Indefinite Integration (part 7)

WEBVTT
Kind: captions
Language: en

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I'm now going to do it
integration by parts problems.

00:00:03.730 --> 00:00:07.870
I think it's just a fun problem
to see because one, it's the

00:00:07.870 --> 00:00:10.300
example a lot of people use,
sometimes even a trick problem

00:00:10.300 --> 00:00:13.920
that's given on a really hard
math exam, or if you go to

00:00:13.920 --> 00:00:17.560
calculus competitions like
I used to in high school.

00:00:17.560 --> 00:00:21.470
Not to make myself too-- I was
actually not that geeky as a

00:00:21.470 --> 00:00:25.650
high school student, but I have
to admit, I was a mathlete.

00:00:25.650 --> 00:00:29.830
But anyway, this is just a fun
integration by parts problem

00:00:29.830 --> 00:00:34.770
because you actually never have
to evaluate the final integral.

00:00:34.770 --> 00:00:36.830
So let's say we want to
take the integral-- it's

00:00:36.830 --> 00:00:38.690
a bit of a classic.

00:00:38.690 --> 00:00:41.660
I wouldn't be surprised if your
math teacher does the same

00:00:41.660 --> 00:00:43.790
problem for you, just to show
you integration by parts.

00:00:43.790 --> 00:00:47.080
Let's take the integral of e to
the x-- you probably never

00:00:47.080 --> 00:00:49.960
heard of someone call a math
problem a classic, but

00:00:49.960 --> 00:00:53.810
hopefully I will instill in you
this love for mathematics and

00:00:53.810 --> 00:00:57.540
you will also consider this
to be a classic problem.

00:00:57.540 --> 00:00:59.760
e to the x times cosine of x.

00:01:02.670 --> 00:01:04.490
I think you might already see
where I'm going with this,

00:01:04.490 --> 00:01:07.330
because these are both fun
functions, because e to the x

00:01:07.330 --> 00:01:08.870
you can take the derivative,
you could take the

00:01:08.870 --> 00:01:11.280
anti-derivative and it
still stays e to the x.

00:01:11.280 --> 00:01:14.510
Cosine of x you take the
derivative, you go to minus

00:01:14.510 --> 00:01:16.980
sign of x, you take the
derivative again then you to

00:01:16.980 --> 00:01:19.270
minus cosine of x, then you
take the derivative again then

00:01:19.270 --> 00:01:20.420
you get a plus sign of x.

00:01:20.420 --> 00:01:21.450
It's like this cycle.

00:01:21.450 --> 00:01:23.540
The same thing happens when
you take the anti-derivative.

00:01:23.540 --> 00:01:26.340
It's not as cool as e to the x,
it doesn't stay exactly the

00:01:26.340 --> 00:01:29.390
same, but it kind of cycles.

00:01:29.390 --> 00:01:31.690
If you take two
anti-derivatives you get back

00:01:31.690 --> 00:01:33.550
to the negative of itself.

00:01:33.550 --> 00:01:35.400
And if you take two
derivatives, you get back

00:01:35.400 --> 00:01:37.780
to the negative of itself.

00:01:37.780 --> 00:01:40.320
It's also a pretty cool
function and I think you can

00:01:40.320 --> 00:01:45.850
start to see how integration
by parts might be cool here.

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Whenever I do integration by
parts I always like to assume

00:01:48.730 --> 00:01:50.880
that this is the g prime of x.

00:01:50.880 --> 00:01:54.070
That e to the x is g prime
of x, because e to the x

00:01:54.070 --> 00:01:55.090
literally doesn't change.

00:01:55.090 --> 00:01:58.690
Although we could do this
problem the other way.

00:01:58.690 --> 00:02:00.820
Maybe I'll experiment
doing it the other way.

00:02:00.820 --> 00:02:03.190
but let's assume this is
g prime of x, and let's

00:02:03.190 --> 00:02:06.410
assume this f of x.

00:02:06.410 --> 00:02:08.450
So this is derivative.

00:02:08.450 --> 00:02:11.420
So integration by parts, as we
take the original functions,

00:02:11.420 --> 00:02:14.430
g of x and f of x.

00:02:14.430 --> 00:02:16.590
If this is g prime of
x, what's go of x.

00:02:16.590 --> 00:02:20.130
What's the anti-derivative
of e to the x.

00:02:20.130 --> 00:02:21.270
It's just use e to the x.

00:02:21.270 --> 00:02:23.460
I'm going to switch colors,
I don't like this blue.

00:02:23.460 --> 00:02:27.200
So this is g of x.

00:02:27.200 --> 00:02:29.450
I actually took the
anti-derivative of it, but

00:02:29.450 --> 00:02:31.060
it's the same exact thing.

00:02:31.060 --> 00:02:35.320
And then times f of x.

00:02:35.320 --> 00:02:41.320
Then I want to subtract
the indefinite integral

00:02:41.320 --> 00:02:46.100
of f prime of x.

00:02:46.100 --> 00:02:47.715
One, g of x.

00:02:51.200 --> 00:02:54.220
This is the same as this, which
are both the anti-derivative of

00:02:54.220 --> 00:02:56.370
this, although they
are all the same.

00:02:56.370 --> 00:03:01.830
So this is g of x and then I
would take the derivative

00:03:01.830 --> 00:03:04.050
of f of x. f prime of x.

00:03:04.050 --> 00:03:05.770
What's the derivative
of cosine of x?

00:03:05.770 --> 00:03:07.740
It's minus sine of x.

00:03:07.740 --> 00:03:13.410
So sine of x d x, it's
minus sine of x.

00:03:13.410 --> 00:03:15.490
I could put the minus here,
that'll make it look messy, I

00:03:15.490 --> 00:03:16.970
could put the minus here
that'll make it messy or I

00:03:16.970 --> 00:03:19.240
could just put minus here and
make these minuses cancel

00:03:19.240 --> 00:03:21.340
out and I get a plus here.

00:03:21.340 --> 00:03:25.370
So I get the integral of e to
the x cosine of x d x is equal

00:03:25.370 --> 00:03:29.886
to e to the x cosine of x plus
the integral of e to

00:03:29.886 --> 00:03:32.680
the x sine of x d x.

00:03:32.680 --> 00:03:34.410
Hopefully I haven't
confused you too much.

00:03:34.410 --> 00:03:36.430
I should actually do some
integration by parts problems

00:03:36.430 --> 00:03:37.280
without e to the x.

00:03:37.280 --> 00:03:40.460
It's very hard to keep track
of what I've done here.

00:03:40.460 --> 00:03:41.250
This is the anti-derivative.

00:03:44.900 --> 00:03:47.000
This is the anti-derivative
and this is also the

00:03:47.000 --> 00:03:48.000
anti-derivative.

00:03:48.000 --> 00:03:50.565
This is g prime of
x, this is g of x.

00:03:55.670 --> 00:03:58.520
So once again we are not
clear whether we've

00:03:58.520 --> 00:03:59.920
made any progress.

00:03:59.920 --> 00:04:00.070
We've

00:04:00.070 --> 00:04:03.780
gone from e to the x cosine of
x to e to the x sine of x.

00:04:03.780 --> 00:04:08.150
Let's take integration by parts
again, see what happens.

00:04:08.150 --> 00:04:10.550
I'm just going to write on the
right side of the equal sign,

00:04:10.550 --> 00:04:13.900
because this might
get a little long.

00:04:13.900 --> 00:04:18.620
I'm just going to write this
first part x to the x cosine

00:04:18.620 --> 00:04:23.945
of x plus-- and now let's do
integration by parts again.

00:04:32.520 --> 00:04:34.880
For this round of integration
by parts this was g of x, but

00:04:34.880 --> 00:04:39.730
now, for this around, I'm going
to assume it's g prime of x.

00:04:39.730 --> 00:04:42.080
Which doesn't really make a
difference because whenever I

00:04:42.080 --> 00:04:44.440
take the anti-derivative of it
to g of x, it stays the same.

00:04:44.440 --> 00:04:45.870
And then I'm going to assume
that this is f of x.

00:04:48.750 --> 00:04:54.370
So integration by parts tells
us we take f of x times g of x,

00:04:54.370 --> 00:04:57.270
so I take this function and the
anti-derivative of

00:04:57.270 --> 00:04:59.130
this function.

00:04:59.130 --> 00:05:02.170
The anti-derivative of this
function is once again just e

00:05:02.170 --> 00:05:06.940
to the x and then f times that
function unchanged

00:05:06.940 --> 00:05:09.630
time sine of x.

00:05:09.630 --> 00:05:16.160
From that I subtract the
integral of the anti-derivative

00:05:16.160 --> 00:05:21.410
of this or I take g of x which
is e to the x, and then the

00:05:21.410 --> 00:05:24.390
derivative of f of
x, f prime of x.

00:05:24.390 --> 00:05:25.680
What's the derivative
of sine of x?

00:05:25.680 --> 00:05:28.605
It's cosine of x.

00:05:28.605 --> 00:05:31.590
Cosine of x d of x.

00:05:31.590 --> 00:05:32.650
Let's see if we're
getting anywhere.

00:05:32.650 --> 00:05:35.560
It seems like I just keep
adding terms, making it

00:05:35.560 --> 00:05:36.760
more and more complicated.

00:05:36.760 --> 00:05:38.980
In order to see if we're
getting anywhere, let me just

00:05:38.980 --> 00:05:41.590
rewrite the whole thing and
maybe get rid of these

00:05:41.590 --> 00:05:43.430
parenthesis, because it's
just a plus, so we can get

00:05:43.430 --> 00:05:44.155
rid of the parenthesis.

00:05:48.250 --> 00:05:51.220
Let me use a new color.

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OK.

00:05:51.980 --> 00:06:00.020
So this is the original
problem, e to the x cosine of x

00:06:00.020 --> 00:06:05.820
d x equals, and now let me
switch back to this color, it

00:06:05.820 --> 00:06:11.920
equals e to the x cosine of x,
and then I can just-- this

00:06:11.920 --> 00:06:13.970
parentheses doesn't matter
because I'm just adding

00:06:13.970 --> 00:06:18.730
everything in the parentheses--
e to x cosine of x plus e to

00:06:18.730 --> 00:06:35.520
the x sine of x minus e to
the x cosine x access d x.

00:06:35.520 --> 00:06:38.330
Now you might think that I
arbitrarily switched colors

00:06:38.330 --> 00:06:42.360
here when I rewrote this,
but if you look you might

00:06:42.360 --> 00:06:45.800
see why I actually did
switch colors here.

00:06:45.800 --> 00:06:47.740
See anything interesting?

00:06:47.740 --> 00:06:48.890
Exactly.

00:06:48.890 --> 00:06:51.590
This is the same thing as
this, just a minus right?

00:06:51.590 --> 00:06:55.060
So we're going to do something
what I think to be fairly cool.

00:06:55.060 --> 00:06:59.110
Let's add this term to both
sides of the equation.

00:06:59.110 --> 00:07:01.140
Let's take this and
let's put it on to this

00:07:01.140 --> 00:07:02.420
side of the equation.

00:07:02.420 --> 00:07:03.800
If I take this and put
it on this side of the

00:07:03.800 --> 00:07:05.500
equation, what happens?

00:07:05.500 --> 00:07:08.100
I then have two of these on the
left side equation, so that

00:07:08.100 --> 00:07:16.250
becomes-- I mean I could write
it out it's e to the x cosine

00:07:16.250 --> 00:07:19.250
of x d x plus, right?

00:07:19.250 --> 00:07:20.710
Because I'm taking this and I'm
putting it on that side of the

00:07:20.710 --> 00:07:26.620
equation, e to the
x cosine of x d x.

00:07:26.620 --> 00:07:30.470
That's just the same thing as
2 times the integral of e

00:07:30.470 --> 00:07:34.770
to the x cosine of x d x.

00:07:34.770 --> 00:07:37.560
And then that equals this term.

00:07:37.560 --> 00:07:45.300
Which equals e to the x cosine
of x plus e to the x sine of x.

00:07:45.300 --> 00:07:47.020
I know it's really messy.

00:07:47.020 --> 00:07:49.810
All I have to do now to solve
this integral is divide both

00:07:49.810 --> 00:07:52.670
sides by 2 and I'm done.

00:07:52.670 --> 00:07:55.240
So let me write it out, this
is very exciting, it's

00:07:55.240 --> 00:07:56.800
the home stretch.

00:07:56.800 --> 00:07:59.750
If I divide both sides by 2, I
get-- and I'm going to try to

00:07:59.750 --> 00:08:05.740
write it so you can see
everything-- e to the x cosine

00:08:05.740 --> 00:08:16.730
of x d x equals and on that
side I have e to the x cosine

00:08:16.730 --> 00:08:26.200
of x plus e to the x
sine of x over 2.

00:08:26.200 --> 00:08:28.480
I think that's pretty neat.

00:08:28.480 --> 00:08:31.720
It's neat how integration by
parts allowed us to do this.

00:08:31.720 --> 00:08:33.510
We actually never even have
to evaluate this integral.

00:08:33.510 --> 00:08:36.080
We said, this integral is just
the original problem again.

00:08:36.080 --> 00:08:37.700
And you can think about
why that happened, right?

