
and we'll get used to
using it over time.
00:01:02.040 --> 00:01:10.290
If you have a function
which is y = f(x),
00:01:10.290 --> 00:01:27.650
then the differential of y
is going to be denoted dy,
00:01:27.650 --> 00:01:29.500
and it's by definition f'(x) dx.
00:01:34.160 --> 00:01:41.410
So here's the notation.
00:01:41.410 --> 00:01:44.800
And because y is really
equal to f, sometimes
00:01:44.800 --> 00:01:50.140
we also call it the
differential of f.
00:01:50.140 --> 00:01:58.700
It's also called the
differential of f.
00:01:58.700 --> 00:02:06.750
That's the notation,
and it's the same thing
00:02:06.750 --> 00:02:11.630
as what happens if you
formally just take this dx,
00:02:11.630 --> 00:02:14.910
act like it's a number
and divide it into dy.
00:02:14.910 --> 00:02:22.670
So it means the same thing
as this statement here.
00:02:22.670 --> 00:02:29.160
And this is more or less
the Leibniz interpretation
00:02:29.160 --> 00:02:38.360
of derivatives.
00:02:38.360 --> 00:02:50.990
Of a derivative as a ratio of
these so called differentials.
00:02:50.990 --> 00:03:04.290
It's a ratio of what are
known as infinitesimals.
00:03:04.290 --> 00:03:09.050
Now, this is kind of a vague
notion, this little bit
00:03:09.050 --> 00:03:12.830
here being an infinitesimal.
00:03:12.830 --> 00:03:16.110
It's sort of like an
infinitely small quantity.
00:03:16.110 --> 00:03:21.300
And Leibniz perfected
the idea of dealing
00:03:21.300 --> 00:03:23.050
with these intuitively.
00:03:23.050 --> 00:03:26.730
And subsequently, mathematicians
use them all the time.
00:03:26.730 --> 00:03:33.350
They're way more effective than
the notation that Newton used.
00:03:33.350 --> 00:03:36.250
You might think that
notations are a small matter,
00:03:36.250 --> 00:03:40.960
but they allow you to think
much faster, sometimes.
00:03:40.960 --> 00:03:43.565
When you have the right
names and the right symbols
00:03:43.565 --> 00:03:44.190
for everything.
00:03:44.190 --> 00:03:47.860
And in this case it made
it very big difference.
00:03:47.860 --> 00:03:52.850
Leibniz's notation was adopted
on the continent and Newton
00:03:52.850 --> 00:03:56.650
dominated in Britain
and, as a result,
00:03:56.650 --> 00:03:58.800
the British fell
behind by one or two
00:03:58.800 --> 00:04:01.850
hundred years in the
development of calculus.
00:04:01.850 --> 00:04:03.540
It was really a serious matter.
00:04:03.540 --> 00:04:05.860
So it's really well
worth your while to get
00:04:05.860 --> 00:04:08.940
used to this idea of ratios.
00:04:08.940 --> 00:04:12.140
And it comes up all over the
place, both in this class
00:04:12.140 --> 00:04:14.320
and also in
multivariable calculus.
00:04:14.320 --> 00:04:17.380
It's used in many contexts.
00:04:17.380 --> 00:04:20.030
So first of all, just
to go a little bit easy.
00:04:20.030 --> 00:04:25.320
We'll illustrate it by its
use in linear approximations,
00:04:25.320 --> 00:04:36.420
which we've already done.
00:04:36.420 --> 00:04:38.780
The picture here, which we've
drawn a number of times,
00:04:38.780 --> 00:04:41.020
is that you have some function.
00:04:41.020 --> 00:04:44.300
And here's a value
of the function.
00:04:44.300 --> 00:04:47.060
And it's coming up like that.
00:04:47.060 --> 00:04:48.330
So here's our function.
00:04:48.330 --> 00:04:51.090
And we go forward
a little increment
00:04:51.090 --> 00:04:56.540
to a place which is
dx further along.
00:04:56.540 --> 00:04:59.350
The idea of this
notation is that dx
00:04:59.350 --> 00:05:05.260
is going to replace
the symbol delta x,
00:05:05.260 --> 00:05:07.370
which is the change in x.
00:05:07.370 --> 00:05:10.270
And we won't think
too hard about-- well,
00:05:10.270 --> 00:05:12.560
this is a small quantity,
this is a small quantity,
00:05:12.560 --> 00:05:16.030
we're not going to think too
hard about what that means.
00:05:16.030 --> 00:05:20.780
Now, similarly, if you see
how much we've gone up - well,
00:05:20.780 --> 00:05:26.600
this is kind of low, so
it's a small bit here.
00:05:26.600 --> 00:05:31.240
So this distance
here is, previously
00:05:31.240 --> 00:05:36.040
we called it delta y.
00:05:36.040 --> 00:05:41.810
But now we're just
going to call it dy.
00:05:41.810 --> 00:05:51.200
So dy replaces delta y.
00:05:51.200 --> 00:05:57.450
So this is the change in
level of the function.
00:05:57.450 --> 00:05:59.760
And we'll represent it
symbolically this way.
00:05:59.760 --> 00:06:04.050
Very frequently, this just
saves a little bit of notation.
00:06:04.050 --> 00:06:05.930
For the purposes
of this, we'll be
00:06:05.930 --> 00:06:09.620
doing the same things we did
with delta x and delta y,
00:06:09.620 --> 00:06:12.540
but this is the way that
Leibniz thought of it.
00:06:12.540 --> 00:06:14.690
And he would just have
drawn it with this.
00:06:14.690 --> 00:06:24.670
So this distance here is dx
and this distance here is dy.
00:06:24.670 --> 00:06:30.320
So for an example of
linear approximation,
00:06:30.320 --> 00:06:35.970
we'll say what's 64.1,
say, to the 1/3 power,
00:06:35.970 --> 00:06:39.500
approximately equal to?
00:06:39.500 --> 00:06:43.470
Now, I'm going to carry this
out in this new notation here.
00:06:43.470 --> 00:06:47.810
The function involved is x^1/3.
00:06:47.810 --> 00:06:50.990
And then it's a
differential, dy.
00:06:50.990 --> 00:06:53.670
Now, I want to use this
rule to get used to it.
00:06:53.670 --> 00:06:56.870
Because this is what we're going
to be doing all of today is,
00:06:56.870 --> 00:07:00.010
we're differentiating, or
taking the differential of y.
00:07:00.010 --> 00:07:02.490
So that is going to be
just the derivative.
