WEBVTT

00:00:06.920 --> 00:00:08.580
Welcome back to recitation.

00:00:08.580 --> 00:00:11.021
In this video, I want us to
practice doing integration

00:00:11.021 --> 00:00:11.520
again.

00:00:11.520 --> 00:00:14.360
And so what we're going
to do is two problems.

00:00:14.360 --> 00:00:16.640
One is a definite integral,
one is indefinite.

00:00:16.640 --> 00:00:20.680
So this integral,
we're going to have,

00:00:20.680 --> 00:00:24.400
find the value of the integral
minus pi over 4 to pi over 4

00:00:24.400 --> 00:00:28.000
of secant cubed u du,
and then this one,

00:00:28.000 --> 00:00:31.660
I just want you to actually just
find an antiderivative, then,

00:00:31.660 --> 00:00:35.030
for 1 over 2 sine u cosine u.

00:00:35.030 --> 00:00:37.760
So why don't you take a while to
work on this, pause the video,

00:00:37.760 --> 00:00:40.430
and then when you're
ready, restart the video,

00:00:40.430 --> 00:00:42.920
and I will come back and
show you how I did it.

00:00:52.220 --> 00:00:52.720
OK.

00:00:52.720 --> 00:00:53.820
Welcome back.

00:00:53.820 --> 00:00:56.970
So again, we're going
to work on finding,

00:00:56.970 --> 00:00:59.690
in this one, a specific
number, and in this one,

00:00:59.690 --> 00:01:01.230
an antiderivative.

00:01:01.230 --> 00:01:03.120
So we'll start off
with the integral

00:01:03.120 --> 00:01:07.410
from minus pi over 4 to pi
over 4 of secant cubed u du.

00:01:07.410 --> 00:01:08.930
And the first thing
I'm going to do,

00:01:08.930 --> 00:01:13.460
is I'm going to drop
the bounds and just find

00:01:13.460 --> 00:01:15.990
the function I need, and then
I'll bring the bounds back in.

00:01:15.990 --> 00:01:19.230
I don't want to write the
bounds down at every step.

00:01:19.230 --> 00:01:22.356
I'm not doing any
substitution, I don't think.

00:01:22.356 --> 00:01:24.730
So I don't need to worry about
if I'm changing the bounds

00:01:24.730 --> 00:01:25.080
or not.

00:01:25.080 --> 00:01:26.454
So I'll keep the
bounds the same,

00:01:26.454 --> 00:01:29.660
but I won't write them down
again until near the end.

00:01:29.660 --> 00:01:30.430
All right.

00:01:30.430 --> 00:01:35.660
So let's look at this integral.

00:01:35.660 --> 00:01:40.130
So the easiest way, I think,
to start an integral like this,

00:01:40.130 --> 00:01:45.290
is to split it up into secant
u and then secant squared u.

00:01:45.290 --> 00:01:47.090
Maybe there are some
other ways you did it,

00:01:47.090 --> 00:01:49.840
but when I was looking at
this problem, the way I saw

00:01:49.840 --> 00:01:55.190
was to start off and write this
as secant u times 1-- oops,

00:01:55.190 --> 00:02:01.230
let me make sure, yes-- times
1 plus tan squared u du.

00:02:01.230 --> 00:02:03.380
Because secant squared
u is equal to 1

00:02:03.380 --> 00:02:05.210
plus tangent squared u.

00:02:05.210 --> 00:02:06.500
And what does it do for us?

00:02:06.500 --> 00:02:08.560
Well, it partially
answers the question,

00:02:08.560 --> 00:02:10.450
but not completely, we'll see.

00:02:10.450 --> 00:02:13.080
So what it does first,
is it gives us something

00:02:13.080 --> 00:02:16.680
we can find pretty easily.

00:02:16.680 --> 00:02:19.040
The first one, we just get
the integral of secant u du.

00:02:19.040 --> 00:02:20.740
We know that one.

00:02:20.740 --> 00:02:22.270
The second one is
a little harder,

00:02:22.270 --> 00:02:29.650
because we get the integral of
secant u tangent squared u du.

00:02:29.650 --> 00:02:34.040
And this has
promise, but it's not

00:02:34.040 --> 00:02:36.300
going to work for us right away.