00:08:37.700 --> 00:08:39.540
Because these trick
functions cycle.

00:08:39.540 --> 00:08:42.030
So we had to do integration
by parts twice to get back

00:08:42.030 --> 00:08:43.560
to where we were before.

00:08:43.560 --> 00:08:50.010
And then we use that to solve
it without actually having

00:08:50.010 --> 00:08:51.330
to evaluate the integral.

00:08:51.330 --> 00:08:53.520
And what I also think is cool
is even if you just look

00:08:53.520 --> 00:08:57.410
at this solution, it's
kind of neat, right?

00:08:57.410 --> 00:09:01.240
The anti-derivative of e to the
x and-- actually never forget

00:09:01.240 --> 00:09:05.560
the plus c, that would've given
me minus 1 point on the exam.

00:09:05.560 --> 00:09:08.320
What's kind of cool, the
integral of e to the x cosine

00:09:08.320 --> 00:09:13.423
of x is this expression that's
e to the x cosine of x plus e

00:09:13.423 --> 00:09:15.290
to the x sine of
x divided by 2.

00:09:15.290 --> 00:09:19.520
It's the average of e to
the x cosine of x and

00:09:19.520 --> 00:09:21.160
e to the x sine of x.

00:09:21.160 --> 00:09:24.790
I think that's a pretty neat
property, and you might want

00:09:24.790 --> 00:09:30.210
to graph them and play with
them, but it's kind of neat.

00:09:30.210 --> 00:09:33.980
Hopefully I have convinced you
that is a classic of a problem,

00:09:33.980 --> 00:09:37.130
and you also find it neat, and
I'll see you in the

00:09:37.130 --> 00:09:38.640
next presentation.


Integration by Parts (part 6 of Indefinite Integration)

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:01.900
Welcome back.

00:00:01.900 --> 00:00:05.920
Well I'm now going to do just a
bunch of integration by parts

00:00:05.920 --> 00:00:09.110
problems, as many as I can do
in ten minutes without

00:00:09.110 --> 00:00:09.750
confusing you.

00:00:09.750 --> 00:00:11.400
So let me just write the
formula for integration by

00:00:11.400 --> 00:00:13.910
parts, and if you ever forget
it-- I mean, it doesn't hurt to

00:00:13.910 --> 00:00:15.840
memorize it, but if you ever
forget it-- you just really

00:00:15.840 --> 00:00:18.240
have to just derive it from the
product rule of

00:00:18.240 --> 00:00:19.480
differentiation.

00:00:19.480 --> 00:00:27.080
But it just says that if we
have an integral of f of x

00:00:27.080 --> 00:00:32.320
times g prime of x-- so if you
see, within the integral, sign,

00:00:32.320 --> 00:00:34.590
one function and then you see
the derivative of another

00:00:34.590 --> 00:00:38.640
function, and I think with
practice-- integration by parts

00:00:38.640 --> 00:00:40.150
is really a bit of an art.

00:00:40.150 --> 00:00:45.480
It's not systematic-- that's g
prime of x-- that equals f of

00:00:45.480 --> 00:00:51.650
access times g of x-- this is
how it's the product rule in

00:00:51.650 --> 00:00:56.200
reverse-- minus the integral of
the derivative of the first

00:00:56.200 --> 00:01:01.650
function, f prime of x,
times the second function.

00:01:01.650 --> 00:01:03.570
And it kind of easy to
memorize, because there's this

00:01:03.570 --> 00:01:04.605
symmetry to the formula.

00:01:08.110 --> 00:01:09.490
So let's see if we
can apply this.

00:01:09.490 --> 00:01:12.140
And really, once you know that
you should use integration by

00:01:12.140 --> 00:01:15.140
parts, I think you'll find that
it's not that hard to do it.

00:01:15.140 --> 00:01:17.510
The hard part is to recognize
when you should use

00:01:17.510 --> 00:01:18.890
integration by parts.

00:01:18.890 --> 00:01:22.940
From my point of view, it's
kind of my last resort, or once

00:01:22.940 --> 00:01:24.650
you have a lot of practice, you
might recognize, well, if

00:01:24.650 --> 00:01:27.250
there's an e to the x in it, or
if there's a trig function in

00:01:27.250 --> 00:01:32.320
it, and I can't do the reverse
chain rule or integration by

00:01:32.320 --> 00:01:37.370
substitution, then integration
by parts is probably my best

00:01:37.370 --> 00:01:39.580
option, assuming I'm seeing
this on an exam and

00:01:39.580 --> 00:01:40.070
not in real life.

00:01:40.070 --> 00:01:42.150
In real life, it might be an
unsolvable integral, and

00:01:42.150 --> 00:01:44.800
you'd have to use a computer
or some other technique.

00:01:44.800 --> 00:01:46.550
But if you're seeing it on an
exam, you know it's a solvable

00:01:46.550 --> 00:01:48.410
integral, and if you can't
solve it any other way, it's

00:01:48.410 --> 00:01:50.830
probably integration by parts.

00:01:50.830 --> 00:01:52.690
But let's just do
some problems.

00:01:52.690 --> 00:01:59.600
Let's say I want to take the
integral of x squared e to dx.

00:02:03.890 --> 00:02:05.920
So if I saw this out of the
blue, and I didn't know that

00:02:05.920 --> 00:02:08.660
this was a presentation on the
integration by parts, I would

00:02:08.660 --> 00:02:11.440
first-- clearly this isn't a
polynomial, so I can't just do

00:02:11.440 --> 00:02:14.440
a simple polynomial
antiderivative.

00:02:14.440 --> 00:02:18.190
Then I would try to see, is
there the derivative of

00:02:18.190 --> 00:02:23.310
something, of one function, of
kind of a composite function

00:02:23.310 --> 00:02:25.310
here so I can do the
reverse chain rule.

00:02:25.310 --> 00:02:28.450
The derivative of x here is 1,
so I can't do anything here.

00:02:28.450 --> 00:02:30.290
So I use the chain rule.

00:02:30.290 --> 00:02:33.700
And the way I think about
the chain rule is I

00:02:33.700 --> 00:02:35.310
want to simplify it.

00:02:35.310 --> 00:02:38.040
So when I go into this term
right here, I'm going to

00:02:38.040 --> 00:02:40.490
have to pick up my f of x.

00:02:40.490 --> 00:02:43.300
I have to pick my f of x out
of probably one of these two

00:02:43.300 --> 00:02:47.500
functions, so that the f
prime of x is simpler.

00:02:47.500 --> 00:02:53.700
And I need to pick my g prime
of x, I would guess that either

00:02:53.700 --> 00:02:58.230
x squared is going to be my g
prime of x, or e to the x is

00:02:58.230 --> 00:03:00.790
going to be my g prime of x,
and I want to pick that so that

00:03:00.790 --> 00:03:02.855
when I take the antiderivative
of it, it's going

00:03:02.855 --> 00:03:03.530
to be simpler.

00:03:03.530 --> 00:03:06.030
Or at least, not
more complicated.

00:03:06.030 --> 00:03:07.570
I know that if I take the
derivative of x squared,

00:03:07.570 --> 00:03:08.850
that simplifies it.

00:03:08.850 --> 00:03:11.820
And I also know that the-- and
once again, this is one of my,

00:03:11.820 --> 00:03:15.510
to me, a very mind blowing
idea-- but that the

00:03:15.510 --> 00:03:18.740
antiderivative of e to
the x is e to the x.

00:03:18.740 --> 00:03:23.860
So it's probably a good idea to
say that f of x is equal to--

00:03:23.860 --> 00:03:27.280
it's probably, let me change
colors-- it's probably a good

00:03:27.280 --> 00:03:30.650
idea to make f of x is equal to
x squared, because later I'm

00:03:30.650 --> 00:03:32.310
going to take the derivative of
it, and the derivative of it

00:03:32.310 --> 00:03:35.850
simplifies it, and it's
probably a good idea to make g

00:03:35.850 --> 00:03:39.830
prime of x e to the x, because
later, I'm going to take the

00:03:39.830 --> 00:03:41.760
antiderivative of it, and the
antiderivative of e to

00:03:41.760 --> 00:03:42.770
the x is e to the x.

00:03:42.770 --> 00:03:45.240
It's not going to become
any more complicated.

00:03:45.240 --> 00:03:52.120
So if we assume what I'm doing
is right, then what did we say?

00:03:52.120 --> 00:03:56.890
Well here, we just multiply the
two real functions, right?

00:03:56.890 --> 00:03:59.550
So when I say the real
functions, I mean not

00:03:59.550 --> 00:04:01.030
the derivatives of
e to one of them.

00:04:01.030 --> 00:04:04.460
So f of x, we're saying
f of x is x squared.

00:04:04.460 --> 00:04:07.290
Let me try to stay
color consistent.

00:04:09.840 --> 00:04:13.230
And we said g of x-- now don't
get confused-- we're saying g

00:04:13.230 --> 00:04:16.790
prime of x-- let me write it in
a corner down here-- we're

00:04:16.790 --> 00:04:22.360
saying that g prime of x
is equal to e to the x.

00:04:22.360 --> 00:04:26.050
And of course, if g prime of x
is equal to ex, then g of x

00:04:26.050 --> 00:04:30.520
is also equal to e to the x.

00:04:30.520 --> 00:04:32.840
So g of x-- I don't want you
to think that I'm somehow

00:04:32.840 --> 00:04:34.420
putting g prime of x here.

00:04:34.420 --> 00:04:36.550
I've taken the antiderivative,
it just happens to be

00:04:36.550 --> 00:04:37.705
the same function.

00:04:41.900 --> 00:04:47.500
And then from that, we subtract
the integral, we take the

00:04:47.500 --> 00:04:56.890
derivative of x squared, so you
get 2x, and then times the

00:04:56.890 --> 00:04:59.980
antiderivative of g prime of x.

00:04:59.980 --> 00:05:02.590
Well, the g prime of x is
e to the x, you take the

00:05:02.590 --> 00:05:06.020
antiderivative, it's
still e to the x.

00:05:06.020 --> 00:05:08.955
Actually I should probably stay
at least keep the colors kind

00:05:08.955 --> 00:05:11.870
of consistent, so you know
what I'm hopefully doing.

00:05:14.520 --> 00:05:16.540
The example using e to the x
might be a little complicated,

00:05:16.540 --> 00:05:18.030
because it's hard to tell
whether I've taken the

00:05:18.030 --> 00:05:19.130
derivative or not.

00:05:19.130 --> 00:05:21.330
And you can kind of keep
reverting back to the top

00:05:21.330 --> 00:05:22.405
formula if you get confused.

00:05:29.130 --> 00:05:31.650
So it looks like I've
simplified it a little bit.

00:05:31.650 --> 00:05:37.460
This integral looks easier to
solve than this integral.

00:05:37.460 --> 00:05:39.760
But once again, when I look
at this, I'm like, well,

00:05:39.760 --> 00:05:40.670
how do I solve this?

00:05:40.670 --> 00:05:45.230
I can't use the integration by
substitution, because there's

00:05:45.230 --> 00:05:48.300
not an embedded function and
then I have the derivative of

00:05:48.300 --> 00:05:50.190
it sitting right
next to it, so.

00:05:50.190 --> 00:05:52.900
Maybe I need to do
integration by parts again.

00:05:52.900 --> 00:05:54.030
So let's do that.

00:05:54.030 --> 00:05:59.260
So let's say-- let me do it
separately-- let's assume-- I

00:05:59.260 --> 00:06:01.390
think you getting a little bit
of the hang of it-- along the

00:06:01.390 --> 00:06:04.990
same vein, that this is f of x,
that this is f of x, and that

00:06:04.990 --> 00:06:06.630
this is g prime of x now.

00:06:06.630 --> 00:06:09.320
We're kind of doing
integration by parts within

00:06:09.320 --> 00:06:10.730
integration by parts.

00:06:10.730 --> 00:06:14.070
So if that is the case, that
this integral is going to

00:06:14.070 --> 00:06:17.080
equal-- because we have the
minus sign out front, this

00:06:17.080 --> 00:06:21.500
minus sign out front-- this
integral is going to equal f

00:06:21.500 --> 00:06:24.540
of x times g of x where
f of x is just 2x.

00:06:28.060 --> 00:06:30.530
Our g of x, this is
g prime of x now.

00:06:30.530 --> 00:06:32.710
Remember, we're kind of doing
a new problem within the

00:06:32.710 --> 00:06:34.560
original big problem.

00:06:34.560 --> 00:06:38.540
So this is g prime of x, but
g of x is still just e of x.

00:06:38.540 --> 00:06:41.550
I took the
antiderivative of it.

00:06:41.550 --> 00:06:47.810
And that's minus the integral
of the derivative of

00:06:47.810 --> 00:06:50.050
the first function.

00:06:50.050 --> 00:06:51.830
That prime of x.

00:06:51.830 --> 00:06:53.530
So that's just 2.

00:06:53.530 --> 00:06:57.680
And then the antiderivative
of the second function.

00:06:57.680 --> 00:06:58.290
Well, that's easy.

00:06:58.290 --> 00:07:00.010
The antiderivative of e to
the x is just e to the x.

