Matrices: Reduced row echelon form 3

WEBVTT
Kind: captions
Language: en

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I have here three linear
equations of four unknowns.

00:00:03.810 --> 00:00:07.050
And like the first video, where
I talked about reduced

00:00:07.050 --> 00:00:10.690
row echelon form, and solving
systems of linear equations

00:00:10.690 --> 00:00:13.510
using augmented matrices, at
least my gut feeling says,

00:00:13.510 --> 00:00:16.720
look, I have fewer equations
than variables, so I probably

00:00:16.720 --> 00:00:18.470
won't be able to constrain
this enough.

00:00:18.470 --> 00:00:20.370
Or maybe I'll have an infinite
number of solutions.

00:00:20.370 --> 00:00:21.690
But let's see if I'm right.

00:00:21.690 --> 00:00:24.720
So let's construct the augmented
matrix for this

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system of equations.

00:00:27.310 --> 00:00:33.190
My coefficients on the x1
terms are 1, 1, and 2.

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Coefficients on the x2
are 2, 2, and 4.

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Coefficients on x3
are 1, 2, and 0.

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There's of course no x3 term,
so we can view it as a 0

00:00:46.190 --> 00:00:47.060
coefficient.

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Coefficients on the x4 are
1, minus 1, and 6.

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And then on the right-hand side
of the equals sign, I

00:00:56.080 --> 00:01:00.670
have 8, 12, and 4.

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There's my augmented matrix,
now let's put this guy into

00:01:04.360 --> 00:01:08.430
reduced row echelon form.

00:01:08.430 --> 00:01:10.640
The first thing I want to do is,
I want to zero out these

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two rows right here.

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So what can we do?

00:01:14.910 --> 00:01:18.550
I'm going to keep my first row
the same for now, so it's 1,

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2, 1, 1, 8.

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That line essentially represents
my equals sign.

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What I can do is, let me
subtract-- let me replace the

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second row with the second
row minus the first row.

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So 1 minus 1 is 0, 2 minus 2
is 0, 2 minus 1 is 1, 1--

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negative 1 minus 1 is minus 2,
and then 12 minus 8 is 4.

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There you go, that looks good so
far, it looks like column,

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or x2 which is represented by
column two, looks like it

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might be a free variable, but
we're not 100% sure yet.

00:02:04.590 --> 00:02:07.240
Let's do all of our rows.

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So let's take-- to get rid of
this guy right here, let's

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replace our third equation with
the third equation minus

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two times our first equation.

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So we get 2 minus 2 times 1 is
0, 4 minus 2 times 2, well

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that's 0, 0 minus 2 times
1, that's minus 2.

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6 minus 2 times 1, well
that's 4, right?

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6 minus 2.

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And then 4 minus 2 times
8, is minus 16, 4

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minus 16 is minus 12.

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Now what can we do?

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Well, let's see if we can
get rid of this minus

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2 term right there.

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So let me rewrite my
augmented matrix.

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I'm going to keep row two the
same this time, so I get a 0,

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0, 1, minus 2, and essentially
my equals sign, or the

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augmented part of the matrix.

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And now let's see
what I can do.

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Let me get rid of this 0 up
here, because I want to get

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into reduced row echelon form.

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So any of my pivot entries,
which are always going to have

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the coefficient 1, or the entry
1, it should be the only

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non-zero term in my row.

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How do I get rid of
this one here?

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Well I can subtract-- I can
replace row one with row 1

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minus row 2.

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So 1 minus 0 is 1, 2 minus 0 is
2, 1 minus 1 is 0, 1 minus

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minus 2, that's 1 plus
2, which is 3.

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And then 8 minus 4 is 4.

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And now how can I get
rid of this guy?

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Well let me replace row 3 with
row 3 plus 2 times row 1.

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Sorry, with row 3 plus
2 times row 2.

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Right?

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Because then you'd have minus
2, plus 2 times this, and

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they'd cancel out.

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So let's see the zeros.

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0 plus 2 times 0, that's 0.

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0 plus 2 times 0, that's 0,
minus 2 plus 2 times 1 is 0.

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4 plus 2 times minus 2, that's
4 minus 4, that's 0.

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And then I have minus
12, plus 2 times 4.

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That's minus 12 plus
8, that's minus 4.

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Now, this is interesting right
now-- this is interesting.

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I have essentially put this in
reduced row echelon form.