00:07:02.490 --> 00:07:11.980
That's 1/3 x^(-2/3) dx.
00:07:11.980 --> 00:07:18.060
And now I'm just going to
fill in exactly what this is.
00:07:18.060 --> 00:07:25.450
At x = 64, which is the natural
place close by where it's easy
00:07:25.450 --> 00:07:36.110
to do the evaluations, we have
y = 64^(1/3), which is just 4.
00:07:36.110 --> 00:07:39.740
And how about dy?
00:07:39.740 --> 00:07:42.360
Well, so this is a little
bit more complicated.
00:07:42.360 --> 00:07:43.540
Put it over here.
00:07:43.540 --> 00:07:50.690
So dy = 1/3 64^(-2/3) dx.
00:07:55.800 --> 00:08:16.680
And that is 1/3 * 1/16
dx, which is 1/48 dx.
00:08:16.680 --> 00:08:19.810
And now I'm going
to work out what
00:08:19.810 --> 00:08:26.320
64 to the, whatever it is
here, this strange fraction.
00:08:26.320 --> 00:08:31.210
I just want to be very careful
to explain to you one more
00:08:31.210 --> 00:08:33.170
thing.
00:08:33.170 --> 00:08:37.780
Which is that
we're using x = 64,
00:08:37.780 --> 00:08:45.240
and so we're thinking of x
+ dx is going to be 64.1.
00:08:45.240 --> 00:08:53.700
So that means that dx
is going to be 1/10.
00:08:53.700 --> 00:08:59.760
So that's the increment
that we're interested in.
00:08:59.760 --> 00:09:03.910
And now I can carry
out the approximation.
00:09:03.910 --> 00:09:11.230
The approximation says
that 64.1^(1/3) is, well,
00:09:11.230 --> 00:09:14.710
it's approximately what
I'm going to call y + dy.
00:09:14.710 --> 00:09:18.060
Because really, the dy
that I'm determining here
00:09:18.060 --> 00:09:26.450
is determined by this linear
relation. dy = 1/48 dx.
00:09:26.450 --> 00:09:29.820
And so this is only
approximately true.
00:09:29.820 --> 00:09:33.430
Because what's really
true is that this
00:09:33.430 --> 00:09:37.690
is equal to y + delta y.
00:09:37.690 --> 00:09:39.590
In our previous notation.
00:09:39.590 --> 00:09:41.600
So this is in disguise.
00:09:41.600 --> 00:09:42.960
What this is equal to.
00:09:42.960 --> 00:09:44.500
And that's the
only approximately
00:09:44.500 --> 00:09:47.470
equal to what the linear
approximation would give you.
00:09:47.470 --> 00:09:51.730
So, really, even though I wrote
dy is this increment here,
00:09:51.730 --> 00:09:54.830
what it really is if
dx is exactly that,
00:09:54.830 --> 00:09:57.410
is it's the amount
it would go up
00:09:57.410 --> 00:10:00.270
if you went straight
up the tangent line.
00:10:00.270 --> 00:10:02.300
So I'm not going to do
that because that's not
00:10:02.300 --> 00:10:03.370
what people write.
00:10:03.370 --> 00:10:06.040
And that's not even
what they think.
00:10:06.040 --> 00:10:08.110
They're really thinking
of both dx and dy
00:10:08.110 --> 00:10:10.750
as being infinitesimally small.
00:10:10.750 --> 00:10:15.900
And here we're going to the
finite level and doing it.
00:10:15.900 --> 00:10:20.350
So this is just something
you have to live with,
00:10:20.350 --> 00:10:28.120
is a little ambiguity
in this notation.
00:10:28.120 --> 00:10:29.490
This is the approximation.
00:10:29.490 --> 00:10:32.970
And now I can just calculate
these numbers here.
00:10:32.970 --> 00:10:36.130
y at this value is 4.
00:10:36.130 --> 00:10:43.630
And dy, as I said, is 1/48 dx.
00:10:43.630 --> 00:10:50.440
And that turns out to be 4
+ 1/480, because dx is 1/10.
00:10:50.440 --> 00:10:54.390
So that's approximately 4.002.
00:10:54.390 --> 00:11:04.550
And that's our approximation.
00:11:04.550 --> 00:11:20.760
Now, let's just compare it
to our previous notation.
00:11:20.760 --> 00:11:22.970
This will serve as
a review of, if you
00:11:22.970 --> 00:11:35.870
like, of linear approximation.
00:11:35.870 --> 00:11:38.990
But what I want to emphasize
is that these things
00:11:38.990 --> 00:11:43.210
are supposed to be the same.
00:11:43.210 --> 00:11:45.520
Just that it's really
the same thing.
00:11:45.520 --> 00:11:52.280
It's just a different
notation for the same thing.
00:11:52.280 --> 00:11:56.360
I remind you the basic formula
for linear approximation is
00:11:56.360 --> 00:12:00.980
that f(x) is approximately
f(a) + f'(a) (x-a).
00:12:05.160 --> 00:12:11.620
And we're applying it in the
situation that a = 64 and f(x)
00:12:11.620 --> 00:12:12.910
= x^(1/3).
00:12:17.980 --> 00:12:27.580
And so f(a), which is
f(64), is of course 4.
00:12:27.580 --> 00:12:43.210
And f'(a), which is 1/3
a^(-2/3), is in our case 1/16.
00:12:43.210 --> 00:12:49.600
No, 1/48.
00:12:49.600 --> 00:12:52.980
OK, that's the same
calculation as before.
00:12:52.980 --> 00:12:59.490
And then our relationship
becomes x^(1/3) is
00:12:59.490 --> 00:13:08.420
approximately equal to 4 plus
1/48 times x minus a, which is
00:13:08.420 --> 00:13:12.250
64.
00:13:12.250 --> 00:13:14.820
So look, every single number
that I've written over here
00:13:14.820 --> 00:13:20.030
has a corresponding number
for this other method.
00:13:20.030 --> 00:13:23.250
And now if I plug in the
value we happen to want,
00:13:23.250 --> 00:13:32.500
which is the 64.1, this
would be 4 + 1/48 1/10,
00:13:32.500 --> 00:13:38.580
which is just the same
thing we had before.
00:13:38.580 --> 00:13:45.270
So again, same answer.
00:13:45.270 --> 00:13:55.300
Same method, new notation.
00:13:55.300 --> 00:14:02.300
Well, now I get to use this
notation in a novel way.
00:14:02.300 --> 00:14:04.640
So again, here's the notation.