00:07:03.210 --> 00:07:04.310
Interesting.

00:07:04.310 --> 00:07:05.890
Now I think you see
where we're going.

00:07:05.890 --> 00:07:08.850
This is actually-- let me
write out the whole thing.

00:07:08.850 --> 00:07:11.870
Because this is-- x
squared, e to the x.

00:07:11.870 --> 00:07:14.680
Just so we don't lose track
of our original problem.

00:07:14.680 --> 00:07:15.310
Interesting.

00:07:15.310 --> 00:07:18.510
Now I think we have an
integral that is pretty

00:07:18.510 --> 00:07:20.400
straightforward to solve.

00:07:20.400 --> 00:07:22.430
Don't want to forget my dx's.

00:07:22.430 --> 00:07:25.840
What's the integral of-- we
could take this 2 out of

00:07:25.840 --> 00:07:27.880
this, and I think it becomes
pretty obvious-- what's the

00:07:27.880 --> 00:07:29.120
integral of e to the x?

00:07:29.120 --> 00:07:30.880
This is a scratch-out,
this says dex.

00:07:30.880 --> 00:07:32.830
Just so-- and it's
a little messy.

00:07:32.830 --> 00:07:34.980
I don't like this color.

00:07:34.980 --> 00:07:36.230
Magenta.

00:07:36.230 --> 00:07:38.640
Well the integral of e to the
x, or the antiderivative of e

00:07:38.640 --> 00:07:40.970
to the x is e to the x, right?

00:07:40.970 --> 00:07:41.780
So let's write that.

00:07:41.780 --> 00:07:43.940
So I'm going to rewrite
everything we've done it.

00:07:43.940 --> 00:07:58.260
So there's x squared e to the x
minus 2xe to the x, and then

00:07:58.260 --> 00:08:01.150
this minus, you distribute it
so it becomes a plus, so then

00:08:01.150 --> 00:08:05.940
it's plus 2-- I just took the
minus, I multiplied it times

00:08:05.940 --> 00:08:08.780
this minus, so I got a plus 2--
and then the antiderivative of

00:08:08.780 --> 00:08:12.210
e to the x is just e to the x.

00:08:12.210 --> 00:08:17.140
And then of course, we should
never forget the plus c.

00:08:17.140 --> 00:08:18.560
Pretty fancy, no?

00:08:18.560 --> 00:08:24.130
We've figured out the
antiderivative, the indefinite

00:08:24.130 --> 00:08:27.550
integral of x squared e to the
x is this big fancy thing.

00:08:27.550 --> 00:08:29.520
I bet you, before listening to
this video, you would never

00:08:29.520 --> 00:08:34.260
imagine that you could tackle
integration like this.

00:08:34.260 --> 00:08:36.970
You could actually try x to
the n times e to the x.

00:08:36.970 --> 00:08:38.580
You can try x to the
tenth times e to the x.

00:08:38.580 --> 00:08:41.480
It actually turns out you'll
just have to do this many many

00:08:41.480 --> 00:08:44.550
many many times, but every time
you do integration by parts,

00:08:44.550 --> 00:08:48.670
that the exponent on the x-term
just becomes smaller and

00:08:48.670 --> 00:08:51.490
smaller and smaller until you
get to something that's really

00:08:51.490 --> 00:08:53.640
easy to integrate, and then you
can do it, and you'll have kind

00:08:53.640 --> 00:08:55.430
of this big long expression.

00:08:55.430 --> 00:08:58.550
It might be tedious, it might
be hairy, but at least you have

00:08:58.550 --> 00:09:00.840
a tool kit-- or you have
something in your tool kit--

00:09:00.840 --> 00:09:05.180
that you can tackle integral
problems like this.

00:09:05.180 --> 00:09:07.850
I'll probably do one more video
on integration by parts, just

00:09:07.850 --> 00:09:11.620
because I think this is one of
the harder concepts to really

00:09:11.620 --> 00:09:14.930
grasp and feel comfortable
with, and then I'll try to

00:09:14.930 --> 00:09:16.590
do a bunch of examples.

00:09:16.590 --> 00:09:19.560
Maybe not soon, but in the
next couple of weeks, on

00:09:19.560 --> 00:09:20.670
just a lot of integration.

00:09:20.670 --> 00:09:23.210
And I'm going to mix it up so
that hopefully you can get a

00:09:23.210 --> 00:09:27.680
sense of how I try to figure
out which of my integration

00:09:27.680 --> 00:09:29.650
tools I should use for a
certain problem when

00:09:29.650 --> 00:09:30.870
I see the problem.

00:09:30.870 --> 00:09:32.720
See you in the next
presentation.


Definite Integrals (area under a curve) (part III)

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:01.410
Welcome back.

00:00:01.410 --> 00:00:05.400
I'm just continuing on with
hopefully giving you, one, how

00:00:05.400 --> 00:00:07.660
to actually solve indefinite
integrals and also giving you a

00:00:07.660 --> 00:00:10.950
sense of why you solve
it the way you do.

00:00:10.950 --> 00:00:16.150
And I think that's often
missing in some textbooks.

00:00:16.150 --> 00:00:17.780
But anyway, let's say that this
is the distance and let me give

00:00:17.780 --> 00:00:21.250
you a formula, actually, for
the distance, just for fun.

00:00:21.250 --> 00:00:23.200
Oh, my phone is ringing.

00:00:23.200 --> 00:00:26.590
Let me lower the volume,
because you're more important.

00:00:26.590 --> 00:00:30.610
So, let's say that the
distance, s -- this time I'll

00:00:30.610 --> 00:00:34.845
write it as a function -- let's
say the distance is -- I said

00:00:34.845 --> 00:00:40.220
it started at five, so let's
say it's 2t -- let's say this

00:00:40.220 --> 00:00:42.440
is actually a cubic function.

00:00:42.440 --> 00:00:47.290
You're not only accelerating,
your rate of acceleration

00:00:47.290 --> 00:00:47.970
is increasing.

00:00:47.970 --> 00:00:50.300
I think, actually, the rate
of acceleration, if I'm not

00:00:50.300 --> 00:00:52.340
mistaken, is actually called
jerk, but I might have

00:00:52.340 --> 00:00:54.330
to Wikipedia that.

00:00:54.330 --> 00:00:57.690
Let's say its 2t to
the third plus 5.

00:01:00.240 --> 00:01:08.980
And let's say I wanted to know
how far I travel between t

00:01:08.980 --> 00:01:17.540
equals 2 seconds and
t equals 5 seconds.

00:01:17.540 --> 00:01:18.100
Right?

00:01:18.100 --> 00:01:20.130
I'm not looking for the total
distance I've traveled.

00:01:20.130 --> 00:01:23.570
I just want to know how far do
I travel between time equals 2

00:01:23.570 --> 00:01:25.790
seconds and time
equal 5 seconds.

00:01:25.790 --> 00:01:26.040
Right?

00:01:26.040 --> 00:01:30.610
So this might be
2 and this is 5.

00:01:30.610 --> 00:01:33.082
So an easy way to do that is
I could just evaluate this

00:01:33.082 --> 00:01:36.920
function at t equals 5 -- let
me use a different color.

00:01:36.920 --> 00:01:39.750
I think it's getting messy --
I could just evaluate this

00:01:39.750 --> 00:01:40.910
function at t equals 5.

00:01:44.890 --> 00:01:53.200
If t equals 5, 5 to the third
power is 125, 250, it's 255,

00:01:53.200 --> 00:01:58.640
so the object has gone 255
feet at t equals 5, right?

00:01:58.640 --> 00:02:04.070
And then, at time equals 2,
the object has gone how far?

00:02:04.070 --> 00:02:05.110
2 to the third is 8.

00:02:05.110 --> 00:02:05.550
16.

00:02:05.550 --> 00:02:07.180
It's gone 21 feet.

00:02:07.180 --> 00:02:09.590
Right?

00:02:09.590 --> 00:02:12.450
To figure out how far I
travelled between time equals 2

00:02:12.450 --> 00:02:18.720
and time equals 5, I just say s
of 5 minus s of 2, right?

00:02:18.720 --> 00:02:22.825
How far did I go after 5
seconds minus how far I

00:02:22.825 --> 00:02:24.000
already was after 2 seconds.

00:02:24.000 --> 00:02:33.520
And this is just 255 minus
21 and that's, what, 234.

00:02:33.520 --> 00:02:36.480
234 feet is how far I
travelled between 2

00:02:36.480 --> 00:02:38.070
seconds and 5 seconds.

00:02:38.070 --> 00:02:39.040
Interesting.

00:02:39.040 --> 00:02:40.950
And I think you're starting to
get a little intuition about

00:02:40.950 --> 00:02:43.420
why we evaluated that previous
indefinite integral in the

00:02:43.420 --> 00:02:45.120
previous video the way we did.

00:02:45.120 --> 00:02:47.820
So let's actually draw the
derivative of this function.

00:02:47.820 --> 00:02:49.220
So what's the derivative?

00:02:49.220 --> 00:02:53.950
So let me just call
that v of t, I guess.

00:02:53.950 --> 00:02:56.690
v of t is just the derivative,
right, because it's the rate

00:02:56.690 --> 00:02:59.420
of change of distance
with respect to time.

00:02:59.420 --> 00:03:05.530
3 times 2 is 6t squared and the
constant disappears, right?

00:03:05.530 --> 00:03:07.010
So it's just 6t squared.

00:03:07.010 --> 00:03:08.140
And that makes sense, right?

00:03:08.140 --> 00:03:10.010
Because your velocity doesn't
care about where you

00:03:10.010 --> 00:03:11.500
started off from, right?

00:03:11.500 --> 00:03:13.210
You're going to be the same
velocity if you started

00:03:13.210 --> 00:03:15.010
from 10 feet or if you
started from 2 feet.

00:03:15.010 --> 00:03:17.520
Your velocity doesn't really
matter about where your

00:03:17.520 --> 00:03:18.710
starting position is.

00:03:18.710 --> 00:03:20.420
So let's graph this.

00:03:20.420 --> 00:03:20.570
See?

00:03:20.570 --> 00:03:23.250
You're actually learning
a little physics while

00:03:23.250 --> 00:03:24.310
you're learning calculus.

00:03:24.310 --> 00:03:25.800
Actually, I think it's silly
that they're taught as

00:03:25.800 --> 00:03:26.630
two separate classes.

00:03:26.630 --> 00:03:29.060
I think physics and calculus
should just be one

00:03:29.060 --> 00:03:30.960
fun 2-hour class.

00:03:30.960 --> 00:03:33.330
But I'll talk about
that at another time.

00:03:36.040 --> 00:03:38.100
So, going back to this.

00:03:38.100 --> 00:03:38.730
Let me graph that.

00:03:38.730 --> 00:03:39.090
6t squared.

00:03:39.090 --> 00:03:41.120
Well, that's just going
to look like a parabola.

00:03:41.120 --> 00:03:42.110
Right?

00:03:42.110 --> 00:03:45.440
It's going to look
something like this.

00:03:45.440 --> 00:03:46.970
This is t.

00:03:46.970 --> 00:03:49.090
This is the velocity.

00:03:49.090 --> 00:03:53.990
And now, if we just had this
velocity graph, if we didn't

00:03:53.990 --> 00:03:57.320
know all of this over
here and I asked you the

00:03:57.320 --> 00:03:58.070
same question, though.

00:03:58.070 --> 00:04:02.370
I said, how far does this
thing travel between 2

00:04:02.370 --> 00:04:08.420
seconds and 5 seconds?

00:04:08.420 --> 00:04:08.815
Right?

00:04:12.270 --> 00:04:14.990
Well, I could do it the way
that we learned in the previous

00:04:14.990 --> 00:04:19.230
video where I draw a bunch of
small rectangles, each of a

00:04:19.230 --> 00:04:23.530
really small width, and I
multiply it times its

00:04:23.530 --> 00:04:27.565
instantaneous velocity at
that exact moment, right?

00:04:30.630 --> 00:04:35.350
And then I sum up all of those
rectangles -- look how pretty

00:04:35.350 --> 00:04:40.350
that is -- I sum up all
of the rectangles.

00:04:40.350 --> 00:04:45.930
And I'll get a pretty good
approximation for how far

00:04:45.930 --> 00:04:48.850
I've travelled between
2 and 5 seconds.

00:04:48.850 --> 00:04:54.070
Because remember, the area of
each of these rectangles

00:04:54.070 --> 00:04:57.960
represents how far I
traveled in that little

00:04:57.960 --> 00:04:59.960
amount of time, dt.

00:04:59.960 --> 00:05:05.310
Because time times a constant
velocity is equal to distance.

00:05:05.310 --> 00:05:17.480
But as you can see this also
tells me the area between

00:05:17.480 --> 00:05:20.130
t equals 2 and t equals 5.

00:05:20.130 --> 00:05:23.440
So, not only did I figure out
the distance between how far I

00:05:23.440 --> 00:05:27.280
traveled from 2 seconds to 5
seconds, I also figured out the

00:05:27.280 --> 00:05:31.060
area under this curve from
2 seconds to 5 seconds.