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I have two pivot entries, that's
a pivot entry right

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there, and that's a pivot
entry right there.

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They're the only non-zero term
in their respective columns.

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And this is just kind of a style
issue, but this pivot

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entry is in a lower
row than that one.

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So it's in a column to the right
of this one right there.

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And I just inspected, this looks
like a-- this column two

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looks kind of like a free
variable-- there's no pivot

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entry there, no pivot
entry there.

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But let's see, let's
map this back to

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our system of equations.

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These are just numbers to
me and I just kind of

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mechanically, almost like a
computer, put this in reduced

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row echelon form.

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Actually, almost exactly
like a computer.

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But let me put it back to my
system of linear equations, to

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see what our result is.

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So we get 1 times x1, let
me write it in yellow.

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So I get 1 times x1, plus 2
times x2, plus 0 times x3,

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plus 3 times x4 is equal to 4.

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Obviously I could ignore this
term right there, I didn't

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even have to write it.

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Actually.

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I'm not going to write that.

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Then I get 0 times x1, plus 0
times x2, plus 1 times x3, so

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I can just write that.

00:06:05.860 --> 00:06:06.850
I'll just write the one.

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1 times x3, minus 2 times
x4, is equal to 4.

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And then this last term,
what do I get?

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I get 0 x1, plus 0 x2 plus 0
x3 plus 0 x4, well, all of

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that's equal to 0, and I've got
to write something on the

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left-hand side.

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So let me just write a 0,
and that's got to be

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equal to minus 4.

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Well this doesn't make
any sense whatsoever.

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0 equals minus 4.

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This is this is a nonsensical
constraint, this is

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impossible.

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0 can never equal minus 4.

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This is impossible.

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Which means that it is
essentially impossible to find

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an intersection of these three
systems of equations, or a

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solution set that satisfies
all of them.

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When we looked at this
initially, at the beginning of

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the of the video, we said
there are only three

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equations, we have four
unknowns, maybe there's going

00:07:02.170 --> 00:07:03.980
to be an infinite set
of solutions.

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But turns out that these three--
I guess you can call

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them these three surfaces--
don't intersect in r4.

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Right?

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These are all four dimensional,
we're dealing in

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r4 right here, because we have--
I guess each vector has

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four components, or we have four
variables, is the way you

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could think about it.

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And it's always hard to
visualize things in r4.

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But if we were doing things
in r3, we can imagine the

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situation where, let's say
we had two planes in r3.

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So that's one plane right there,
and then I had another

00:07:36.680 --> 00:07:39.930
completely parallel
plane to that one.

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So I had another completely
parallel plane

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to that first one.

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Even though these would be two
planes in r3, so let me give

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an example.

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So let's say that this first
plane was represented by the

00:07:51.640 --> 00:08:02.680
equation 3x plus 6y plus 9z is
equal to 5, and the second

00:08:02.680 --> 00:08:08.400
plane was represented by the
equation 3x plus 6y plus 9z is

00:08:08.400 --> 00:08:11.170
equal to 2.

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These two planes in r3-- this is
the case of r3, so this is

00:08:15.630 --> 00:08:17.540
r3 right here.

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These two planes, clearly
they'll never intersect.

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Because obviously this one has
the same coefficients adding

00:08:23.520 --> 00:08:26.430
up to 5, this one has the same
coefficient adding up to 2.

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And when, if we just looked at
this initially, if it wasn't

00:08:28.690 --> 00:08:31.090
so obvious, we would have
said, we have only two

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equations with three unknowns,
maybe this has an infinite set

00:08:33.530 --> 00:08:34.230
of solutions.

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But it won't be the case,
because you can actually just

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subtract this equation, from the
bottom equation, from the

00:08:39.799 --> 00:08:40.460
top equation.

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And what do you get?

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You would get a very familiar--
so if you just

00:08:44.190 --> 00:08:46.220
subtract the bottom equation
from the top equation, and you

00:08:46.220 --> 00:08:50.183
get 3x minus 3x, 6y minus 6y, 9z
minus 9z-- actually, let me

00:08:50.183 --> 00:08:51.500
do it right here.

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So for that minus that, you get
zero is equal to 5 minus

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2, which is 3.

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Which is a very similar result
that we got up there.

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So when you have two parallel
planes, in this case in r3, or

00:09:03.110 --> 00:09:07.150
really to any kind of two
parallel equations, or a set

00:09:07.150 --> 00:09:08.620
of parallel equations,
they won't intersect.