00:14:04.640 --> 00:14:16.410
This notation of differential.
00:14:16.410 --> 00:14:20.520
The way I'm going to use it
is in discussing something
00:14:20.520 --> 00:14:32.560
called antiderivative Again,
this is a new notation now.
00:14:32.560 --> 00:14:33.680
But it's also a new idea.
00:14:33.680 --> 00:14:37.850
It's one that we
haven't discussed yet.
00:14:37.850 --> 00:14:42.500
Namely, the notation that
I want to describe here
00:14:42.500 --> 00:14:48.090
is what's called the
integral of g(x) dx.
00:14:48.090 --> 00:14:51.560
And I'll denote that by a
function capital G of x.
00:14:51.560 --> 00:14:53.820
So it's, you start
with a function g(x)
00:14:53.820 --> 00:14:55.380
and you produce a
function capital
00:14:55.380 --> 00:15:12.190
G(x), which is called
the antiderivative of g.
00:15:12.190 --> 00:15:15.330
Notice there's a
differential sitting in here.
00:15:15.330 --> 00:15:31.380
This symbol, this guy here,
is called an integral sign.
00:15:31.380 --> 00:15:34.600
Or an integral, or this whole
thing is called an integral.
00:15:34.600 --> 00:15:37.830
And another name for
the antiderivative of g
00:15:37.830 --> 00:15:50.700
is the indefinite integral of g.
00:15:50.700 --> 00:15:58.050
And I'll explain to you why
it's indefinite in just--
00:15:58.050 --> 00:16:04.330
very shortly here.
00:16:04.330 --> 00:16:13.330
Well, so let's carry
out some examples.
00:16:13.330 --> 00:16:16.840
Basically what I'd like
to do is as many examples
00:16:16.840 --> 00:16:18.510
along the lines of
all the derivatives
00:16:18.510 --> 00:16:21.329
that we derived at the
beginning of the course.
00:16:21.329 --> 00:16:22.870
In other words, in
principle you want
00:16:22.870 --> 00:16:26.550
to be able to integrate as
many things as possible.
00:16:26.550 --> 00:16:34.680
We're going to start out with
the integral of sin x dx.
00:16:34.680 --> 00:16:40.670
That's a function whose
derivative is sin x.
00:16:40.670 --> 00:16:44.990
So what function would that be?
00:16:44.990 --> 00:16:48.400
Cosine x, minus, right.
00:16:48.400 --> 00:16:49.590
It's -cos x.
00:16:52.700 --> 00:16:56.240
So -cos x differentiated
gives you sin x.
00:16:56.240 --> 00:17:00.540
So that is an
antiderivative of sine.
00:17:00.540 --> 00:17:02.210
And it satisfies this property.
00:17:02.210 --> 00:17:09.520
So this function,
G(x) = - cos x,
00:17:09.520 --> 00:17:15.170
has the property that
its derivative is sin x.
00:17:15.170 --> 00:17:20.700
On the other hand, if you
differentiate a constant,
00:17:20.700 --> 00:17:22.230
you get 0.
00:17:22.230 --> 00:17:25.090
So this answer is what's
called indefinite.
00:17:25.090 --> 00:17:28.910
Because you can also
add any constant here.
00:17:28.910 --> 00:17:33.610
And the same thing will be true.
00:17:33.610 --> 00:17:38.060
So, c is constant.
00:17:38.060 --> 00:17:41.660
And as I said, the integral
is called indefinite.
00:17:41.660 --> 00:17:45.410
So that's an explanation
for this modifier
00:17:45.410 --> 00:17:46.810
in front of the "integral".
00:17:46.810 --> 00:17:48.960
It's indefinite because
we actually didn't
00:17:48.960 --> 00:17:50.540
specify a single function.
00:17:50.540 --> 00:17:52.170
We don't get a single answer.
00:17:52.170 --> 00:17:54.400
Whenever you take the
antiderivative of something
00:17:54.400 --> 00:18:08.110
it's ambiguous up to a constant.
00:18:08.110 --> 00:18:12.340
Next, let's do some
other standard functions
00:18:12.340 --> 00:18:13.890
from our repertoire.
00:18:13.890 --> 00:18:17.820
We have the integral of x^a dx.
00:18:17.820 --> 00:18:20.730
Some power, the
integral of a power.
00:18:20.730 --> 00:18:24.680
And if you think about it, what
you should be differentiating
00:18:24.680 --> 00:18:27.590
is one power larger than that.
00:18:27.590 --> 00:18:33.250
But then you have to
divide by 1/(a+1),
00:18:33.250 --> 00:18:36.780
in order that the
differentiation be correct.
00:18:36.780 --> 00:18:44.360
So this just is the fact
that d/dx of x^(a+1),
00:18:44.360 --> 00:18:46.600
or maybe I should
even say it this way.
00:18:46.600 --> 00:18:49.620
Maybe I'll do it in
differential notation.
00:18:49.620 --> 00:18:54.230
d(x^(a+1)) = (a+1) x^a dx.
00:18:57.370 --> 00:19:02.950
So if I divide that
through by a+1,
00:19:02.950 --> 00:19:06.280
then I get the relation above.
00:19:06.280 --> 00:19:11.800
And because this is
ambiguous up to a constant,
00:19:11.800 --> 00:19:15.820
it could be any
additional constant
00:19:15.820 --> 00:19:20.830
added to that function.
00:19:20.830 --> 00:19:26.290
Now, the identity that I
wrote down below is correct.
00:19:26.290 --> 00:19:30.700
But this one is
not always correct.
00:19:30.700 --> 00:19:35.200
What's the exception?
00:19:35.200 --> 00:19:38.211
Yeah. a equals--
00:19:38.211 --> 00:19:38.710
STUDENT: 0.
00:19:38.710 --> 00:19:42.060
PROFESSOR: Negative 1.
00:19:42.060 --> 00:19:47.440
So this one is OK for all a.
00:19:47.440 --> 00:19:52.550
But this one fails because
we've divided by 0 when a = -1.
00:19:52.550 --> 00:20:04.770
So this is only true when
a is not equal to -1.
00:20:04.770 --> 00:20:07.790
And in fact, of course,
what's happening when a = 0,
00:20:07.790 --> 00:20:11.670
you're getting 0 when you
differentiate the constant.
00:20:11.670 --> 00:20:15.510
So there's a third case
that we have to carry out.
00:20:15.510 --> 00:20:25.230
Which is the exceptional case,
namely the integral of dx/x.