00:05:31.060 --> 00:05:38.580
So, interestingly enough, if I
just changed this from a to b,

00:05:38.580 --> 00:05:41.920
and, in general, if you want to
figure out the area under a

00:05:41.920 --> 00:05:51.680
curve from a to b, it's just
the indefinite integral from a

00:05:51.680 --> 00:05:53.790
to b -- actually, from b to a.

00:05:53.790 --> 00:05:55.650
The b should be the larger one.

00:05:55.650 --> 00:05:56.920
b to a.

00:05:56.920 --> 00:05:59.090
I guess a to b, depending
on how you say it.

00:05:59.090 --> 00:06:00.360
Let me write that in a
different color because I

00:06:00.360 --> 00:06:02.100
think I'm making it messier.

00:06:02.100 --> 00:06:04.950
From a to b of this
velocity function.

00:06:04.950 --> 00:06:10.460
So, in this case, 6t
squared d t, right?

00:06:10.460 --> 00:06:13.200
If these weren't 2 and 5,
if this was just a and b.

00:06:13.200 --> 00:06:16.370
And the way you evaluate this
is you figure out the

00:06:16.370 --> 00:06:21.760
antiderivative of this inside
function, and then you evaluate

00:06:21.760 --> 00:06:25.010
the antiderivative at b,
and then from that, you

00:06:25.010 --> 00:06:28.040
subtract it out at a.

00:06:28.040 --> 00:06:32.590
So in this case, the
antiderivative of this is 2t to

00:06:32.590 --> 00:06:40.900
the third and we evaluated at
b, and we evaluated at a.

00:06:40.900 --> 00:06:42.770
Actually, let me stick
to the old numbers.

00:06:42.770 --> 00:06:46.510
We evaluated it at 5 and
you evaluated it at 2.

00:06:46.510 --> 00:06:51.240
So if you evaluated
it at 5, that's 255.

00:06:51.240 --> 00:06:52.750
If you evaluate it
at 2, that's 21.

00:06:52.750 --> 00:06:55.020
So you're doing the exact same
thing we did here when we

00:06:55.020 --> 00:06:57.480
actually had this graph.

00:06:57.480 --> 00:07:01.560
So I did all of this, not to
confuse you further, but really

00:07:01.560 --> 00:07:04.580
just to give you an intuition
of why one, why the

00:07:04.580 --> 00:07:08.290
antiderivative is the area
under the curve, and then two,

00:07:08.290 --> 00:07:21.390
why-- let's say that a, b-- and
then why we evaluate

00:07:21.390 --> 00:07:23.560
it this way.

00:07:23.560 --> 00:07:25.950
You might see this
in your books.

00:07:29.040 --> 00:07:32.560
This is just saying, if I want
to figure out the area under a

00:07:32.560 --> 00:07:37.820
curve from a to b of f of
x, that we figure out

00:07:37.820 --> 00:07:38.600
the antiderivative.

00:07:38.600 --> 00:07:42.660
This capital F is just
the antiderivative.

00:07:42.660 --> 00:07:45.740
We just figure out the
antiderivative and we evaluated

00:07:45.740 --> 00:07:49.940
at b and we evaluated at a, and
then we subtract

00:07:49.940 --> 00:07:50.540
the difference.

00:07:50.540 --> 00:07:51.680
And that's what we
did here, right?

00:07:51.680 --> 00:07:53.150
This is what we did here
intuitively when we

00:07:53.150 --> 00:07:56.110
worked with distance.

00:07:56.110 --> 00:07:58.220
The derivative and the
antiderivative don't only apply

00:07:58.220 --> 00:08:00.020
to distance and velocity.

00:08:00.020 --> 00:08:03.870
But I did this to give you an
intuition of why this works

00:08:03.870 --> 00:08:07.690
and why the antiderivative
is the area under a curve.

00:08:07.690 --> 00:08:10.710
So let me clear this up and
just rewrite that last thing

00:08:10.710 --> 00:08:13.490
I wrote, but maybe a
little bit cleaner.

00:08:17.680 --> 00:08:18.650
OK.

00:08:18.650 --> 00:08:24.380
So let's say that F of x with a
big, fat capital F is equal to

00:08:24.380 --> 00:08:30.440
-- actually, let me do it a
better way -- let me say that

00:08:30.440 --> 00:08:36.930
the derivative of big fat F
of x is equal to f of x.

00:08:36.930 --> 00:08:39.200
Right?

00:08:39.200 --> 00:08:40.920
I think, actually, this is the
fundamental theorem of

00:08:40.920 --> 00:08:43.450
calculus, but I don't want to
throw out things without

00:08:43.450 --> 00:08:43.920
knowing for sure.

00:08:43.920 --> 00:08:46.170
I have to go make sure.

00:08:46.170 --> 00:08:47.770
See, I haven't done
math in a long time.

00:08:47.770 --> 00:08:50.230
I'm giving you all this based
on intuition, not necessarily

00:08:50.230 --> 00:08:52.400
what I'm reading.

00:08:52.400 --> 00:08:56.020
So the derivative of big F is
small f, and all we're saying

00:08:56.020 --> 00:09:06.470
is that if we take the integral
of small f of x from a to b,

00:09:06.470 --> 00:09:13.480
dx, that this is big F, it's
antiderivative, at b minus

00:09:13.480 --> 00:09:16.320
the antiderivative at a.

00:09:16.320 --> 00:09:18.190
In the next presentation,
I'll use this.

00:09:18.190 --> 00:09:19.846
This is actually pretty easy
to use once you know how

00:09:19.846 --> 00:09:20.580
to use antiderivatives.

00:09:20.580 --> 00:09:22.990
And we did these three videos
really just to give you -- or

00:09:22.990 --> 00:09:25.165
actually, is this the third or
the second -- just to give you

00:09:25.165 --> 00:09:28.720
an intuition of why this is,
because I think that's really

00:09:28.720 --> 00:09:30.830
important if you're ever going
to really use calculus in your

00:09:30.830 --> 00:09:33.160
life or write a computer
program or whatever.

00:09:33.160 --> 00:09:36.280
And in the next couple videos
I'll actually apply this to a

00:09:36.280 --> 00:09:38.640
bunch of problems and you'll
hopefully see that it's a

00:09:38.640 --> 00:09:42.860
pretty straightforward
thing to actually compute.

00:09:42.860 --> 00:09:44.880
I'll see you in the
next presentation.


Definite integrals (part II)

WEBVTT
Kind: captions
Language: en

00:00:01.000 --> 00:00:02.110
Welcome back.

00:00:02.110 --> 00:00:06.800
So where I left off, we said
that we had this, I guess you

00:00:06.800 --> 00:00:09.450
could call it, equation or this
function, although I didn't

00:00:09.450 --> 00:00:11.110
write it with the function
notation, where I said, the

00:00:11.110 --> 00:00:13.540
distance is equal to 16 t
squared, and I graphed it, it's

00:00:13.540 --> 00:00:16.100
like a parabola, right,
for positive time.

00:00:16.100 --> 00:00:17.960
And then we said, well, the
velocity, if we know the

00:00:17.960 --> 00:00:21.190
distance, the velocity is just
the change of the distance

00:00:21.190 --> 00:00:22.360
with respect to time.

00:00:22.360 --> 00:00:24.080
It's just, the velocity is
always changing, you can't

00:00:24.080 --> 00:00:25.560
just take the slope, you
actually have to take

00:00:25.560 --> 00:00:26.920
the derivative, right?

00:00:26.920 --> 00:00:29.760
So we took the derivative with
respect to time of this

00:00:29.760 --> 00:00:32.915
function, or this equation, and
we got 32t, and this

00:00:32.915 --> 00:00:33.210
is the velocity.

00:00:33.210 --> 00:00:35.060
And then we graphed it.

00:00:35.060 --> 00:00:36.130
And then I asked a question.

00:00:36.130 --> 00:00:39.160
I was like, well, we want to
figure out, if we were given

00:00:39.160 --> 00:00:47.080
this, if we were given just
this, and I asked you, what is

00:00:47.080 --> 00:00:49.920
the distance that this object
travels after time, you

00:00:49.920 --> 00:00:52.130
know, after 10 seconds?

00:00:52.130 --> 00:00:56.240
Let's, you know, let's say this
t0 is equal to 10 seconds.

00:00:56.240 --> 00:00:59.150
I want to know how far is this
thing gone after 10 seconds.

00:00:59.150 --> 00:01:01.450
And let's say you didn't know
that you could just take the

00:01:01.450 --> 00:01:03.605
antiderivative, let's say we
didn't know this at all, and

00:01:03.605 --> 00:01:04.840
let's say you didn't know that
you could just take the

00:01:04.840 --> 00:01:07.860
antiderivative, because we just
showed that, you know, the

00:01:07.860 --> 00:01:10.990
derivative of distance is
velocity, so the antiderivative

00:01:10.990 --> 00:01:12.670
of velocity is distance.

00:01:12.670 --> 00:01:14.700
So let's say you couldn't just
take the antiderivative.

00:01:14.700 --> 00:01:18.140
What's a way that you could
start to try to approximate

00:01:18.140 --> 00:01:20.780
how far you've traveled
after, say, 10 seconds?

00:01:20.780 --> 00:01:21.330
Well [? as I said, ?]

00:01:21.330 --> 00:01:24.520
you graph this, and you say,
let's assume over some

00:01:24.520 --> 00:01:31.580
change in time, velocity is
roughly constant, right?

00:01:31.580 --> 00:01:33.390
Let's say velocity
is right here.

00:01:33.390 --> 00:01:37.370
So you could approximate how
far you travel over that small

00:01:37.370 --> 00:01:40.020
change in time by multiplying
that change in time, let's say

00:01:40.020 --> 00:01:43.900
that's like, you know, a
millionth of a second, times

00:01:43.900 --> 00:01:46.940
the velocity at roughly that
time, or maybe even the average

00:01:46.940 --> 00:01:51.940
velocity over that time, and
you'd get the distance you've

00:01:51.940 --> 00:01:58.390
traveled over that very small
fraction of time, right?

00:01:58.390 --> 00:02:02.770
But if you look at it visually,
that also happens to be the

00:02:02.770 --> 00:02:06.470
area of this rectangle, right?

00:02:06.470 --> 00:02:08.320
And what we said, is if you
want to know how far you travel

00:02:08.320 --> 00:02:12.160
after 10 seconds, you just draw
a bunch of these rectangles,

00:02:12.160 --> 00:02:14.520
and you sum up the area, right?

00:02:14.520 --> 00:02:16.550
And you could imagine, and you
don't have to imagine, it's

00:02:16.550 --> 00:02:21.380
actually true, the smaller the
bases of these rectangles, and

00:02:21.380 --> 00:02:26.610
the more of these rectangles
you have, the more accurate

00:02:26.610 --> 00:02:30.840
your approximation will be,
and you'll approach 2 things.

00:02:30.840 --> 00:02:34.640
You'll approach the area under
this curve, right, almost the

00:02:34.640 --> 00:02:37.740
exact area under this curve,
and you'd also get almost the

00:02:37.740 --> 00:02:47.030
exact value of the distance
after, say, 10 seconds

00:02:47.030 --> 00:02:48.130
in this case, right?

00:02:48.130 --> 00:02:50.100
But 10 didn't have to
be an exact number.

00:02:50.100 --> 00:02:51.290
It could have been a variable.

00:02:51.290 --> 00:02:53.310
So this is something
pretty interesting.

00:02:53.310 --> 00:02:57.100
All of a sudden, we see
that the antiderivative

00:02:57.100 --> 00:03:00.450
is pretty darn similar to
the area under the curve.

00:03:00.450 --> 00:03:01.800
And it actually turns out
that they're the same thing.

00:03:01.800 --> 00:03:04.600
And this is where I'm
going to teach you the

00:03:04.600 --> 00:03:05.930
indefinite integral.

00:03:05.930 --> 00:03:07.860
So the indefinite integral, I
don't know how comfortable you

00:03:07.860 --> 00:03:10.010
are with summation, I remember
the first time l learned

00:03:10.010 --> 00:03:13.170
calculus, I wasn't that
comfortable with summation, but

00:03:13.170 --> 00:03:17.030
it's really, all the indefinite
integral, is is you can kind of

00:03:17.030 --> 00:03:19.590
view it as a sum, right?

00:03:19.590 --> 00:03:23.130
So now, you'll maybe understand
a little bit more why this

00:03:23.130 --> 00:03:24.860
symbol looks kind
of like a sigma.

00:03:24.860 --> 00:03:26.270
That's actually how I view it.

00:03:26.270 --> 00:03:28.320
And please look it up so
you can see properly

00:03:28.320 --> 00:03:30.250
drawn integrals.

00:03:30.250 --> 00:03:33.830
But in this case, the
indefinite integral is just

00:03:33.830 --> 00:03:38.700
saying, well, I'm going to take
the sum from t equals 0, right,

00:03:38.700 --> 00:03:43.910
so from t equals 0, to let's
say in this example, t equals

00:03:43.910 --> 00:03:45.560
10, right, because I said 10.

00:03:45.560 --> 00:03:47.660
From t equals 0 to t equals 10.