00:09:08.620 --> 00:09:12.290
And you're going to get, when
you put it in reduced row

00:09:12.290 --> 00:09:14.490
echelon form, or you just do
basic elimination, or you

00:09:14.490 --> 00:09:17.250
solve the systems, you're going
to get a statement that

00:09:17.250 --> 00:09:19.720
zero is equal to something, and
that means that there are

00:09:19.720 --> 00:09:20.970
no solutions.

00:09:23.990 --> 00:09:26.030
So the general take-away,
if you have zero equals

00:09:26.030 --> 00:09:27.840
something, no solutions.

00:09:27.840 --> 00:09:31.770
If you have the same number of
pivot variables, the same

00:09:31.770 --> 00:09:36.210
number of pivot entries as you
do columns, so if you get the

00:09:36.210 --> 00:09:39.040
situations-- let me write this
down, this is good to know.

00:09:39.040 --> 00:09:41.600
if you have zero is equal
to anything, then

00:09:41.600 --> 00:09:44.760
that means no solution.

00:09:44.760 --> 00:09:46.710
If you're dealing with r3,
then you probably have

00:09:46.710 --> 00:09:50.680
parallel planes, in r2 you're
dealing with parallel lines.

00:09:50.680 --> 00:09:55.930
If you have the situation where
you have the same number

00:09:55.930 --> 00:10:02.960
of pivot entries as columns, so
it's just 1, 1, 1, 1, this

00:10:02.960 --> 00:10:05.350
is the case of r4 right there.

00:10:05.350 --> 00:10:07.390
I think you get the idea.

00:10:07.390 --> 00:10:10.440
That equals a, b, c, d.

00:10:10.440 --> 00:10:11.690
Then you have a unique
solution.

00:10:15.730 --> 00:10:19.010
Now if, you have any free
variables-- so free variables

00:10:19.010 --> 00:10:23.190
look like this, so let's say we
have 1, 0, 1, 0, and then I

00:10:23.190 --> 00:10:30.010
have the entry 1, 1,
let me be careful.

00:10:30.010 --> 00:10:33.320
0, let me do it like this.

00:10:33.320 --> 00:10:41.040
1, 0, 0, and then I have the
entry 1, 2, and then I have a

00:10:41.040 --> 00:10:43.520
bunch of zeroes over here.

00:10:43.520 --> 00:10:45.370
And then this has to equal
zero-- remember, if this was a

00:10:45.370 --> 00:10:47.760
bunch of zeroes equaling some
variable, then I would have no

00:10:47.760 --> 00:10:50.330
solution, or equalling some
constant, let's say this is

00:10:50.330 --> 00:10:52.820
equal to 5, this
is equal to 2.

00:10:52.820 --> 00:10:55.120
If this is our reduced row
echelon form that we

00:10:55.120 --> 00:10:58.860
eventually get to, then we have
a few free variables.

00:10:58.860 --> 00:11:02.750
This is a free, or I guess we
could call this column a free

00:11:02.750 --> 00:11:04.730
column, to some degree this
one would be too.

00:11:04.730 --> 00:11:07.390
Because it has no
pivot entries.

00:11:07.390 --> 00:11:09.280
These are the pivot entries.

00:11:09.280 --> 00:11:13.420
So this is variable x2 and
that's variable x4.

00:11:13.420 --> 00:11:14.910
Then these would be
free, we can set

00:11:14.910 --> 00:11:16.240
them equal to anything.

00:11:16.240 --> 00:11:19.170
So then here we have unlimited
solutions,

00:11:19.170 --> 00:11:21.670
or no unique solutions.

00:11:21.670 --> 00:11:23.510
And that was actually the
first example we saw.

00:11:23.510 --> 00:11:25.540
And these are really the three
cases that you're going to see

00:11:25.540 --> 00:11:27.880
every time, and it's good to
get familiar with them so

00:11:27.880 --> 00:11:29.570
you're never going to get
stumped up when you have

00:11:29.570 --> 00:11:32.930
something like 0 equals minus
4, or 0 equals 3.

00:11:32.930 --> 00:11:35.440
Or if you have just a bunch of
zeros and a bunch of rows.

00:11:35.440 --> 00:11:36.870
I want to make that
very clear.