00:20:25.230 --> 00:20:31.265
And this time, if
we just think back
00:20:31.265 --> 00:20:32.640
to what our-- So
what we're doing
00:20:32.640 --> 00:20:35.080
is thinking backwards here,
which a very important thing
00:20:35.080 --> 00:20:38.690
to do in math at all stages.
00:20:38.690 --> 00:20:41.800
We got all of our formulas, now
we're reading them backwards.
00:20:41.800 --> 00:20:49.890
And so this one, you
may remember, is ln x.
00:20:49.890 --> 00:20:53.370
The reason why I want to do
this carefully and slowly now,
00:20:53.370 --> 00:20:57.299
is right now I also want to
write the more standard form
00:20:57.299 --> 00:20:58.090
which is presented.
00:20:58.090 --> 00:21:01.540
So first of all, first we
have to add a constant.
00:21:01.540 --> 00:21:04.160
And please don't put
the parentheses here.
00:21:04.160 --> 00:21:10.360
The parentheses go there.
00:21:10.360 --> 00:21:14.700
But there's another formula
hiding in the woodwork
00:21:14.700 --> 00:21:16.500
here behind this one.
00:21:16.500 --> 00:21:19.260
Which is that you can also
get the correct formula
00:21:19.260 --> 00:21:20.870
when x is negative.
00:21:20.870 --> 00:21:27.130
And that turns out
to be this one here.
00:21:27.130 --> 00:21:32.670
So I'm treating the case, x
positive, as being something
00:21:32.670 --> 00:21:34.460
that you know.
00:21:34.460 --> 00:21:43.530
But let's check the
case, x negative.
00:21:43.530 --> 00:21:45.810
In order to check
the case x negative,
00:21:45.810 --> 00:21:51.750
I have to differentiate the
logarithm of the absolute value
00:21:51.750 --> 00:21:55.680
of x in that case.
00:21:55.680 --> 00:21:57.612
And that's the
same thing, again,
00:21:57.612 --> 00:22:02.170
for x negative as the derivative
of the logarithm of negative x.
00:22:02.170 --> 00:22:08.410
That's the formula,
when x is negative.
00:22:08.410 --> 00:22:10.980
And if you carry
that out, what you
00:22:10.980 --> 00:22:18.400
get, maybe I'll put
this over here, is,
00:22:18.400 --> 00:22:20.800
well, it's the chain rule.
00:22:20.800 --> 00:22:27.160
It's 1/(-x) times the derivative
with respect to x of -x.
00:22:27.160 --> 00:22:30.730
So see that there
are two minus signs.
00:22:30.730 --> 00:22:32.750
There's a -x in the
denominator and then
00:22:32.750 --> 00:22:35.610
there's the derivative
of -x in the numerator.
00:22:35.610 --> 00:22:38.070
That's just -1.
00:22:38.070 --> 00:22:39.160
This part is -1.
00:22:39.160 --> 00:22:43.100
So this -1 over
-x, which is 1/x.
00:22:43.100 --> 00:22:53.200
So the negative signs cancel.
00:22:53.200 --> 00:23:00.090
If you just keep track of this
in terms of ln(-x) and its
00:23:00.090 --> 00:23:05.480
graph, that's a function
that looks like this.
00:23:05.480 --> 00:23:08.370
For x negative.
00:23:08.370 --> 00:23:14.170
And its derivative
is 1/x, I claim.
00:23:14.170 --> 00:23:17.120
And if you just look at
it a little bit carefully,
00:23:17.120 --> 00:23:23.431
you see that the slope
is always negative.
00:23:23.431 --> 00:23:23.930
Right?
00:23:23.930 --> 00:23:26.950
So here the slope is negative.
00:23:26.950 --> 00:23:30.480
So it's going to
be below the axis.
00:23:30.480 --> 00:23:32.890
And, in fact, it's getting
steeper and steeper negative
00:23:32.890 --> 00:23:34.980
as we go down.
00:23:34.980 --> 00:23:37.980
And it's getting less and less
negative as we go horizontally.
00:23:37.980 --> 00:23:40.790
So it's going like
this, which is indeed
00:23:40.790 --> 00:23:43.770
the graph of this
function, for x negative.
00:23:43.770 --> 00:23:53.320
Again, x negative.
00:23:53.320 --> 00:23:56.850
So that's one other
standard formula.
00:23:56.850 --> 00:24:00.501
And very quickly, very often,
we won't put the absolute value
00:24:00.501 --> 00:24:01.000
signs.
00:24:01.000 --> 00:24:03.180
We'll only consider the
case x positive here.
00:24:03.180 --> 00:24:06.280
But I just want you to
have the tools to do it
00:24:06.280 --> 00:24:08.710
in case we want to
use, we want to handle,
00:24:08.710 --> 00:24:14.040
both positive and negative x.
00:24:14.040 --> 00:24:28.620
Now, let's do two more examples.
00:24:28.620 --> 00:24:35.870
The integral of sec^2 x dx.
00:24:35.870 --> 00:24:38.270
These are supposed to
get you to remember
00:24:38.270 --> 00:24:41.130
all of your differentiation
formulas, the standard ones.
00:24:41.130 --> 00:24:48.200
So this one, integral of
sec^2 dx is what? tan x.
00:24:48.200 --> 00:24:50.690
And here we have + c, all right?
00:24:50.690 --> 00:24:55.520
And then the last one of, a
couple of, this type would be,
00:24:55.520 --> 00:24:56.800
let's see.
00:24:56.800 --> 00:25:04.139
I should do at least this one
here, square root of 1 - x^2.
00:25:04.139 --> 00:25:05.680
This is another
notation, by the way,
00:25:05.680 --> 00:25:07.240
which is perfectly acceptable.
00:25:07.240 --> 00:25:10.280
Notice I've put the
dx in the numerator
00:25:10.280 --> 00:25:13.400
and the function in
the denominator here.
00:25:13.400 --> 00:25:18.880
So this one turns
out to be sin^(-1) x.
00:25:18.880 --> 00:25:23.710
And, finally, let's see.
00:25:23.710 --> 00:25:28.470
About the integral
of dx / (1 + x^2).
00:25:28.470 --> 00:25:41.320
That one is tan^(-1) x.
00:25:41.320 --> 00:25:43.652
For a little while, because
you're reading these things
00:25:43.652 --> 00:25:45.110
backwards and
forwards, you'll find
00:25:45.110 --> 00:25:46.970
this happens to you on exams.
00:25:46.970 --> 00:25:49.540
It gets slightly worse
for a little while.