00:02:36.300 --> 00:02:39.310
Because, you know, the
derivative of tangent

00:02:39.310 --> 00:02:42.340
is secant squared, and
the derivative of secant

00:02:42.340 --> 00:02:43.920
is secant tangent.

00:02:43.920 --> 00:02:47.267
So a substitution won't
work in this case,

00:02:47.267 --> 00:02:48.850
because neither one
of these functions

00:02:48.850 --> 00:02:50.390
is a derivative
of the other one.

00:02:50.390 --> 00:02:52.920
If I even move off
a tangent, and I

00:02:52.920 --> 00:02:55.480
have secant tangent
times another tangent,

00:02:55.480 --> 00:02:56.830
it's not the right derivative.

00:02:56.830 --> 00:02:58.820
A tangent's derivative
is secant squared.

00:02:58.820 --> 00:03:01.660
So my main point here is that
a u-substitution doesn't work,

00:03:01.660 --> 00:03:04.147
or the substitution
strategy doesn't work.

00:03:04.147 --> 00:03:05.980
So what we're going to
do, is we're actually

00:03:05.980 --> 00:03:10.030
going to take this part, and
make it into an integration

00:03:10.030 --> 00:03:11.710
by parts problem.

00:03:11.710 --> 00:03:13.185
So this is going
to be, I'm going

00:03:13.185 --> 00:03:14.925
to stop writing
equals signs, I'm

00:03:14.925 --> 00:03:17.110
going to figure out the
integral of this one.

00:03:17.110 --> 00:03:20.190
So this is a little
sidebar, and I'm

00:03:20.190 --> 00:03:24.830
going to look at the integral
of secant u tangent squared u.

00:03:27.830 --> 00:03:29.330
And the way I'm
going to write that,

00:03:29.330 --> 00:03:32.450
is I'm going to write
that as secant u tangent

00:03:32.450 --> 00:03:36.340
u will be one function I want.

00:03:36.340 --> 00:03:41.980
We'll call that w--
I guess, v prime.

00:03:41.980 --> 00:03:44.700
We can use v prime.

00:03:44.700 --> 00:03:48.340
And then the usual thing
we call u, we'll call w.

00:03:48.340 --> 00:03:52.097
So tangent u we'll call w.

00:03:52.097 --> 00:03:54.180
And I want to write down,
I want to explain to you

00:03:54.180 --> 00:03:56.100
why I'm picking these,
and sort of where

00:03:56.100 --> 00:03:57.840
this is going to get us.

00:03:57.840 --> 00:04:00.820
So this is an easy thing to
take an antiderivative of.

00:04:00.820 --> 00:04:01.320
Right?

00:04:01.320 --> 00:04:01.990
I can take this.

00:04:01.990 --> 00:04:05.270
I know v is going
to be secant u.

00:04:05.270 --> 00:04:07.980
Because the derivative of
secant u is secant u tangent u.

00:04:07.980 --> 00:04:12.540
And then w is tangent u, w
prime is secant squared u.

00:04:12.540 --> 00:04:16.420
So if you think about, what does
an integration by parts have?

00:04:16.420 --> 00:04:22.520
It's going to have v*w minus
the integral of v w prime.

00:04:22.520 --> 00:04:23.582
That's a lot to take in.

00:04:23.582 --> 00:04:25.040
But the point is
that that integral

00:04:25.040 --> 00:04:28.900
is going to have the
antiderivative of this, which

00:04:28.900 --> 00:04:32.280
is secant, and the derivative of
this, which is secant squared.

00:04:32.280 --> 00:04:34.700
So it's going to
have a secant cubed.

00:04:34.700 --> 00:04:36.470
Which might seem weird,
because now we're

00:04:36.470 --> 00:04:38.590
getting back to what
we started with.

00:04:38.590 --> 00:04:41.630
But the sign on the secant
cubed is going to be opposite.

00:04:41.630 --> 00:04:43.470
Again, this is a lot of talking.

00:04:43.470 --> 00:04:44.590
But let's figure out now.

00:04:44.590 --> 00:04:47.970
I just want to show you
where we're headed with this,

00:04:47.970 --> 00:04:50.780
why I picked the things I did.

00:04:50.780 --> 00:04:54.880
So as I mentioned-- let me
just write these down-- secant

00:04:54.880 --> 00:05:03.160
u is equal to v, and secant
squared u is equal to w prime.

00:05:03.160 --> 00:05:05.861
Sorry if that looks a little
weird, but that's a u.