00:11:36.870 --> 00:11:40.550
Sometimes, you see a bunch of
zeroes here, on the left-hand

00:11:40.550 --> 00:11:44.360
side of the augmented divide,
and you might say, oh maybe I

00:11:44.360 --> 00:11:45.770
have no unique solutions,
I have an

00:11:45.770 --> 00:11:46.730
infinite number of solutions.

00:11:46.730 --> 00:11:50.380
But you have to look at
this entry right here.

00:11:50.380 --> 00:11:52.880
Only if this whole thing is
zero and you have free

00:11:52.880 --> 00:11:56.210
variables, then you have an
infinite number of solutions.

00:11:56.210 --> 00:11:59.050
If you have a statement like,
0 is equal to a, if this is

00:11:59.050 --> 00:12:01.380
equal to 7 right here, then all
of the sudden you would

00:12:01.380 --> 00:12:03.000
have no solution to this.

00:12:03.000 --> 00:12:05.390
That you're dealing with
parallel surfaces.

00:12:05.390 --> 00:12:07.850
Anyway, hopefully you
found that helpful.


Matrices: Reduced row echelon form 1

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:04.340
I have here three equations
of four unknowns.

00:00:04.340 --> 00:00:07.530
You can already guess, or you
already know, that if you have

00:00:07.530 --> 00:00:09.540
more unknowns than equations,
you are probably not

00:00:09.540 --> 00:00:11.340
constraining it enough.

00:00:11.340 --> 00:00:11.800
You actually are going
to have an

00:00:11.800 --> 00:00:13.360
infinite number of solutions.

00:00:13.360 --> 00:00:15.320
Those infinite number of
solutions could still be

00:00:15.320 --> 00:00:18.180
constrained.

00:00:18.180 --> 00:00:20.400
Let's say we're in four
dimensions, in this case,

00:00:20.400 --> 00:00:21.810
because we have four
variables.

00:00:21.810 --> 00:00:24.350
Maybe we were constrained into a
plane in four dimensions, or

00:00:24.350 --> 00:00:25.740
if we were in three dimensions,
maybe we're

00:00:25.740 --> 00:00:26.750
constrained to a line.

00:00:26.750 --> 00:00:29.640
A line is an infinite number of
solutions, but it's a more

00:00:29.640 --> 00:00:31.100
constrained set.

00:00:31.100 --> 00:00:33.700
Let's solve this set of
linear equations.

00:00:33.700 --> 00:00:36.830
We've done this by elimination
in the past. What I want to do

00:00:36.830 --> 00:00:38.950
is I want to introduce
the idea of matrices.

00:00:38.950 --> 00:00:41.770
The matrices are really just
arrays of numbers that are

00:00:41.770 --> 00:00:44.390
shorthand for this system
of equations.

00:00:44.390 --> 00:00:46.800
Let me create a matrix here.

00:00:46.800 --> 00:00:49.880
I could just create a
coefficient matrix, where the

00:00:49.880 --> 00:00:55.280
coefficient matrix would just
be, let me write it neatly,

00:00:55.280 --> 00:01:01.730
the coefficient matrix would
just be the coefficients on

00:01:01.730 --> 00:01:05.340
the left hand side of these
linear equations.

00:01:05.340 --> 00:01:07.570
The coefficient there is 1.

00:01:07.570 --> 00:01:08.980
The coefficient there is 1.

00:01:08.980 --> 00:01:10.930
The coefficient there is 2.

00:01:10.930 --> 00:01:12.750
You have 2, 2, 4.

00:01:12.750 --> 00:01:14.860
2, 2, 4.

00:01:14.860 --> 00:01:16.820
1, 2, 0.

00:01:16.820 --> 00:01:20.360
1, 2, there is no coefficient
the x3 term here, because

00:01:20.360 --> 00:01:21.920
there is no x3 term there.

00:01:21.920 --> 00:01:25.710
We'll say the coefficient on
the x3 term there is 0.

00:01:25.710 --> 00:01:31.210
And then we have 1,
minus 1, and 6.

00:01:31.210 --> 00:01:34.360
Now if I just did this right
there, that would be the

00:01:34.360 --> 00:01:37.210
coefficient matrix for
this system of

00:01:37.210 --> 00:01:38.600
equations right there.

00:01:38.600 --> 00:01:41.240
What I want to do is I want to
augment it, I want to augment

00:01:41.240 --> 00:01:44.780
it with what these equations
need to be equal to.