00:03:47.660 --> 00:03:53.150
and I'm going to take the sum
of each of the heights, the

00:03:53.150 --> 00:03:58.810
height at any given point,
which is the velocity.

00:04:02.280 --> 00:04:03.720
And then, what's the
formula for the velocity?

00:04:03.720 --> 00:04:13.140
It's 32t and then I'm at
times the base at each

00:04:13.140 --> 00:04:16.620
of these rectangles, dt.

00:04:16.620 --> 00:04:18.440
And so this is the
definite integral.

00:04:18.440 --> 00:04:21.620
The definite integral is
literally, and they never do

00:04:21.620 --> 00:04:24.050
this in math texts, and that's
what always kind of confused

00:04:24.050 --> 00:04:26.560
me, is that you can kind of
view it like a sum, like this.

00:04:29.140 --> 00:04:33.090
It's kind of the sum of each of
these rectangles, but it's the

00:04:33.090 --> 00:04:38.680
limit, as-- if these were
discrete rectangles, you could

00:04:38.680 --> 00:04:41.110
just do a sum, and you could
make the rectangle bases

00:04:41.110 --> 00:04:42.870
smaller and smaller, and have
more and more rectangles,

00:04:42.870 --> 00:04:44.210
and just do a regular sum.

00:04:44.210 --> 00:04:45.870
And actually, that's how, if
you ever write a computer

00:04:45.870 --> 00:04:49.770
program to approximate an
integral, or approximate the

00:04:49.770 --> 00:04:51.640
area under a curve, that's
the way a computer program

00:04:51.640 --> 00:04:53.020
would actually do it.

00:04:53.020 --> 00:04:57.480
But the actual indefinite
integral says, well, this is a

00:04:57.480 --> 00:05:01.310
sum, but it's the limit as the
bases of these rectangles get

00:05:01.310 --> 00:05:05.230
smaller and smaller and smaller
and smaller, and we have more

00:05:05.230 --> 00:05:06.940
and more and more of
these rectangles.

00:05:06.940 --> 00:05:11.310
So as these dt's approach 0,
the number of rectangles

00:05:11.310 --> 00:05:12.880
actually approach infinity.

00:05:12.880 --> 00:05:14.950
So I'm actually going to, I'll
do that more rigorously later,

00:05:14.950 --> 00:05:16.780
but I think it's very important
to get this intuitive feel

00:05:16.780 --> 00:05:17.680
of just what an integral is.

00:05:17.680 --> 00:05:20.930
It isn't just this voodoo
that happens to be there.

00:05:20.930 --> 00:05:24.160
But anyway, so going
back to the problem.

00:05:24.160 --> 00:05:29.360
So the integral from-- this
is now a definite integral,

00:05:29.360 --> 00:05:32.670
extending from t equals
0 to t equals 10.

00:05:32.670 --> 00:05:34.020
This tells us 2 things.

00:05:34.020 --> 00:05:38.300
This tells us the area of the
curve from t equals zero to t

00:05:38.300 --> 00:05:41.360
equals 10, right, it tells us
this whole area, and it also

00:05:41.360 --> 00:05:46.580
tells us how far the object
has gone after 10 seconds.

00:05:46.580 --> 00:05:47.320
Right?

00:05:47.320 --> 00:05:48.450
So it's very interesting.

00:05:48.450 --> 00:05:50.770
The indefinite integral
tells us 2 things.

00:05:50.770 --> 00:05:55.160
It tells us area, and it also
tells us the antiderivative.

00:05:55.160 --> 00:05:55.360
Right?

00:05:55.360 --> 00:05:58.130
We're already familiar with
it as an antiderivative.

00:05:58.130 --> 00:06:00.220
So let me give you
another example.

00:06:00.220 --> 00:06:01.760
Actually, maybe I'll stick
with this example, but

00:06:01.760 --> 00:06:03.650
I'll clear it a bit.

00:06:03.650 --> 00:06:05.310
Actually, maybe I should erase.

00:06:05.310 --> 00:06:08.810
Erasing might be a good
option with this one,

00:06:08.810 --> 00:06:12.190
since it's fairly messy.

00:06:12.190 --> 00:06:13.830
I think you know all
this stuff now.

00:06:13.830 --> 00:06:14.760
I just need space.

00:06:17.830 --> 00:06:20.372
Maybe, OK, so we have that
indefinite integral.

00:06:20.372 --> 00:06:22.040
And we could actually
figure it out, too.

00:06:22.040 --> 00:06:25.470
I mean, well, after t
seconds, [UNINTELLIGIBLE].

00:06:25.470 --> 00:06:27.920
So and the way you evaluate an
indefinite integral, and let me

00:06:27.920 --> 00:06:31.780
show you that first, is that
you figure out the integral.

00:06:31.780 --> 00:06:33.740
So let me just say, let
me continue with the

00:06:33.740 --> 00:06:35.060
problem, actually.

00:06:35.060 --> 00:06:38.790
As you can tell, I don't plan
much for these presentations.

00:06:38.790 --> 00:06:41.170
So the way you figure out the
indefinite integral, is you

00:06:41.170 --> 00:06:46.300
say, and sometimes they won't
write t equals 0 to t equals t.

00:06:46.300 --> 00:06:54.610
They'll just say from
0 to 10 of 32t dt.

00:06:54.610 --> 00:06:55.370
Right?

00:06:55.370 --> 00:06:57.770
And the way you evaluate
this, is you figure out the

00:06:57.770 --> 00:06:59.780
antiderivative, and you really
don't have to do the plus c

00:06:59.780 --> 00:07:03.330
here, so the antiderivative, we
know, is 16t squared, right?

00:07:03.330 --> 00:07:05.440
It's one half t
squared times 32.

00:07:05.440 --> 00:07:06.690
So that's 16t squared.

00:07:09.300 --> 00:07:14.010
And we evaluate this at ten,
and we evaluate it at 0, and

00:07:14.010 --> 00:07:15.680
then we subtract
the difference.

00:07:15.680 --> 00:07:20.250
So we evaluate this at 10,
so 16 times 100, right?

00:07:20.250 --> 00:07:23.060
That's evaluated at 10,
and then we subtract

00:07:23.060 --> 00:07:24.155
it, evaluate at 0.

00:07:24.155 --> 00:07:26.320
So 16 times 0 is 0.

00:07:26.320 --> 00:07:30.960
So after 10 seconds, we
would have gone 1600 feet.

00:07:30.960 --> 00:07:41.230
And also, the area under
this curve is 1600.

00:07:41.230 --> 00:07:44.200
So let's use this to do
a couple more examples.

00:07:44.200 --> 00:07:47.990
And actually, I want to show
you why we do this subtraction.

00:07:47.990 --> 00:07:51.560
Actually, I'm going to
do that right now.

00:07:51.560 --> 00:07:53.110
Let me clear it.

00:08:00.600 --> 00:08:01.850
Oh, that's ugly.

00:08:07.520 --> 00:08:09.100
I'll now do it more
general, actually.

00:08:13.700 --> 00:08:20.600
Let me draw this twice,
once for the distance, and

00:08:20.600 --> 00:08:21.480
once for its derivative.

00:08:28.860 --> 00:08:34.580
So let's say that the distance,
yeah, well, let's just say it

00:08:34.580 --> 00:08:36.670
looks something like this.

00:08:36.670 --> 00:08:40.580
Let's say you start at some
distance, and then it

00:08:40.580 --> 00:08:42.350
goes off like that.

00:08:42.350 --> 00:08:43.410
Right?

00:08:43.410 --> 00:08:47.190
So let's say we call
this distance b.

00:08:47.190 --> 00:08:51.160
Well, let's just call this,
you know, I don't know, 5.

00:08:51.160 --> 00:08:51.450
Right?

00:08:51.450 --> 00:08:56.600
We start at 5 feet, and then
we moved forward from there.

00:08:56.600 --> 00:09:00.600
And this axis is of course
time, this axis, maybe I

00:09:00.600 --> 00:09:02.590
shouldn't do 5, because
it looks so much like s.

00:09:02.590 --> 00:09:04.280
That's 5, 5 feet.

00:09:04.280 --> 00:09:09.420
And this is the s,
or distance, axis.

00:09:09.420 --> 00:09:11.270
And actually, I just
looked at the clock.

00:09:11.270 --> 00:09:12.310
I'm running out of time.

00:09:12.310 --> 00:09:15.120
So let me continue this in
the next presentation.


Introduction to definite integrals

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.040
Welcome back.

00:00:02.040 --> 00:00:03.980
In this presentation, I
actually want to show you how

00:00:03.980 --> 00:00:06.730
we can use the antiderivative
to figure out the

00:00:06.730 --> 00:00:08.340
area under a curve.

00:00:08.340 --> 00:00:09.695
Actually I'm going to focus
more a little bit more

00:00:09.695 --> 00:00:10.770
on the intuition.

00:00:10.770 --> 00:00:12.960
So let actually use an
example from physics.

00:00:12.960 --> 00:00:15.640
I'll use distance and velocity.

00:00:15.640 --> 00:00:17.550
And actually this could be a
good review for derivatives,

00:00:17.550 --> 00:00:19.590
or actually an application
of derivatives.

00:00:19.590 --> 00:00:22.610
So let's say that I
described the position

00:00:22.610 --> 00:00:23.500
of something moving.

00:00:23.500 --> 00:00:26.160
Let's say it's x.

00:00:26.160 --> 00:00:35.520
Let's say that x is equal to,
I don't know, 16t squared.

00:00:35.520 --> 00:00:35.850
Right?

00:00:35.850 --> 00:00:36.880
So s is distance.

00:00:36.880 --> 00:00:38.120
Let me write this
in the corner.

00:00:38.120 --> 00:00:41.220
I don't know why the
convention is to use s as

00:00:41.220 --> 00:00:42.250
the variable for distance.

00:00:42.250 --> 00:00:45.400
One would think, well actually,
I know, why won't they use d?

00:00:45.400 --> 00:00:48.740
Because d is the letter used
for differential, I guess.

00:00:48.740 --> 00:00:55.800
So s is equal to distance,
and then t is equal to time.

00:00:58.990 --> 00:01:02.580
So this is just a formula that
tells us the position, kind of

00:01:02.580 --> 00:01:06.210
how far has something
gone, after x many, let's

00:01:06.210 --> 00:01:07.140
say, seconds, right?

00:01:07.140 --> 00:01:10.620
So after like, 4 seconds, we
would have gone, let's say

00:01:10.620 --> 00:01:12.750
the distance is in feet,
this is in seconds.

00:01:12.750 --> 00:01:15.690
After 4 seconds, we would
have gone 256 feet.

00:01:15.690 --> 00:01:16.660
That's all that says.

00:01:16.660 --> 00:01:21.250
And let me graph that as well.

00:01:21.250 --> 00:01:23.120
Graph it.

00:01:23.120 --> 00:01:28.620
That's a horrible line.

00:01:28.620 --> 00:01:30.430
Have to use the line tool,
might have better luck.

00:01:33.400 --> 00:01:35.730
It's slightly better.

00:01:35.730 --> 00:01:38.110
Actually, let me undo that too,
because I just want to do

00:01:38.110 --> 00:01:40.200
it for positive t, right?

00:01:40.200 --> 00:01:42.480
Because you can't really
go back in time.

00:01:42.480 --> 00:01:45.340
For the purposes of this
lecture, you can't

00:01:45.340 --> 00:01:47.520
go back in time.

00:01:47.520 --> 00:01:51.810
So that'll have to do.

00:01:51.810 --> 00:01:55.820
So this curve will essentially
just be a parabola, right?

00:01:55.820 --> 00:01:56.720
It'll look something like this.

00:02:01.700 --> 00:02:02.790
So actually, if you
look at it, I mean you

00:02:02.790 --> 00:02:04.100
could just eyeball it.

00:02:04.100 --> 00:02:06.590
The object, every second you
go, it's going a little

00:02:06.590 --> 00:02:07.410
bit further, right?

00:02:07.410 --> 00:02:08.990
So it's actually accelerating.

00:02:08.990 --> 00:02:11.880
And so what if we wanted to
figure out what the velocity

00:02:11.880 --> 00:02:13.960
of this object, right?

00:02:13.960 --> 00:02:18.580
This is, let's see, this
is d, this is t, right?

00:02:18.580 --> 00:02:20.630
And this is, I don't know
if it's clear, but this is

00:02:20.630 --> 00:02:22.780
kind of 1/2 a parabola.

00:02:22.780 --> 00:02:24.900
So this is the
distance function.

00:02:24.900 --> 00:02:26.330
What would the velocity be?

00:02:26.330 --> 00:02:29.170
Well the velocity is
just, what's velocity?

00:02:29.170 --> 00:02:31.590
It's distance divided
by time, right?

00:02:31.590 --> 00:02:33.490
And since this velocity
is always changing, we

00:02:33.490 --> 00:02:35.570
want to figure out the
instantaneous velocity.

00:02:35.570 --> 00:02:38.620
And that's actually one of the
initial uses of what made

00:02:38.620 --> 00:02:39.930
derivatives so useful.

00:02:39.930 --> 00:02:43.430
So we want to find the change,
the instantaneous change

00:02:43.430 --> 00:02:45.450
with respect to time of
this formula, right?