00:05:05.861 --> 00:05:06.360
OK.

00:05:06.360 --> 00:05:09.520
So now let's write this with
an integration by parts.

00:05:09.520 --> 00:05:18.350
I get v*w, which is secant u
tangent u minus the integral

00:05:18.350 --> 00:05:22.830
of v*dw, which is
secant cubed u du.

00:05:30.030 --> 00:05:32.970
So this is what I was trying to
show you we were anticipating.

00:05:32.970 --> 00:05:36.920
So when I put this all together,
I replace this integral

00:05:36.920 --> 00:05:38.926
by these two things.

00:05:38.926 --> 00:05:40.050
And what's the point there?

00:05:40.050 --> 00:05:41.070
Notice what I have.

00:05:41.070 --> 00:05:43.320
If I actually look at
the pieces, I have,

00:05:43.320 --> 00:05:45.670
up here, I have
a secant cubed u.

00:05:45.670 --> 00:05:51.030
It's going to equal
secant u plus this--

00:05:51.030 --> 00:05:55.430
I have to evaluate that at
the bounds-- minus this.

00:05:55.430 --> 00:05:57.120
So I have the same
thing on this side

00:05:57.120 --> 00:05:59.560
that I had on the other
side, but with a minus sign.

00:05:59.560 --> 00:06:01.220
So if I add it to
the other side,

00:06:01.220 --> 00:06:02.880
I'm going to get two of them.

00:06:02.880 --> 00:06:04.960
So this was sort of
where we're headed.

00:06:04.960 --> 00:06:07.920
Now let's put it all together.

00:06:07.920 --> 00:06:09.760
Let's take it back.

00:06:09.760 --> 00:06:12.229
So this stuff here
is that, right?

00:06:12.229 --> 00:06:13.020
That's what we did.

00:06:13.020 --> 00:06:18.830
I'm going to write just the
important stuff right here.

00:06:18.830 --> 00:06:23.250
I've got the integral
of secant cubed u du is

00:06:23.250 --> 00:06:26.590
equal to secant u tangent u.

00:06:30.130 --> 00:06:32.470
Plus that secant-- I
forgot to put that one in,

00:06:32.470 --> 00:06:33.880
so let me write in that one.

00:06:33.880 --> 00:06:42.834
Plus the integral of secant
u du minus secant cubed u du.

00:06:42.834 --> 00:06:44.250
The integral of
secant cubed u du.

00:06:47.770 --> 00:06:49.300
OK?

00:06:49.300 --> 00:06:50.880
And now I'm going
to work some magic.

00:06:50.880 --> 00:06:53.510
I'm actually going to
erase something and move it

00:06:53.510 --> 00:06:54.720
to the other side.

00:06:54.720 --> 00:06:57.690
So let me sneak an
eraser off screen.

00:06:57.690 --> 00:06:59.480
I'm going to add this
to the other side.

00:06:59.480 --> 00:07:00.605
And what's going to happen?

00:07:03.510 --> 00:07:07.120
I come over here and
I get two of them.

00:07:07.120 --> 00:07:07.660
Right?

00:07:07.660 --> 00:07:09.576
There was one on this
side, with a minus sign.

00:07:09.576 --> 00:07:11.330
I added it, and now
I have two of them.

00:07:11.330 --> 00:07:12.440
And now what's the magic?

00:07:12.440 --> 00:07:14.630
Well, I just divide
everything by 2.

00:07:14.630 --> 00:07:16.530
And so this is
going to be over 2,

00:07:16.530 --> 00:07:19.100
and this is going to be over 2.

00:07:19.100 --> 00:07:21.720
And now I know what
an antiderivative is.

00:07:21.720 --> 00:07:23.230
Notice I haven't
put in the plus c,

00:07:23.230 --> 00:07:26.590
because I'm about to
put in some bounds.

00:07:26.590 --> 00:07:27.840
All right?

00:07:27.840 --> 00:07:30.700
And by the way, I know an
antiderivative of secant u.

00:07:30.700 --> 00:07:32.989
So we'll get to
that in one second.

00:07:32.989 --> 00:07:34.280
But hopefully everyone follows.

00:07:34.280 --> 00:07:36.321
I had an integral of secant
cubed u on this side.

00:07:36.321 --> 00:07:38.820
I had a minus integral of
secant cubed u over here.