00:25:49.540 --> 00:25:53.120
You will antidifferentiate when
you meant to differentiate.
00:25:53.120 --> 00:25:54.620
And you'll differentiate
when you're
00:25:54.620 --> 00:25:57.230
meant to antidifferentiate.
00:25:57.230 --> 00:26:00.190
Don't get too
frustrated by that.
00:26:00.190 --> 00:26:05.270
But eventually, you'll
get them squared away.
00:26:05.270 --> 00:26:08.130
And it actually helps
to do a lot of practice
00:26:08.130 --> 00:26:15.410
with antidifferentiations,
or integrations,
00:26:15.410 --> 00:26:17.100
as they're sometimes called.
00:26:17.100 --> 00:26:19.790
Because that will
solidify your remembering
00:26:19.790 --> 00:26:25.820
all of the
differentiation formulas.
00:26:25.820 --> 00:26:30.080
So, last bit of
information that I
00:26:30.080 --> 00:26:32.730
want to emphasize before
we go on some more
00:26:32.730 --> 00:26:44.130
complicated examples is this.
00:26:44.130 --> 00:26:49.250
It's obvious because the
derivative of a constant is 0.
00:26:49.250 --> 00:26:55.330
That the antiderivative is
ambiguous up to a constant.
00:26:55.330 --> 00:26:56.920
But it's very
important to realize
00:26:56.920 --> 00:27:01.410
that this is the only
ambiguity that there is.
00:27:01.410 --> 00:27:04.390
So the last thing that
I want to tell you about
00:27:04.390 --> 00:27:24.070
is uniqueness of antiderivatives
up to a constant.
00:27:24.070 --> 00:27:30.290
The theorem is the following.
00:27:30.290 --> 00:27:41.230
The theorem is if F' =
G', then F equals G--
00:27:41.230 --> 00:27:43.620
so F(x) equals G(x)
plus a constant.
00:27:48.420 --> 00:27:54.050
But that means, not only that
these are antiderivatives,
00:27:54.050 --> 00:27:56.640
all these things with these
plus c's are antiderivatives.
00:27:56.640 --> 00:28:02.080
But these are the only ones.
00:28:02.080 --> 00:28:03.300
Which is very reassuring.
00:28:03.300 --> 00:28:06.020
And that's a kind of uniqueness,
although its uniqueness up
00:28:06.020 --> 00:28:09.660
to a constant, it's
acceptable to us.
00:28:09.660 --> 00:28:12.850
Now, the proof of
this is very quick.
00:28:12.850 --> 00:28:18.216
But this is a fundamental fact.
00:28:18.216 --> 00:28:19.340
The proof is the following.
00:28:19.340 --> 00:28:29.340
If F' = G', then if you take
the difference between the two
00:28:29.340 --> 00:28:33.520
functions, its derivative,
which of course is F' -
00:28:33.520 --> 00:28:40.110
G', is equal to 0.
00:28:40.110 --> 00:28:55.300
Hence, F(x) - G(x)
is a constant.
00:28:55.300 --> 00:28:58.940
Now, this is a key fact.
00:28:58.940 --> 00:28:59.870
Very important fact.
00:28:59.870 --> 00:29:03.840
We deduced it last time
from the mean value theorem.
00:29:03.840 --> 00:29:05.370
It's not a small matter.
00:29:05.370 --> 00:29:06.850
It's a very, very
important thing.
00:29:06.850 --> 00:29:08.690
It's the basis for calculus.
00:29:08.690 --> 00:29:11.550
It's the reason why
calculus make sense.
00:29:11.550 --> 00:29:14.120
If we didn't have the fact
that the derivative is
00:29:14.120 --> 00:29:18.890
0 implied that the function
was constant, we would be done.
00:29:18.890 --> 00:29:23.012
We would have-- Calculus
would be just useless for us.
00:29:23.012 --> 00:29:24.470
The point is, the
rate of change is
00:29:24.470 --> 00:29:27.330
supposed to determine
the function up
00:29:27.330 --> 00:29:29.650
to this starting value.
00:29:29.650 --> 00:29:32.330
So this conclusion
is very important.
00:29:32.330 --> 00:29:35.010
And we already checked it
last time, this conclusion.
00:29:35.010 --> 00:29:37.710
And now just by
algebra, I can rearrange
00:29:37.710 --> 00:30:03.340
this to say that F(x) is
equal to G(x) plus a constant.
00:30:03.340 --> 00:30:07.530
Now, maybe I should leave
differentials up here.
00:30:07.530 --> 00:30:21.390
Because I want to
illustrate-- So let's
00:30:21.390 --> 00:30:29.050
go on to some trickier,
slightly trickier, integrals.
00:30:29.050 --> 00:30:35.740
Here's an example.
00:30:35.740 --> 00:30:42.010
The integral of, say,
x^3 (x^4 + 2)^5 dx.
00:30:51.210 --> 00:30:53.480
This is a function
which you actually
00:30:53.480 --> 00:30:56.500
do know how to integrate,
because we already
00:30:56.500 --> 00:30:59.840
have a formula for all powers.
00:30:59.840 --> 00:31:03.280
Namely, the integral of
x^a is equal to this.
00:31:03.280 --> 00:31:06.520
And even if it were a negative
power, we could do it.
00:31:06.520 --> 00:31:08.630
So it's OK.
00:31:08.630 --> 00:31:14.290
On the other hand, to expand the
5th power here is quite a mess.
00:31:14.290 --> 00:31:18.480
And this is just a
very, very bad idea.
00:31:18.480 --> 00:31:21.320
There's another trick for
doing this that evaluates this
00:31:21.320 --> 00:31:23.060
much more efficiently.
00:31:23.060 --> 00:31:26.950
And it's the only
device that we're going
00:31:26.950 --> 00:31:31.550
to learn now for integrating.
00:31:31.550 --> 00:31:36.690
Integration actually is much
harder than differentiation,
00:31:36.690 --> 00:31:37.520
symbolically.
00:31:37.520 --> 00:31:42.610
It's quite difficult. And
occasionally impossible.
00:31:42.610 --> 00:31:45.550
And so we have to
go about it gently.
00:31:45.550 --> 00:31:47.450
But for the purposes
of this unit,
00:31:47.450 --> 00:31:50.069
we're only going
to use one method.
00:31:50.069 --> 00:31:50.860
Which is very good.
00:31:50.860 --> 00:31:52.526
That means whenever
you see an integral,
00:31:52.526 --> 00:31:56.180
either you'll be able to divine
immediately what the answer is,
00:31:56.180 --> 00:31:57.830
or you'll use this method.