00:02:45.450 --> 00:02:47.350
Because this is the
distance formula.

00:02:47.350 --> 00:02:50.410
So if we know the instant rate
of change of distance with

00:02:50.410 --> 00:02:53.310
respect to time, we'll
know the velocity, right?

00:02:53.310 --> 00:03:02.040
So ds, dt, is equal to?

00:03:02.040 --> 00:03:03.550
So what's the derivative here?

00:03:03.550 --> 00:03:09.280
It's 32t, right?

00:03:09.280 --> 00:03:10.320
And this is the velocity.

00:03:14.060 --> 00:03:16.660
Maybe I should switch back
to, let me write that,

00:03:16.660 --> 00:03:20.360
v equals velocity.

00:03:20.360 --> 00:03:21.880
I don't know why I switched
colors, but I'll stick

00:03:21.880 --> 00:03:23.250
with the yellow.

00:03:23.250 --> 00:03:24.510
So let's graph this function.

00:03:24.510 --> 00:03:28.680
This will actually be a fairly
straightforward graph to draw.

00:03:33.670 --> 00:03:35.270
It's pretty straight.

00:03:35.270 --> 00:03:37.160
And then we draw the x-axis.

00:03:41.910 --> 00:03:43.390
I'm doing pretty good.

00:03:43.390 --> 00:03:43.790
OK.

00:03:48.010 --> 00:03:56.370
So this, I'll draw it in
red, this is this going

00:03:56.370 --> 00:03:57.420
to be a line, right?

00:03:57.420 --> 00:03:59.450
32t it's a line with slope 32.

00:03:59.450 --> 00:04:00.530
So it's actually a
pretty steep line.

00:04:00.530 --> 00:04:02.640
I won't draw it that steep
because I'm going to use

00:04:02.640 --> 00:04:05.880
this for an illustration.

00:04:05.880 --> 00:04:06.855
So this is the velocity.

00:04:09.990 --> 00:04:11.580
This is velocity.

00:04:11.580 --> 00:04:17.330
This is that graph, and
this is distance, right?

00:04:17.330 --> 00:04:19.970
So in case you hadn't learned
already, and maybe I'll do a

00:04:19.970 --> 00:04:22.470
whole presentation on kind of
using calculus for physics, and

00:04:22.470 --> 00:04:24.000
using derivatives for physics.

00:04:24.000 --> 00:04:27.460
But if you have to distance
formula, it's derivative

00:04:27.460 --> 00:04:28.730
is just velocity.

00:04:28.730 --> 00:04:30.830
And I guess if you view
it the other way, if you

00:04:30.830 --> 00:04:33.920
have the velocity, it's
antiderivative is distance.

00:04:33.920 --> 00:04:37.800
Although you won't know
where, at what position,

00:04:37.800 --> 00:04:38.770
the object started.

00:04:38.770 --> 00:04:42.080
In this case, the object
started at position of 0,

00:04:42.080 --> 00:04:44.420
but it could be, you know,
at any constant, right?

00:04:44.420 --> 00:04:46.210
You could have started
here and then curved up.

00:04:46.210 --> 00:04:48.140
But anyway, let's just
assume we started at 0.

00:04:48.140 --> 00:04:51.170
So the derivative of distance
is velocity, the antiderivative

00:04:51.170 --> 00:04:52.350
of velocity is distance.

00:04:52.350 --> 00:04:54.020
Keep that in mind.

00:04:54.020 --> 00:04:56.130
Well let's look at this.

00:04:56.130 --> 00:05:03.880
Let's assume that we were
only given this graph.

00:05:03.880 --> 00:05:05.520
And we said, you know,
this is the graph of the

00:05:05.520 --> 00:05:08.850
velocity of some object.

00:05:08.850 --> 00:05:11.930
And we want to figure out what
the distance is after, you

00:05:11.930 --> 00:05:13.220
know, t seconds, right?

00:05:13.220 --> 00:05:17.340
So this is the t-axis, this
is the velocity axis, right?

00:05:17.340 --> 00:05:19.490
So let's say we were only given
this, and let's say we didn't

00:05:19.490 --> 00:05:22.590
know that the antiderivative of
the velocity function is

00:05:22.590 --> 00:05:23.250
the distance function.

00:05:23.250 --> 00:05:27.340
How would we figure out, how
would we figure out what

00:05:27.340 --> 00:05:29.360
the distance would be
at any given time?

00:05:29.360 --> 00:05:31.530
Well let's think about it.

00:05:31.530 --> 00:05:34.080
If we have a constant, this
red is kind of bloody.

00:05:34.080 --> 00:05:37.150
Let me switch to
something more pleasant.

00:05:37.150 --> 00:05:40.340
If we have, over any small
period of time, right, or if we

00:05:40.340 --> 00:05:44.090
have a constant velocity, when
you have a constant velocity,

00:05:44.090 --> 00:05:46.990
distance is just velocity
times time, right?

00:05:46.990 --> 00:05:50.030
So let's say we had
a very small time

00:05:50.030 --> 00:05:52.090
fragment here, right?

00:05:52.090 --> 00:05:54.190
I'll draw it big, but let's
say this time fragment

00:05:54.190 --> 00:05:55.640
it is really small.

00:05:55.640 --> 00:05:59.330
And let's called this very
small time fragment, let call

00:05:59.330 --> 00:06:02.480
this delta t, or dt actually.

00:06:02.480 --> 00:06:05.120
The way I've used dt is like,
it's like a change in time

00:06:05.120 --> 00:06:07.040
that's unbelievably
small, right?

00:06:07.040 --> 00:06:09.490
So it's like almost
instantaneous, but not quite.

00:06:09.490 --> 00:06:11.410
Or you can actually view
it as instantaneous.

00:06:11.410 --> 00:06:13.710
So this is how much
time goes by.

00:06:13.710 --> 00:06:16.390
You can kind of view this as
a very small change in time.

00:06:16.390 --> 00:06:20.040
So if we have a very small
change of time, and over that

00:06:20.040 --> 00:06:22.510
very small change in time,
we have a roughly constant

00:06:22.510 --> 00:06:26.500
velocity, let's say the roughly
constant velocity is this.

00:06:31.250 --> 00:06:34.600
Right, this is the velocity, so
say we had over this very small

00:06:34.600 --> 00:06:37.210
change in time, we have this
roughly constant velocity

00:06:37.210 --> 00:06:38.210
that's on this graph.

00:06:38.210 --> 00:06:41.720
Actually, let me
take do it here.

00:06:41.720 --> 00:06:43.400
We have this roughly
constant velocity.

00:06:43.400 --> 00:06:47.870
So the distance that the object
travels over the small time

00:06:47.870 --> 00:06:50.650
would be the small time
times the velocity, right?

00:06:50.650 --> 00:06:54.150
It would be whatever the value
of this red line is, times the

00:06:54.150 --> 00:06:57.340
width of this distance, right?

00:06:57.340 --> 00:06:59.230
So what's another way?

00:06:59.230 --> 00:07:01.950
Visually I kind of did
it ahead of time, but

00:07:01.950 --> 00:07:02.900
what's happening here?

00:07:02.900 --> 00:07:08.120
If I take this change in time,
right, which is kind of the

00:07:08.120 --> 00:07:12.890
base of this rectangle, and I
multiply it times the velocity

00:07:12.890 --> 00:07:15.750
which is really just the height
of this rectangle, what

00:07:15.750 --> 00:07:16.510
have I figured out?

00:07:16.510 --> 00:07:20.790
Well I figured out the area
of this rectangle, right?

00:07:20.790 --> 00:07:23.390
Right, the velocity this
moment, times the change in

00:07:23.390 --> 00:07:26.040
time at this moment, is
nothing but the area of

00:07:26.040 --> 00:07:28.130
this very skinny rectangle.

00:07:28.130 --> 00:07:29.210
Skinny and tall, right?

00:07:29.210 --> 00:07:33.080
It's almost infinitely skinny,
but it's, we're assuming for

00:07:33.080 --> 00:07:37.040
these purposes it has some very
notional amount of width.

00:07:37.040 --> 00:07:39.990
So there we figured out the
area of this column, right?

00:07:39.990 --> 00:07:44.510
Well, if we wanted to figure
out the distance that you

00:07:44.510 --> 00:07:50.960
travel after, let's say, you
know, I don't know, let's say

00:07:50.960 --> 00:07:54.010
t, let's say t sub
nought, right?

00:07:54.010 --> 00:07:55.710
This is just a particular t.

00:07:55.710 --> 00:07:57.980
After t sub nought
seconds, right?

00:07:57.980 --> 00:08:00.840
Well then, all we would have to
do is, we would have to just

00:08:00.840 --> 00:08:04.010
figure, we would just do
a bunch of dt's, right?

00:08:04.010 --> 00:08:08.900
You'd do another one here,
you'd figure out the area of

00:08:08.900 --> 00:08:12.630
this column, you'd figure out
the area of this column, the

00:08:12.630 --> 00:08:15.490
area of this column, right?

00:08:15.490 --> 00:08:18.970
Because each of these areas
of each of these columns

00:08:18.970 --> 00:08:21.690
represents the distance
that the object travels

00:08:21.690 --> 00:08:24.610
over that dt, right?

00:08:24.610 --> 00:08:28.506
So if you wanted to know how
far you traveled after t sub

00:08:28.506 --> 00:08:33.340
zero seconds, you'd essentially
get, or an approximation would

00:08:33.340 --> 00:08:36.210
be, the sum of all
of these areas.

00:08:36.210 --> 00:08:40.110
And as you got more and more,
as you made the dt's smaller

00:08:40.110 --> 00:08:41.430
and smaller, skinnier,
skinnier, skinnier.

00:08:41.430 --> 00:08:43.810
And you had more and more and
more and more of these

00:08:43.810 --> 00:08:47.930
rectangles, then your
approximation will get pretty

00:08:47.930 --> 00:08:50.700
close to, well, two things.

00:08:50.700 --> 00:08:53.320
It'll get pretty close to, as
you can imagine, the area

00:08:53.320 --> 00:08:56.230
under this curve, or
in this case a line.

00:08:56.230 --> 00:09:01.870
But it would also get you
pretty much the exact amount

00:09:01.870 --> 00:09:06.720
of distance you've traveled
after t sub nought seconds.

00:09:06.720 --> 00:09:12.410
So I think I'm running into the
ten minute wall, so I'm just

00:09:12.410 --> 00:09:15.600
going to pause here, and I'm
going to continue this in

00:09:15.600 --> 00:09:17.280
the next presentation.


Indefinite Integration (part IV)

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:01.850
Welcome back.

00:00:01.850 --> 00:00:04.510
In the last presentation, I
showed you how to essentially

00:00:04.510 --> 00:00:07.660
reverse the chain rule when
you're doing an integral.

00:00:07.660 --> 00:00:10.370
And you could also do this
by integral, it's called

00:00:10.370 --> 00:00:11.890
integration by substitution.

00:00:11.890 --> 00:00:12.690
And I'll show you why.

00:00:12.690 --> 00:00:15.030
And this is essentially just
a reverse of the chain rule.

00:00:15.030 --> 00:00:23.770
The last problem we did in that
last video, I said, sine of x

00:00:23.770 --> 00:00:29.490
to the third power times cosine
of x, and I took the integral

00:00:29.490 --> 00:00:30.980
of the whole thing.

00:00:30.980 --> 00:00:34.850
And I did it kind of, you know,
just telling you that, well,

00:00:34.850 --> 00:00:36.090
we're just reversing
the chain rule.

00:00:36.090 --> 00:00:41.110
So we see the derivative of the
sine x here, right, which is

00:00:41.110 --> 00:00:44.170
cosine of x, so we can just
treat sine of x like a variable

00:00:44.170 --> 00:00:45.550
and take its integral.

00:00:45.550 --> 00:00:51.890
And I said that that is
equal to sine of x to

00:00:51.890 --> 00:00:54.260
the fourth, times 1/4.

00:00:54.260 --> 00:00:54.600
Right?

00:00:54.600 --> 00:00:56.940
And the reason why we could
just treat the sine of x like

00:00:56.940 --> 00:00:59.880
it's just like kind of a
variable instead of a function,

00:00:59.880 --> 00:01:02.120
is because we had its
derivative sitting right here.

00:01:02.120 --> 00:01:04.640
And if you keep doing it back
and forth between the chain

00:01:04.640 --> 00:01:06.100
rule and what I just did, I
think it'll make

00:01:06.100 --> 00:01:07.080
a lot of sense.

00:01:07.080 --> 00:01:08.430
So this might have been
a little confusing.

00:01:08.430 --> 00:01:11.595
So I'll show you a technique
called integration by

00:01:11.595 --> 00:01:13.130
substitution, which
is essentially the

00:01:13.130 --> 00:01:14.410
exact same thing.

00:01:14.410 --> 00:01:17.200
So let's start with this exact
same integral here, and let's

00:01:17.200 --> 00:01:19.970
pretend like we don't
know the answer.

00:01:19.970 --> 00:01:26.250
We say, well, we have a
function and its derivative, so

00:01:26.250 --> 00:01:34.510
let me let u equal the function
that we have the derivative of.