00:07:38.820 --> 00:07:40.320
I added it to the other side.

00:07:40.320 --> 00:07:44.040
It gave me two of them, so
then I just divided by 2.

00:07:44.040 --> 00:07:44.890
All right?

00:07:44.890 --> 00:07:49.440
And now let me just explicitly
write down the last thing

00:07:49.440 --> 00:07:49.940
we need.

00:07:53.720 --> 00:07:54.930
We still need this one.

00:07:54.930 --> 00:08:00.180
And this is going to be 1/2
natural log absolute value

00:08:00.180 --> 00:08:03.670
secant u plus tangent u.

00:08:03.670 --> 00:08:04.170
OK.

00:08:04.170 --> 00:08:07.540
So now we just have to evaluate
everywhere and we're done.

00:08:07.540 --> 00:08:09.490
So now we have to
evaluate all of this.

00:08:09.490 --> 00:08:11.630
Remember, I said I
left out the bounds.

00:08:11.630 --> 00:08:16.670
The bounds are pi over
4, minus pi over 4;

00:08:16.670 --> 00:08:21.760
pi over 4 minus pi over 4.

00:08:21.760 --> 00:08:22.985
All right.

00:08:22.985 --> 00:08:25.360
So I'm going to give myself
a little cheat sheet up here,

00:08:25.360 --> 00:08:29.560
and then I'm going to write
down the numbers I get here.

00:08:29.560 --> 00:08:32.900
So my cheat sheet
is to remind myself

00:08:32.900 --> 00:08:37.140
that secant of plus
or minus pi over 4

00:08:37.140 --> 00:08:38.775
is equal to square root 2.

00:08:38.775 --> 00:08:41.060
Let me just make
sure that's right.

00:08:41.060 --> 00:08:44.440
Cosine pi over 4 is
1 over square root 2.

00:08:44.440 --> 00:08:47.690
Cosine is even, so plus or minus
pi over 4 will be the same.

00:08:47.690 --> 00:08:51.160
Secant is 1 over
that, so I'm good.

00:08:51.160 --> 00:08:54.490
Tangent of pi over 4.

00:08:54.490 --> 00:08:57.190
Well, tangent is odd, so I
should say plus or minus here,

00:08:57.190 --> 00:08:59.720
tangent is odd, so it's
going to be, they're going

00:08:59.720 --> 00:09:01.620
to have two different signs.

00:09:01.620 --> 00:09:04.920
Tangent pi over 4 is sine pi
over 4 over cosine pi over 4.

00:09:04.920 --> 00:09:06.530
It's the same value there.

00:09:06.530 --> 00:09:07.430
So it's 1.

00:09:07.430 --> 00:09:10.720
So tangent plus or minus pi
over 4 is plus or minus 1.

00:09:10.720 --> 00:09:12.530
So that's what
we're working with.

00:09:12.530 --> 00:09:15.530
So now let's start plugging in.

00:09:15.530 --> 00:09:18.300
Secant pi over 4
tangent pi over 4

00:09:18.300 --> 00:09:20.860
gives me root 2 times 1 over 2.

00:09:20.860 --> 00:09:23.860
So I get root 2 over
2, is the first thing.

00:09:23.860 --> 00:09:27.760
So I have to write--
this equals is from here.

00:09:27.760 --> 00:09:30.490
So I have root 2 over 2.

00:09:30.490 --> 00:09:34.780
And then secant minus
pi over 4 is again

00:09:34.780 --> 00:09:38.040
root 2, tangent minus
pi over 4 is minus 1.

00:09:38.040 --> 00:09:42.830
So I have minus, I
have a negative here,

00:09:42.830 --> 00:09:47.740
and then I have a 1 here, or a
root 2 here, negative 1 over 2.

00:09:47.740 --> 00:09:51.800
So I get another negative,
so I have a plus.

00:09:51.800 --> 00:09:54.270
So one negative came from,
I was using the lower bound,

00:09:54.270 --> 00:09:56.830
and one negative came
from the tangent.

00:09:56.830 --> 00:09:58.740
That gave me a plus.

00:09:58.740 --> 00:10:00.960
And then I have plus 1/2.

00:10:00.960 --> 00:10:02.190
Well, again.