00:31:57.830 --> 00:31:59.090
So this is it.
00:31:59.090 --> 00:32:09.470
The trick is called the
method of substitution.
00:32:09.470 --> 00:32:17.860
And it is tailor-made for
notion of differentials.
00:32:17.860 --> 00:32:36.510
So tailor-made for
differential notation.
00:32:36.510 --> 00:32:37.840
The idea is the following.
00:32:37.840 --> 00:32:40.260
I'm going to to
define a new function.
00:32:40.260 --> 00:32:43.200
And it's the messiest
function that I see here.
00:32:43.200 --> 00:32:50.290
It's u = x^4 + 2.
00:32:50.290 --> 00:32:56.300
And then, I'm going to take
its differential and what
00:32:56.300 --> 00:32:58.840
I discover, if I
look at its formula,
00:32:58.840 --> 00:33:02.570
and the rule for differentials,
which is right here.
00:33:02.570 --> 00:33:06.070
Its formula is what?
00:33:06.070 --> 00:33:10.180
4x^3 dx.
00:33:10.180 --> 00:33:14.000
Now, lo and behold with
these two quantities,
00:33:14.000 --> 00:33:17.940
I can substitute, I can
plug in to this integral.
00:33:17.940 --> 00:33:21.760
And I will simplify
it considerably.
00:33:21.760 --> 00:33:23.350
So how does that happen?
00:33:23.350 --> 00:33:34.740
Well, this integral is the
same thing as, well, really
00:33:34.740 --> 00:33:36.350
I should combine
it the other way.
00:33:36.350 --> 00:33:41.420
So let me move this over.
00:33:41.420 --> 00:33:43.340
So there are two pieces here.
00:33:43.340 --> 00:33:46.440
And this one is u^5.
00:33:46.440 --> 00:33:54.990
And this one is 1/4 du.
00:33:54.990 --> 00:34:01.840
Now, that makes it the
integral of u^5 du / 4.
00:34:01.840 --> 00:34:04.040
And that's relatively
easy to integrate.
00:34:04.040 --> 00:34:05.460
That is just a power.
00:34:05.460 --> 00:34:06.410
So let's see.
00:34:06.410 --> 00:34:11.250
It's just 1/20 u to
the-- whoops, not 1/20.
00:34:11.250 --> 00:34:15.480
The antiderivative
of u^5 is u^6.
00:34:15.480 --> 00:34:25.480
With the 1/6, so
it's 1/24 u^6 + c.
00:34:25.480 --> 00:34:29.050
Now, that's not the
answer to the question.
00:34:29.050 --> 00:34:32.260
It's almost the answer
to the question.
00:34:32.260 --> 00:34:33.287
Why isn't it the answer?
00:34:33.287 --> 00:34:35.120
It isn't the answer
because now the answer's
00:34:35.120 --> 00:34:37.480
expressed in terms of u.
00:34:37.480 --> 00:34:41.750
Whereas the problem was posed
in terms of this variable x.
00:34:41.750 --> 00:34:45.960
So we must change back
to our variable here.
00:34:45.960 --> 00:34:47.990
And we do that just
by writing it in.
00:34:47.990 --> 00:34:56.190
So it's 1/24 (x^4 + 2)^6 + c.
00:34:56.190 --> 00:35:02.120
And this is the
end of the problem.
00:35:02.120 --> 00:35:02.990
Yeah, question.
00:35:02.990 --> 00:35:16.350
STUDENT: [INAUDIBLE]
00:35:16.350 --> 00:35:19.330
PROFESSOR: The question is,
can you see it directly?
00:35:19.330 --> 00:35:20.160
Yeah.
00:35:20.160 --> 00:35:23.820
And we're going to talk about
that in just one second.
00:35:23.820 --> 00:35:30.290
OK.
00:35:30.290 --> 00:35:35.500
Now, I'm going to
do one more example
00:35:35.500 --> 00:35:44.310
and illustrate this method.
00:35:44.310 --> 00:35:45.405
Here's another example.
00:35:45.405 --> 00:35:51.430
The integral of x dx over
the square root of 1 + x^2.
00:35:51.430 --> 00:35:56.610
Now, here's another example.
00:35:56.610 --> 00:36:03.475
Now, the method of substitution
leads us to the idea u = 1 +
00:36:03.475 --> 00:36:05.190
x^2.
00:36:05.190 --> 00:36:11.540
du = 2x dx, etc.
00:36:11.540 --> 00:36:14.450
It takes about as long as
this other problem did.
00:36:14.450 --> 00:36:15.720
To figure out what's going on.
00:36:15.720 --> 00:36:17.440
It's a very similar
sort of thing.
00:36:17.440 --> 00:36:20.870
You end up integrating u^(-1/2).
00:36:20.870 --> 00:36:23.480
It leads to the
integral of u^(-1/2) du.
00:36:28.350 --> 00:36:31.630
Is everybody seeing
where this...?
00:36:31.630 --> 00:36:37.540
However, there is a
slightly better method.
00:36:37.540 --> 00:36:46.070
So recommended method.
00:36:46.070 --> 00:36:59.250
And I call this method
advanced guessing.
00:36:59.250 --> 00:37:01.077
What advanced guessing
means is that you've
00:37:01.077 --> 00:37:02.660
done enough of these
problems that you
00:37:02.660 --> 00:37:04.750
can see two steps ahead.
00:37:04.750 --> 00:37:08.030
And you know what's
going to happen.
00:37:08.030 --> 00:37:10.440
So the advanced
guessing leads you
00:37:10.440 --> 00:37:12.817
to believe that here
you had a power -1/2,
00:37:12.817 --> 00:37:14.650
here you have the
differential of the thing.
00:37:14.650 --> 00:37:16.790
So it's going to
work out somehow.
00:37:16.790 --> 00:37:19.670
And the advanced guessing allows
you to guess that the answer
00:37:19.670 --> 00:37:23.520
should be something like
this. (1 + x^2)^(1/2).
00:37:26.050 --> 00:37:27.800
So this is your advanced guess.
00:37:27.800 --> 00:37:31.670
And now you just differentiate
it, and see whether it works.
00:37:31.670 --> 00:37:32.550
Well, here it is.
00:37:32.550 --> 00:37:39.330
It's 1/2 (1 + x^2)^(-1/2) 2x,
that's the chain rule here.