00:01:34.510 --> 00:01:34.846
Right?

00:01:38.830 --> 00:01:42.070
u is equal to sine of x, right?

00:01:42.070 --> 00:01:43.510
u is sine of x.

00:01:43.510 --> 00:01:45.180
Well, what's the
derivative of u?

00:01:45.180 --> 00:01:47.000
du, dx.

00:01:49.910 --> 00:01:52.610
Well, we know what
du dx is, right?

00:01:52.610 --> 00:01:55.540
du of dx is equal
to cosine of x.

00:01:55.540 --> 00:01:58.440
We memorized that, and maybe
in a future presentation I'll

00:01:58.440 --> 00:02:00.060
actually prove it to you.

00:02:00.060 --> 00:02:03.400
So what we can now do is
substitute these 2 things

00:02:03.400 --> 00:02:07.240
into this integral.

00:02:07.240 --> 00:02:11.080
So the integral now becomes,
instead of writing sine of x

00:02:11.080 --> 00:02:15.830
to the third power, we can
write u to the third power.

00:02:15.830 --> 00:02:17.590
And what's cosine of x?

00:02:17.590 --> 00:02:21.700
Well, we just showed, cosine
of x is just du dx, right?

00:02:21.700 --> 00:02:32.860
So it's times du dx, and then
we have times dx, right?

00:02:32.860 --> 00:02:37.100
And I know you're probably not
too comfortable with the

00:02:37.100 --> 00:02:40.740
differentials at this point,
but they behave in just the way

00:02:40.740 --> 00:02:42.520
regular numbers do
with a fraction.

00:02:42.520 --> 00:02:46.400
So this dx and this dx actually
do cancel out, and you're left

00:02:46.400 --> 00:02:53.160
with, that this is equal to the
integral of u to the third du.

00:02:53.160 --> 00:02:54.320
And now this is an
easy integral.

00:02:54.320 --> 00:02:56.850
The only thing different than
what you might have seen

00:02:56.850 --> 00:02:59.195
recently is that instead of
an x, we have a u here.

00:02:59.195 --> 00:03:02.060
And while we know that the
answer of this integral, this

00:03:02.060 --> 00:03:09.020
is equal to 1/4 u to the
fourth, and then, of course,

00:03:09.020 --> 00:03:10.600
we should add the plus c.

00:03:10.600 --> 00:03:11.390
Are we done?

00:03:11.390 --> 00:03:13.730
Well, I mean, almost, but it
would be nice to just take that

00:03:13.730 --> 00:03:17.630
u, and unwind it, and put the
sine of x back in for it.

00:03:17.630 --> 00:03:20.840
So let's do that.

00:03:20.840 --> 00:03:22.190
So u is the sine of x.

00:03:22.190 --> 00:03:23.390
That's what we said
at the beginning.

00:03:23.390 --> 00:03:30.180
1/4 sine of x to the
fourth, plus c.

00:03:30.180 --> 00:03:31.520
Done.

00:03:31.520 --> 00:03:33.440
Actually, this might be an
easier way to think about these

00:03:33.440 --> 00:03:36.290
type of integrals than what I
did in the last presentation.

00:03:36.290 --> 00:03:39.000
But you know, every now and
then you have to bear with me.

00:03:39.000 --> 00:03:40.540
I do things in the wrong order.

00:03:40.540 --> 00:03:42.940
Let's do a couple more
problems like this.

00:03:47.060 --> 00:04:06.770
Let's take the integral of 2x
plus 3 times x squared plus 3x

00:04:06.770 --> 00:04:16.140
plus 15 to the fifth power dx.

00:04:16.140 --> 00:04:18.340
That looks complicated
to you, doesn't it?

00:04:18.340 --> 00:04:21.840
Well, just like we said, this
is a pattern, like we saw

00:04:21.840 --> 00:04:23.120
in the previous examples.

00:04:23.120 --> 00:04:28.780
We have this expression here,
x squared plus 3x plus 15,

00:04:28.780 --> 00:04:30.590
and well, what's the
derivative of this?

00:04:30.590 --> 00:04:32.540
x squared plus 3x plus 15?

00:04:32.540 --> 00:04:34.930
Well, it's 2x plus 3, right?

00:04:34.930 --> 00:04:38.700
Notice that I'm engineering it
so it works cleanly, but most

00:04:38.700 --> 00:04:40.850
textbooks and tests
tend to do that.

00:04:40.850 --> 00:04:42.550
So let's make the substitution.

00:04:42.550 --> 00:04:45.430
Because we have a u that we
can use, and then we have

00:04:45.430 --> 00:04:46.630
its derivative, right?

00:04:46.630 --> 00:04:56.380
So we can say u is equal to x
squared plus 3x plus 15, and we

00:04:56.380 --> 00:05:00.320
can say then, the derivative of
u, we know the derivative

00:05:00.320 --> 00:05:04.060
of u is 2x plus 3, right?

00:05:04.060 --> 00:05:05.980
Because the derivative
of 15 is 0.

00:05:05.980 --> 00:05:08.220
So now we can make
our substitutions.

00:05:08.220 --> 00:05:10.480
I'm just going to switch
the orders of these two

00:05:10.480 --> 00:05:11.650
around, no different.

00:05:11.650 --> 00:05:13.770
So this is just u to
the fifth, right?

00:05:13.770 --> 00:05:15.950
Because this is this.

00:05:15.950 --> 00:05:17.300
So this is just u to the fifth.

00:05:20.050 --> 00:05:24.640
And then this is du dx
times du dx, right?

00:05:24.640 --> 00:05:26.940
I just switched the orders.

00:05:26.940 --> 00:05:31.655
And then I multiply
that times dx.

00:05:31.655 --> 00:05:32.730
And these cancel.

00:05:32.730 --> 00:05:35.120
And I know you're not
completely comfortable yet with

00:05:35.120 --> 00:05:38.115
even this integration notation,
why is this dx sitting there in

00:05:38.115 --> 00:05:39.616
the first place, but when we do
the definite integrals it

00:05:39.616 --> 00:05:42.110
will make more sense.

00:05:42.110 --> 00:05:48.730
But this is just equal to the
integral of u to the fifth du.

00:05:48.730 --> 00:05:50.220
And the integral of this,
well, this is easy.

00:05:50.220 --> 00:05:55.640
This is just equal to 1/6 u
to the sixth, right, plus c.

00:05:55.640 --> 00:05:58.590
And now we can
just unwind this.

00:05:58.590 --> 00:06:00.830
I'll do it up here to
make it extra messy.

00:06:00.830 --> 00:06:05.920
This is just equal to 1/6 times
u, which is this right here,

00:06:05.920 --> 00:06:09.570
right, we just set u to equal
this expression, 1/6 x

00:06:09.570 --> 00:06:14.380
squared plus 3x plus 15.

00:06:14.380 --> 00:06:20.010
All of this to the
sixth power, plus c.

00:06:20.010 --> 00:06:21.060
Let's do one more.

00:06:21.060 --> 00:06:23.500
I think we have
time for one more.

00:06:23.500 --> 00:06:27.970
Image, clear image,
image, invert.

00:06:27.970 --> 00:06:28.460
OK.

00:06:28.460 --> 00:06:33.210
I will also switch colors, just
to keep things interesting.

00:06:33.210 --> 00:06:48.210
Let's take the integral
of e to the x times e

00:06:48.210 --> 00:06:53.590
to the x to the fifth.

00:06:53.590 --> 00:06:55.040
I keep using that.

00:06:55.040 --> 00:06:59.250
Let's say to the
minus third power.

00:06:59.250 --> 00:07:00.820
dx.

00:07:00.820 --> 00:07:04.565
Well, once again, we have this
expression e to the x, and

00:07:04.565 --> 00:07:05.630
what's the derivative
of e to the x?

00:07:05.630 --> 00:07:07.550
Well, the derivative of e to
the x, as we learned, which

00:07:07.550 --> 00:07:12.750
is one of these things that
amazes me, is e to the x.

00:07:12.750 --> 00:07:16.050
Actually, that's one definition
for e, is number which, when

00:07:16.050 --> 00:07:17.810
it's raised to the x power,
it's the derivative of

00:07:17.810 --> 00:07:19.600
the same expression.

00:07:19.600 --> 00:07:21.140
But anyway, I don't want
to confuse you too much.

00:07:21.140 --> 00:07:26.070
But we can say then that u is
equal to e to the x, and we

00:07:26.070 --> 00:07:31.090
know that du dx is equal to e
to the x as well, which is,

00:07:31.090 --> 00:07:32.510
once again, mind blowing.

00:07:32.510 --> 00:07:36.150
So if we rewrite this top
integral, this is just equal

00:07:36.150 --> 00:07:38.230
to, I won't switch this time.

00:07:38.230 --> 00:07:39.790
So this is du dx, right?

00:07:42.530 --> 00:07:49.690
du dx times u to
the minus 3 dx.

00:07:49.690 --> 00:07:50.800
And I know what you're
thinking, Sal.

00:07:50.800 --> 00:07:53.310
Well, du dx is e to the x.

00:07:53.310 --> 00:07:54.420
u is also e to the x.

00:07:54.420 --> 00:07:56.390
Why didn't I substitute
it the other way around?

00:07:56.390 --> 00:07:57.830
Why didn't I say this?

00:07:57.830 --> 00:08:04.470
Why didn't I say that this one
is u, and why didn't I say this

00:08:04.470 --> 00:08:14.750
one is du dx to the minus 3?

00:08:14.750 --> 00:08:16.730
Well, as you can see, this
would have been useless, right?

00:08:16.730 --> 00:08:18.590
Because then I can't multiply
it times a dx, and it

00:08:18.590 --> 00:08:19.570
gets all confusing.

00:08:19.570 --> 00:08:22.830
And actually, I just realized,
I constructed a very silly

00:08:22.830 --> 00:08:24.890
problem for you, because you
could simplify this before even

00:08:24.890 --> 00:08:25.870
doing it with substitution.

00:08:25.870 --> 00:08:28.060
But we'll continue doing
it with substitution.

00:08:28.060 --> 00:08:30.260
But anyway, you see, if you did
it in this way, it becomes very

00:08:30.260 --> 00:08:32.420
complicated, so we don't
want to do it like that.

00:08:32.420 --> 00:08:39.060
So this, as we see, simplifies
to the integral of u to

00:08:39.060 --> 00:08:44.100
the minus 3 du, and that
that equals, let's see.

00:08:44.100 --> 00:08:50.510
You raise exponent 1 minus 1/2
u to the minus two, and that's

00:08:50.510 --> 00:09:00.320
the same thing as minus 1/2 e
to the x to the minus 2, or we

00:09:00.320 --> 00:09:06.600
could view that as minus 1/2
e to the minus 2x, and of

00:09:06.600 --> 00:09:09.230
course, plus c at the end.

00:09:09.230 --> 00:09:11.720
Now why was my problem
that I gave you silly?

00:09:11.720 --> 00:09:13.630
Well, I could have simplified
this before even doing

00:09:13.630 --> 00:09:15.070
the substitution, right?

00:09:15.070 --> 00:09:16.890
I could have said that that's
the same thing as the integral

00:09:16.890 --> 00:09:22.950
of e to the x times e to the
minus three x dx, which is the

00:09:22.950 --> 00:09:29.530
same thing as the integral
of e to the minus 2x dx.

00:09:29.530 --> 00:09:32.920
And actually, it's good that it
by substitution, because this

00:09:32.920 --> 00:09:35.550
probably wouldn't have been
completely intuitive for you

00:09:35.550 --> 00:09:37.670
to do just yet, as well.

00:09:37.670 --> 00:09:42.190
But anyway, that's
integration by substitution.

00:09:42.190 --> 00:09:44.340
I might do another presentation
where I do slightly harder

00:09:44.340 --> 00:09:46.780
problems, using this
same technique.

00:09:46.780 --> 00:09:48.750
I'll see you soon.


Indefinite Integration (part III)

WEBVTT
Kind: captions
Language: en

00:00:00.840 --> 00:00:01.890
Welcome back.

00:00:01.890 --> 00:00:04.030
Well I'm now going to do a
presentation on how to

00:00:04.030 --> 00:00:07.330
essentially invert the chain
rule or reverse the chain

00:00:07.330 --> 00:00:09.730
rule, because we're doing
integration, which is the

00:00:09.730 --> 00:00:11.910
opposite of taking
the derivative.

00:00:11.910 --> 00:00:13.570
So let's just take a
review of what the chain

00:00:13.570 --> 00:00:14.610
rule told us before.

00:00:19.220 --> 00:00:23.710
If I were to take the
derivative of f of g of x--

00:00:23.710 --> 00:00:25.180
hopefully this doesn't
confuse you too much.

00:00:25.180 --> 00:00:29.060
I'll give another example
with a concrete f of x

00:00:29.060 --> 00:00:29.986
and a concrete g of x.

00:00:29.986 --> 00:00:31.820
If I want to take the
derivative of that, the chain

00:00:31.820 --> 00:00:34.830
rule just says the derivative
of this composite function is

00:00:34.830 --> 00:00:38.410
just the derivative of
the inside function.