00:10:02.190 --> 00:10:05.720
Natural log of
secant pi over 4 is

00:10:05.720 --> 00:10:07.970
going to be-- so I'm going
to have natural log of root

00:10:07.970 --> 00:10:11.060
2 plus 1, and I'm going
to have a natural log

00:10:11.060 --> 00:10:13.469
of root 2 minus 1.

00:10:13.469 --> 00:10:15.510
And I'm going to have a
negative in between them.

00:10:15.510 --> 00:10:17.218
So I'm going to work
a little magic here.

00:10:17.218 --> 00:10:23.730
It's going to be natural log of
2 plus 1 over root 2 minus 1.

00:10:23.730 --> 00:10:26.750
Now, just to point out,
where did that come from?

00:10:26.750 --> 00:10:30.310
That came from putting in root
2 for both of the pi over 4's

00:10:30.310 --> 00:10:31.830
and minus pi over 4.

00:10:31.830 --> 00:10:35.170
Tangent pi over
4 was the plus 1.

00:10:35.170 --> 00:10:37.590
Tangent of negative pi
over 4 was the minus 1.

00:10:37.590 --> 00:10:39.110
How do I get this division?

00:10:39.110 --> 00:10:41.500
I had natural log of
something minus natural log

00:10:41.500 --> 00:10:42.860
of something else.

00:10:42.860 --> 00:10:49.630
So in the end, I get root 2
plus 1/2 natural log absolute

00:10:49.630 --> 00:10:54.330
root 2 plus 1 over
root 2 minus 1.

00:10:54.330 --> 00:10:56.230
All right.

00:10:56.230 --> 00:11:00.030
That is part (a).

00:11:00.030 --> 00:11:03.550
So part (a), let me just
remind you, what did we do?

00:11:03.550 --> 00:11:06.060
We had secant cubed u.

00:11:06.060 --> 00:11:10.280
We did a substitution for one
of the, for secant squared.

00:11:10.280 --> 00:11:12.090
We got something
we could deal with,

00:11:12.090 --> 00:11:13.944
and something that
didn't look so promising,

00:11:13.944 --> 00:11:15.610
but once we did an
integration by parts,

00:11:15.610 --> 00:11:19.780
we were back to what we started
with, with a different sign.

00:11:19.780 --> 00:11:21.810
So we moved it to
the other side.

00:11:21.810 --> 00:11:24.495
We were basically able to
solve for secant cubed u.

00:11:24.495 --> 00:11:26.120
So we got all the
way through, and then

00:11:26.120 --> 00:11:27.150
we just had to evaluate.

00:11:27.150 --> 00:11:29.770
So the big step was,
once you were here,

00:11:29.770 --> 00:11:32.890
knowing an integration by
parts actually would save you.

00:11:32.890 --> 00:11:34.680
That's sort of the
hard thing to see.

00:11:34.680 --> 00:11:37.010
Takes a little while
to see that one, maybe.

00:11:37.010 --> 00:11:37.510
All right.

00:11:37.510 --> 00:11:38.780
So now the next one.

00:11:38.780 --> 00:11:40.920
If we come back
here, we're trying

00:11:40.920 --> 00:11:44.530
to find an antiderivative
of 1 over 2 sine u cosine u.

00:11:44.530 --> 00:11:47.620
And there may be some
other ways to do this,

00:11:47.620 --> 00:11:50.044
but actually, this
problem, part of the reason

00:11:50.044 --> 00:11:51.460
I wanted to do
this problem, was I

00:11:51.460 --> 00:11:53.460
wanted to remind you that
it's good to know some

00:11:53.460 --> 00:11:55.370
of the basic
trigonometric identities,

00:11:55.370 --> 00:11:57.670
because it'll make
your life a lot easier.

00:11:57.670 --> 00:12:00.610
So this integral is
actually the same

00:12:00.610 --> 00:12:04.470
as, is the integral
of du over sine 2u.

00:12:04.470 --> 00:12:08.010
And the reason is, there's a
trigonometric identity that

00:12:08.010 --> 00:12:12.060
says, 2 sine u cosine
u is equal to sine 2u.

00:12:12.060 --> 00:12:15.900
So I wanted to give
you a reason for why

00:12:15.900 --> 00:12:19.100
we know those, why we know
some of those identities,

00:12:19.100 --> 00:12:21.075
and you end up in
these situations.

00:12:21.075 --> 00:12:22.950
There might be other
ways to handle this one,

00:12:22.950 --> 00:12:27.120
but the easiest, most direct
way is if you do this.