00:37:39.330 --> 00:37:44.770
Which, sure enough, gives you
x over square root of 1 + x^2.
00:37:44.770 --> 00:37:45.480
So we're done.
00:37:45.480 --> 00:37:56.960
And so the answer is square
root of (1 + x^2) + c.
00:37:56.960 --> 00:38:02.160
Let me illustrate this
further with another example.
00:38:02.160 --> 00:38:06.010
I strongly recommend
that you do this,
00:38:06.010 --> 00:38:09.270
but you have to get used to it.
00:38:09.270 --> 00:38:11.490
So here's another example.
00:38:11.490 --> 00:38:18.900
e^(6x) dx.
00:38:18.900 --> 00:38:26.360
My advanced guess is e^(6x).
00:38:26.360 --> 00:38:30.020
And if I check, when
I differentiate it,
00:38:30.020 --> 00:38:33.180
I get 6e^(6x).
00:38:33.180 --> 00:38:35.020
That's the derivative.
00:38:35.020 --> 00:38:38.030
And so I know that
the answer, so now I
00:38:38.030 --> 00:38:39.030
know what the answer is.
00:38:39.030 --> 00:38:46.300
It's 1/6 e^(6x) + c.
00:38:46.300 --> 00:38:54.510
Now, OK, you could,
it's also OK, but slow,
00:38:54.510 --> 00:39:02.550
to use a substitution,
to use u = 6x.
00:39:02.550 --> 00:39:07.920
Then you're going to get
du = 6dx, dot, dot, dot.
00:39:07.920 --> 00:39:23.220
It's going to work, it's
just a waste of time.
00:39:23.220 --> 00:39:26.760
Well, I'm going to give
you a couple more examples.
00:39:26.760 --> 00:39:27.910
So how about this one.
00:39:33.560 --> 00:39:35.120
x e^(-x^2) dx.
00:39:41.340 --> 00:39:45.600
What's the guess?
00:39:45.600 --> 00:39:51.270
Anybody have a guess?
00:39:51.270 --> 00:39:52.480
Well, you could also correct.
00:39:52.480 --> 00:39:54.438
So I don't want you to
bother - yeah, go ahead.
00:39:54.438 --> 00:39:56.787
STUDENT: [INAUDIBLE]
00:39:56.787 --> 00:39:59.120
PROFESSOR: Yeah, so you're
already one step ahead of me.
00:39:59.120 --> 00:40:02.050
Because this is too easy.
00:40:02.050 --> 00:40:04.110
When they get more
complicated, you just
00:40:04.110 --> 00:40:05.690
want to make this guess here.
00:40:05.690 --> 00:40:08.884
So various people have said
1/2, and they understand
00:40:08.884 --> 00:40:10.050
that there's 1/2 going here.
00:40:10.050 --> 00:40:13.890
But let me just show
you what happens, OK?
00:40:13.890 --> 00:40:19.140
If you make this guess
and you differentiate it,
00:40:19.140 --> 00:40:23.940
what you get here is
e^(-x^2) times the derivative
00:40:23.940 --> 00:40:28.130
of negative 2x, so that's -2x.
00:40:28.130 --> 00:40:30.400
x^2, so it's -2x.
00:40:30.400 --> 00:40:37.970
So now you see that you're off
by a factor of not 2, but -2.
00:40:37.970 --> 00:40:39.820
So a number of you
were saying that.
00:40:39.820 --> 00:40:43.120
So the answer is
-1/2 e^(-x^2) + c.
00:40:46.510 --> 00:40:49.090
And I can guarantee
you, having watched
00:40:49.090 --> 00:40:55.290
this on various problems, that
people who don't write this out
00:40:55.290 --> 00:40:57.360
make arithmetic mistakes.
00:40:57.360 --> 00:41:00.140
In other words, there
is a limit to how much
00:41:00.140 --> 00:41:02.760
people can think ahead
and guess correctly.
00:41:02.760 --> 00:41:04.760
Another way of doing
it, by the way,
00:41:04.760 --> 00:41:06.680
is simply to write
this thing in and then
00:41:06.680 --> 00:41:10.160
fix the coefficient by doing
the differentiation here.
00:41:10.160 --> 00:41:14.850
That's perfectly OK as well.
00:41:14.850 --> 00:41:18.920
All right, one more example.
00:41:18.920 --> 00:41:30.840
We're going to integrate
sin x cos x dx.
00:41:30.840 --> 00:41:33.750
So what's a good
guess for this one?
00:41:33.750 --> 00:41:36.520
STUDENT: [INAUDIBLE]
00:41:36.520 --> 00:41:38.850
PROFESSOR: Someone
suggesting sin^2 x.
00:41:38.850 --> 00:41:41.490
So let's try that.
00:41:41.490 --> 00:41:45.020
Over 2 - well, we'll get the
coefficient in just a second.
00:41:45.020 --> 00:41:50.970
So sin^2 x, if I differentiate,
I get 2 sin x cos x.
00:41:50.970 --> 00:41:53.380
So that's off by a factor of 2.
00:41:53.380 --> 00:42:04.540
So the answer is 1/2 sin^2 x.
00:42:04.540 --> 00:42:12.320
But now I want to
point out to you
00:42:12.320 --> 00:42:17.120
that there's another way
of doing this problem.
00:42:17.120 --> 00:42:31.240
It's also true that if
you differentiate cos^2 x,
00:42:31.240 --> 00:42:34.600
you get 2 cos x (-sin x).
00:42:38.130 --> 00:42:51.030
So another answer is that the
integral of sin x cos x dx is
00:42:51.030 --> 00:43:01.840
equal to -1/2 cos^2 x + c.
00:43:01.840 --> 00:43:03.740
So what is going on here?
00:43:03.740 --> 00:43:06.890
What's the problem with this?
00:43:06.890 --> 00:43:10.785
STUDENT: [INAUDIBLE]
00:43:10.785 --> 00:43:11.660
PROFESSOR: Pardon me?
00:43:11.660 --> 00:43:15.060
STUDENT: [INAUDIBLE]
00:43:15.060 --> 00:43:18.130
PROFESSOR: Integrals
aren't unique.
00:43:18.130 --> 00:43:21.390
That's part of the-- but
somehow these two answers still
00:43:21.390 --> 00:43:22.320
have to be the same.
00:43:22.320 --> 00:43:32.910
STUDENT: [INAUDIBLE]
00:43:32.910 --> 00:43:35.910
PROFESSOR: OK.
00:43:35.910 --> 00:43:36.660
What do you think?