00:00:38.410 --> 00:00:45.930
g prime of x times the
derivative of the outer, or

00:00:45.930 --> 00:00:49.840
kind of the parent function,
but still having g

00:00:49.840 --> 00:00:51.610
of x in at times.

00:00:51.610 --> 00:00:55.885
f prime of g of x.

00:00:55.885 --> 00:00:59.850
I know this might seem
complicated if you aren't too

00:00:59.850 --> 00:01:02.693
comfortable with this type of
notation, but done in kind

00:01:02.693 --> 00:01:04.790
of an example form it
makes a lot of sense.

00:01:04.790 --> 00:01:13.790
If I said what is the
derivative of let's

00:01:13.790 --> 00:01:20.100
say sin of x squared.

00:01:22.800 --> 00:01:26.490
Well in this situation, f
of x is sin of x, right?

00:01:26.490 --> 00:01:28.870
And then g of x is
x squared, right?

00:01:28.870 --> 00:01:33.920
And sin of x squared is
essentially f of g of x.

00:01:33.920 --> 00:01:35.680
And this review of chain rule.

00:01:35.680 --> 00:01:38.530
You could go watch the video on
the chain rule as well, but I

00:01:38.530 --> 00:01:40.210
don't mind doing a couple
of problems here.

00:01:40.210 --> 00:01:42.730
All this is saying that the
derivative of this is you take

00:01:42.730 --> 00:01:45.330
the derivative of the inside
function-- g of x in this

00:01:45.330 --> 00:01:49.845
example, which is 2x-- and you
multiply it times the

00:01:49.845 --> 00:01:51.370
derivative of the
outer function or the

00:01:51.370 --> 00:01:52.270
parent function.

00:01:52.270 --> 00:01:56.420
And we memorize I guess that
the derivative of sin of x

00:01:56.420 --> 00:02:02.470
is cosine of x, so it's
times cosine of g of x.

00:02:02.470 --> 00:02:06.900
So we keep the x squared there.

00:02:06.900 --> 00:02:08.840
If it confuses you, just
think about the inside

00:02:08.840 --> 00:02:10.160
and the outside function.

00:02:10.160 --> 00:02:12.700
If you take the derivative of
kind of this composite

00:02:12.700 --> 00:02:16.260
function, it's the same thing
that equals the derivative of

00:02:16.260 --> 00:02:19.970
the inside function, which is
2x times the derivative

00:02:19.970 --> 00:02:21.230
of the outside function.

00:02:21.230 --> 00:02:23.775
But we keep this inside
function in it, and we

00:02:23.775 --> 00:02:24.960
keep this x right there.

00:02:24.960 --> 00:02:27.130
So that's a review
of the chain rule.

00:02:27.130 --> 00:02:31.050
So what happens if we want
to reverse the chain rule?

00:02:31.050 --> 00:02:35.260
Well if we wanted to reverse
it, we're essentially saying

00:02:35.260 --> 00:02:40.560
that we want to take the
integral of something where we

00:02:40.560 --> 00:02:45.650
have the derivative of kind of
the inner function, and then we

00:02:45.650 --> 00:02:52.980
have the derivative of a
larger composite function.

00:02:52.980 --> 00:02:55.300
I'm just rewriting the chain
rule, but I'm writing in

00:02:55.300 --> 00:02:58.330
an integral form that this
is equal to f of g of x.

00:03:00.870 --> 00:03:06.140
This statement up here is
the exact same thing as

00:03:06.140 --> 00:03:08.800
the statement down here.

00:03:08.800 --> 00:03:12.010
All I did is I took the
integral of both sides.

00:03:12.010 --> 00:03:15.240
I'm saying the integral
of this is equal to the

00:03:15.240 --> 00:03:17.680
integral of this right here.

00:03:17.680 --> 00:03:19.700
I probably shouldn't switched
equal signs like that

00:03:19.700 --> 00:03:23.930
with you, but let's use
this formula I guess.

00:03:23.930 --> 00:03:26.285
But all you have to know is
this the reverse of the chain

00:03:26.285 --> 00:03:27.290
rule to solve some problems.

00:03:32.270 --> 00:03:36.125
Image invert colors.

00:03:36.125 --> 00:03:37.440
Let me rewrite that.

00:03:37.440 --> 00:03:45.180
The integral-- if I have g
prime of x times f prime of

00:03:45.180 --> 00:03:52.390
g of x dx, then that is
equal to f of g of x.

00:03:52.390 --> 00:03:54.880
This is just the chain
rule in reverse.

00:03:54.880 --> 00:03:57.010
And I know it's very
complicated sometimes when you

00:03:57.010 --> 00:03:58.420
have it in this notation,
but I'll give you a

00:03:58.420 --> 00:04:00.070
couple of examples.

00:04:00.070 --> 00:04:08.160
What if I had the integral of
let's say-- this is actually

00:04:08.160 --> 00:04:11.400
one that's often kind of viewed
as a trick, but you'll see

00:04:11.400 --> 00:04:15.180
it's actually not that
tricky of a trick.

00:04:15.180 --> 00:04:15.810
OK.

00:04:15.810 --> 00:04:29.650
So let's say I have the natural
log squared over x dx.

00:04:29.650 --> 00:04:33.100
And if you saw an integral
like this, you'd probably be

00:04:33.100 --> 00:04:35.760
daunted, and you'd be
surprised, many people well

00:04:35.760 --> 00:04:38.270
into college calculus courses
are still daunted

00:04:38.270 --> 00:04:39.500
by this problem.

00:04:39.500 --> 00:04:41.060
But all you have to
recognize is this is

00:04:41.060 --> 00:04:41.870
the reverse chain rule.

00:04:41.870 --> 00:04:43.200
Why is this the
reverse chain rule?

00:04:43.200 --> 00:04:51.610
Well, this is the same thing as
the integral of 1/x times the

00:04:51.610 --> 00:04:56.260
natural log-- whoops, this
should be nlx, right-- the

00:04:56.260 --> 00:05:00.770
natural log of x squared dx.

00:05:00.770 --> 00:05:03.400
These are the same thing,
I just took the 1/x out.

00:05:03.400 --> 00:05:05.550
Now this might look
a little familiar.

00:05:05.550 --> 00:05:09.610
Well, what's the derivative
of the natural log of x?

00:05:09.610 --> 00:05:10.960
If you remember from
the derivative module,

00:05:10.960 --> 00:05:12.530
it's 1/x, right?

00:05:12.530 --> 00:05:15.560
Let me write that down
in the corner here.

00:05:15.560 --> 00:05:22.170
The derivative of the natural
log of x is equal to 1/x.

00:05:22.170 --> 00:05:25.130
So right here we have
the derivative of the

00:05:25.130 --> 00:05:26.640
natural log of x.

00:05:26.640 --> 00:05:31.360
So now we can just say that we
could essentially treat this

00:05:31.360 --> 00:05:34.875
natural log of x as kind
of a variable by itself.

00:05:34.875 --> 00:05:37.520
And essentially what I'm
going to be doing if I could

00:05:37.520 --> 00:05:38.900
actually substitute for.

00:05:38.900 --> 00:05:40.440
Actually let's do that.

00:05:40.440 --> 00:05:42.850
Well no, no, no I don't do that
now, that'll confuse you.

00:05:42.850 --> 00:05:44.930
Although my flip-flopping
is probably confusing

00:05:44.930 --> 00:05:46.690
you even more.

00:05:46.690 --> 00:05:49.990
I have the derivative of the
natural log of x, so I can then

00:05:49.990 --> 00:05:52.250
say well I have the derivative
there, so this is a

00:05:52.250 --> 00:05:54.410
composite function.

00:05:54.410 --> 00:05:58.460
This is essentially
f prime of g of x.

00:05:58.460 --> 00:06:02.400
So then I can say well
that integral must be

00:06:02.400 --> 00:06:07.260
equal to this thing.

00:06:07.260 --> 00:06:09.580
This is something
squared, right?

00:06:09.580 --> 00:06:11.330
So what's the integral
of something squared?

00:06:11.330 --> 00:06:16.200
Well the integral of
something squared is 1/3.

00:06:16.200 --> 00:06:17.920
That's something to
the third power.

00:06:17.920 --> 00:06:20.660
We learned in the previous
indefinite integral

00:06:20.660 --> 00:06:22.960
module, right?

00:06:22.960 --> 00:06:26.470
And then it's 1/3 something to
the third power, and then we

00:06:26.470 --> 00:06:30.150
know from the chain rule that
something is the ln of x.

00:06:32.960 --> 00:06:35.120
And I don't know if I've
already forgotten to do it

00:06:35.120 --> 00:06:39.000
once, but don't forget
to do the plus c.

00:06:39.000 --> 00:06:41.810
Now you say, Sal, this
completely confused me,

00:06:41.810 --> 00:06:43.590
because it probably did.

00:06:43.590 --> 00:06:45.360
And if it completely confused
you, let's just take the

00:06:45.360 --> 00:06:47.620
derivative of this and I think
you'll see it happening the

00:06:47.620 --> 00:06:50.120
other way around and it
might make a little sense.

00:06:50.120 --> 00:06:52.960
When you take the derivative,
we just use the chain rule.

00:06:52.960 --> 00:06:55.300
You take the derivative
of the inside first.

00:06:55.300 --> 00:07:01.370
The derivative of the inside is
1/x and you multiply that times

00:07:01.370 --> 00:07:03.670
the derivative of the outside
function, and then you

00:07:03.670 --> 00:07:05.330
keep the inside the same.

00:07:05.330 --> 00:07:08.150
So the derivative of the
outside function is 3 times it

00:07:08.150 --> 00:07:15.090
coefficient, so it's 3 times
1/3 times the whole thing

00:07:15.090 --> 00:07:17.180
to one less exponent.

00:07:17.180 --> 00:07:21.050
So the whole thing is ln of x.

00:07:21.050 --> 00:07:22.420
And then of course
plus 0, right.

00:07:22.420 --> 00:07:24.490
The derivative of c is 0.

00:07:24.490 --> 00:07:27.090
Well this is just equal
3, 3 cancel out.

00:07:27.090 --> 00:07:36.150
This is equal to 1/x times
the ln of x squared, which

00:07:36.150 --> 00:07:39.150
is our original problem.

00:07:39.150 --> 00:07:42.600
Let me do another problem
because I probably started

00:07:42.600 --> 00:07:44.360
off with something a
little bit too hard.

00:07:47.540 --> 00:08:02.070
What is the integral of
let's say sin of x to

00:08:02.070 --> 00:08:05.500
the third power dx.

00:08:05.500 --> 00:08:07.040
That's often written like this.

00:08:07.040 --> 00:08:09.330
That's often written
like sin of x.

00:08:11.960 --> 00:08:15.010
Same thing, but I like to
think of it this way because

00:08:15.010 --> 00:08:18.750
it's not a new notation.

00:08:18.750 --> 00:08:20.610
Actually this is a mistake.

00:08:20.610 --> 00:08:23.310
Clearly I'm making up these
problems on the fly.

00:08:23.310 --> 00:08:25.250
Actually I don't
want to do that.

00:08:25.250 --> 00:08:26.410
That is the wrong problem.

00:08:26.410 --> 00:08:28.380
I want to take the integral--
and actually you can see kind

00:08:28.380 --> 00:08:31.020
of how I'm thinking about these
problems-- I'm going to take

00:08:31.020 --> 00:08:41.880
the integral of cosine of x
times the sin of x to

00:08:41.880 --> 00:08:44.960
the third power dx.

00:08:44.960 --> 00:08:49.790
Well, we have this kind of more
complicated part, the sin of x,

00:08:49.790 --> 00:08:53.410
and we have the derivative sin
of x because we learned the

00:08:53.410 --> 00:08:56.190
derivative sin of
x is cosine of x.

00:08:56.190 --> 00:09:00.420
So if we have a function inside
of a larger composite function,

00:09:00.420 --> 00:09:03.530
and we have it's derivative, we
can just treat this function as

00:09:03.530 --> 00:09:05.720
kind of like a single entity.

00:09:05.720 --> 00:09:09.290
Like if this was just one
variable and then we

00:09:09.290 --> 00:09:10.550
take integral of it.

00:09:10.550 --> 00:09:18.450
So this just equal to sin of x
and we raise this one more

00:09:18.450 --> 00:09:23.880
power to the fourth and
we multiply times 1/4.

00:09:23.880 --> 00:09:25.030
And how did we do that?

00:09:25.030 --> 00:09:28.430
Because we know that the
integral of say x to the fourth

00:09:28.430 --> 00:09:31.570
dx is equal to-- I mean x to
the third dx-- is equal

00:09:31.570 --> 00:09:33.970
to 1/4 x to the fourth.

00:09:33.970 --> 00:09:35.910
Instead of an x we
had a sin here.

00:09:35.910 --> 00:09:38.390
And remember the reason why we
did that is because the

00:09:38.390 --> 00:09:41.330
derivative of the sin function
is sitting right here.

00:09:41.330 --> 00:09:44.660
In the next presentation, I'll
show you why this can also be

00:09:44.660 --> 00:09:47.750
done using substitution, or
why they're the same thing.

00:09:47.750 --> 00:09:49.090
I'll see you in the
next presentation.