00:12:27.120 --> 00:12:29.900
You change it so that (b)
is actually the same thing

00:12:29.900 --> 00:12:32.860
as integral of du over sine 2u.

00:12:32.860 --> 00:12:36.230
So it's just a straight
up double angle formula,

00:12:36.230 --> 00:12:39.690
you can call it if
you need a fancy name.

00:12:39.690 --> 00:12:40.670
And now what is this?

00:12:40.670 --> 00:12:44.430
Well, this is equal to
the integral of-- what's

00:12:44.430 --> 00:12:49.130
1 over sine, is
going to be cosecant.

00:12:49.130 --> 00:12:51.930
Cosecant 2u du.

00:12:51.930 --> 00:12:54.910
And we know the
antiderivative to cosecant u.

00:12:54.910 --> 00:12:57.680
We know that that's going
to be negative natural log

00:12:57.680 --> 00:13:00.500
of cosecant u plus cotangent u.

00:13:00.500 --> 00:13:03.880
But the problem is, when there's
a 2 there, what do we do?

00:13:03.880 --> 00:13:06.940
Well, just think about it as,
if you had the antiderivative,

00:13:06.940 --> 00:13:10.157
you know by the chain
rule, if you just

00:13:10.157 --> 00:13:11.990
put in two u's everywhere
there was a u when

00:13:11.990 --> 00:13:14.580
you took the derivative, you
would end up with an extra 2

00:13:14.580 --> 00:13:15.220
in front.

00:13:15.220 --> 00:13:16.670
So you have to,
basically you have

00:13:16.670 --> 00:13:19.830
to just put in 1/2 in front.

00:13:19.830 --> 00:13:22.270
You could do a
substitution to check this,

00:13:22.270 --> 00:13:24.890
but it's really
straightforward to say,

00:13:24.890 --> 00:13:30.490
this antiderivative is equal
to negative 1/2 natural log

00:13:30.490 --> 00:13:35.860
absolute cosecant 2u
plus cotangent 2u.

00:13:38.331 --> 00:13:40.830
Now that I have that written
out, I'll just point out again,

00:13:40.830 --> 00:13:44.210
if there was no 2 here,
the 1/2 wouldn't be here,

00:13:44.210 --> 00:13:47.720
and we'd just have cosecant
u cotangent u inside here.

00:13:47.720 --> 00:13:52.720
But once we have to have
the 2 to get a cosecant 2u

00:13:52.720 --> 00:13:56.660
in the end, we also need
to divide by 2 to kill it

00:13:56.660 --> 00:13:58.220
off when we take a derivative.

00:13:58.220 --> 00:14:00.070
The chain rule would
give us a 2 in front,

00:14:00.070 --> 00:14:01.940
so the 1/2 kills it off.

00:14:01.940 --> 00:14:04.670
So we don't end up with, you
know, with this not here,

00:14:04.670 --> 00:14:07.590
the derivative of
this is 2 cosecant 2u.

00:14:07.590 --> 00:14:09.980
So we divide by
2, then we get it.

00:14:09.980 --> 00:14:11.190
We get the right answer.

00:14:11.190 --> 00:14:13.910
So this one-- you know,
I'm not intentionally

00:14:13.910 --> 00:14:14.850
trying to trick you.

00:14:14.850 --> 00:14:17.704
I just want to
point out that it's

00:14:17.704 --> 00:14:19.870
good to know some of these
trigonometric identities.

00:14:19.870 --> 00:14:22.470
It makes solving these problems
a lot easier to deal with.

00:14:22.470 --> 00:14:24.670
So the main point of
this one was actually

00:14:24.670 --> 00:14:26.630
knowing the
identity, in my mind.

00:14:26.630 --> 00:14:29.490
Maybe you found
another way to do it.

00:14:29.490 --> 00:14:31.810
Probably it didn't
take two lines, though.

00:14:31.810 --> 00:14:35.350
So if you found other way to
do it, actually, it's good.

00:14:35.350 --> 00:14:36.220
It's creative.

00:14:36.220 --> 00:14:37.250
And I like that.

00:14:37.250 --> 00:14:40.300
But I was hoping to convince
you that sometimes it's

00:14:40.300 --> 00:14:42.950
simple to know a few
of these identities.

00:14:42.950 --> 00:14:45.052
And that is where I will stop.