00:43:36.660 --> 00:43:38.743
STUDENT: If you add them
together, you just get c.
00:43:38.743 --> 00:43:40.900
PROFESSOR: If you add
them together you get c.
00:43:40.900 --> 00:43:44.185
Well, actually,
that's almost right.
00:43:44.185 --> 00:43:45.810
That's not what you
want to do, though.
00:43:45.810 --> 00:43:47.620
You don't want to add them.
00:43:47.620 --> 00:43:50.930
You want to subtract them.
00:43:50.930 --> 00:43:53.750
So let's see what happens
when you subtract them.
00:43:53.750 --> 00:43:56.840
I'm going to ignore the
c, for the time being.
00:43:56.840 --> 00:44:05.520
I get sin^2 x, 1/2 sin^2
x - (-1/2 cos^2 x).
00:44:05.520 --> 00:44:08.880
So the difference between
them, we hope to be 0.
00:44:08.880 --> 00:44:10.760
But actually of
course it's not 0.
00:44:10.760 --> 00:44:17.680
What it is, is it's 1/2 sine
squared plus cosine squared,
00:44:17.680 --> 00:44:18.850
which is 1/2.
00:44:18.850 --> 00:44:24.200
It's not 0, it's a constant.
00:44:24.200 --> 00:44:27.310
So what's really going on here
is that these two formulas
00:44:27.310 --> 00:44:29.290
are the same.
00:44:29.290 --> 00:44:31.740
But you have to understand
how to interpret them.
00:44:31.740 --> 00:44:34.450
The two constants, here's
a constant up here.
00:44:34.450 --> 00:44:37.900
There's a constant, c_1
associated to this one.
00:44:37.900 --> 00:44:43.250
There's a different constant,
c_2 associated to this one.
00:44:43.250 --> 00:44:45.960
And this family of functions
for all possible c_1's
00:44:45.960 --> 00:44:49.860
and all possible c_2's, is
the same family of functions.
00:44:49.860 --> 00:44:52.940
Now, what's the relationship
between c_1 and c_2?
00:44:52.940 --> 00:44:57.210
Well, if you do the
subtraction, c_1 - c_2
00:44:57.210 --> 00:44:59.240
has to be equal to 1/2.
00:44:59.240 --> 00:45:06.610
They're both constants,
but they differ by 1/2.
00:45:06.610 --> 00:45:08.404
So this explains,
when you're dealing
00:45:08.404 --> 00:45:10.820
with families of things, they
don't have to look the same.
00:45:10.820 --> 00:45:12.560
And there are lots
of trig functions
00:45:12.560 --> 00:45:16.280
which look a little different.
00:45:16.280 --> 00:45:19.050
So there can be several formulas
that actually are the same.
00:45:19.050 --> 00:45:21.960
And it's hard to check that
they're actually the same.
00:45:21.960 --> 00:45:28.950
You need some trig
identities to do it.
00:45:28.950 --> 00:45:55.210
Let's do one more example here.
00:45:55.210 --> 00:46:06.250
Here's another one.
00:46:06.250 --> 00:46:13.510
Now, you may be thinking,
and a lot of people
00:46:13.510 --> 00:46:22.250
are, thinking ugh,
it's got a ln in it.
00:46:22.250 --> 00:46:24.400
If you're experienced,
you actually
00:46:24.400 --> 00:46:26.130
can read off the
answer just the way
00:46:26.130 --> 00:46:28.088
there were several people
who were shouting out
00:46:28.088 --> 00:46:31.000
the answers when we were doing
the rest of these problems.
00:46:31.000 --> 00:46:32.970
But, you do need to relax.
00:46:32.970 --> 00:46:35.790
Because in this case, now
this is definitely not
00:46:35.790 --> 00:46:37.490
true in general when
we do integrals.
00:46:37.490 --> 00:46:39.080
But, for now, when
we do integrals,
00:46:39.080 --> 00:46:40.570
they'll all be manageable.
00:46:40.570 --> 00:46:42.670
And there's only one method.
00:46:42.670 --> 00:46:47.390
Which is substitution.
00:46:47.390 --> 00:46:49.680
And in the substitution
method, you
00:46:49.680 --> 00:46:52.200
want to go for the
trickiest part.
00:46:52.200 --> 00:46:55.220
And substitute for that.
00:46:55.220 --> 00:46:57.630
So the substitution
that I proposed
00:46:57.630 --> 00:47:02.200
to you is that this should
be, u should be ln x.
00:47:02.200 --> 00:47:06.270
And the advantage that that
has is that its differential
00:47:06.270 --> 00:47:08.720
is simpler than itself.
00:47:08.720 --> 00:47:15.570
So du = dx / x.
00:47:15.570 --> 00:47:18.500
Remember, we use that in
logarithmic differentiation,
00:47:18.500 --> 00:47:21.550
too.
00:47:21.550 --> 00:47:28.810
So now we can express this
using this substitution.
00:47:28.810 --> 00:47:32.290
And what we get is,
the integral of,
00:47:32.290 --> 00:47:33.990
so I'll divide the
two parts here.
00:47:33.990 --> 00:47:36.515
It's 1 / ln x, and
then it's dx / x.
00:47:36.515 --> 00:47:43.370
And this part is 1 /
u, and this part is du.
00:47:43.370 --> 00:47:49.260
So it's the integral of du / u.
00:47:49.260 --> 00:47:58.490
And that is ln u + c.
00:47:58.490 --> 00:48:11.030
Which altogether, if I put back
in what u is, is ln (ln x) + c.
00:48:11.030 --> 00:48:14.290
And now we see
some uglier things.
00:48:14.290 --> 00:48:15.900
In fact, technically
speaking, we
00:48:15.900 --> 00:48:18.730
could take the
absolute value here.
00:48:18.730 --> 00:48:28.130
And then this would be
absolute values there.
00:48:28.130 --> 00:48:33.090
So this is the type of
example where I really
00:48:33.090 --> 00:48:35.740
would recommend that you
actually use the substitution,
00:48:35.740 --> 00:48:39.030
at least for now.
00:48:39.030 --> 00:48:41.820
All right, tomorrow
we're going to be
00:48:41.820 --> 00:48:43.080
doing differential equations.
00:48:43.080 --> 00:48:45.130
And we're going to
review for the test.
00:48:45.130 --> 00:48:47.649
I'm going to give you a handout
telling you just exactly
00:48:47.649 --> 00:48:48.940
what's going to be on the test.
00:48:48.940 --> 00:48:52.298
So, see you tomorrow.