00:01:44.780 --> 00:01:46.320
Let me augment it.

00:01:46.320 --> 00:01:47.520
What I am going to do is I'm
going to just draw a little

00:01:47.520 --> 00:01:55.000
line here, and write the
7, the 12, and the 4.

00:01:55.000 --> 00:02:00.030
I think you can see that
this is just another

00:02:00.030 --> 00:02:01.800
way of writing this.

00:02:01.800 --> 00:02:03.510
And just by the position, we
know that these are the

00:02:03.510 --> 00:02:05.910
coefficients on the x1 terms.
We know that these are the

00:02:05.910 --> 00:02:08.789
coefficients on the x2 terms.
And what this does, it really

00:02:08.789 --> 00:02:12.790
just saves us from having to
write x1 and x2 every time.

00:02:12.790 --> 00:02:16.140
We can essentially do the same
operations on this that we

00:02:16.140 --> 00:02:18.810
otherwise would have
done on that.

00:02:18.810 --> 00:02:22.560
What we can do is, we can
replace any equation with that

00:02:22.560 --> 00:02:25.020
equation times some
scalar multiple,

00:02:25.020 --> 00:02:26.210
plus another equation.

00:02:26.210 --> 00:02:28.110
We can divide an equation,
or multiply an

00:02:28.110 --> 00:02:29.390
equation by a scalar.

00:02:29.390 --> 00:02:30.810
We can subtract them
from each other.

00:02:30.810 --> 00:02:32.230
We can swap them.

00:02:32.230 --> 00:02:36.320
Let's do that in an attempt
to solve this equation.

00:02:36.320 --> 00:02:38.180
The first thing I want to do,
just like I've done in the

00:02:38.180 --> 00:02:41.220
past, I want to get this
equation into the form of,

00:02:41.220 --> 00:02:43.180
where if I can, I have a 1.

00:02:43.180 --> 00:02:47.910
My leading coefficient in
any of my rows is a 1.

00:02:47.910 --> 00:02:53.270
And that every other entry
in that column is a 0.

00:02:53.270 --> 00:02:55.180
In the past, I made sure
that every other entry

00:02:55.180 --> 00:02:56.300
below it is a 0.

00:02:56.300 --> 00:02:58.950
That's what I was doing in some
of the previous videos,

00:02:58.950 --> 00:03:01.420
when we tried to figure out
of things were linearly

00:03:01.420 --> 00:03:02.620
independent, or not.

00:03:02.620 --> 00:03:05.215
Now I'm going to make sure that
if there is a 1, if there

00:03:05.215 --> 00:03:08.040
is a leading 1 in any of my
rows, that everything else in

00:03:08.040 --> 00:03:09.290
that column is a 0.

00:03:11.670 --> 00:03:16.260
That form I'm doing is called
reduced row echelon form.

00:03:16.260 --> 00:03:18.000
Let me write that.

00:03:18.000 --> 00:03:26.520
Reduced row echelon form.

00:03:26.520 --> 00:03:30.450
If we call this augmented
matrix, matrix A, then I want

00:03:30.450 --> 00:03:35.240
to get it into the reduced row
echelon form of matrix A.

00:03:35.240 --> 00:03:38.410
And matrices, the convention
is, just like vectors, you

00:03:38.410 --> 00:03:41.740
make them nice and bold, but use
capital letters, instead

00:03:41.740 --> 00:03:42.770
of lowercase letters.

00:03:42.770 --> 00:03:45.550
We'll talk more about how
matrices relate to vectors in

00:03:45.550 --> 00:03:46.530
the future.

00:03:46.530 --> 00:03:49.810
Let's just solve this
system of equations.

00:03:49.810 --> 00:03:52.820
The first thing I want to do is,
in an ideal world I would

00:03:52.820 --> 00:03:55.530
get all of these guys
right here to be 0.

00:03:55.530 --> 00:04:02.090
Let me replace this guy with
that guy, with the first entry

00:04:02.090 --> 00:04:03.390
minus the second entry.

00:04:03.390 --> 00:04:04.640
Let me do that.

00:04:08.540 --> 00:04:10.030
The first row isn't
going to change.

00:04:10.030 --> 00:04:13.900
It's going to be 1, 2, 1, 1.

00:04:13.900 --> 00:04:16.560
And then I get a
7 right there.

00:04:16.560 --> 00:04:17.910
That's my first row.

00:04:17.910 --> 00:04:20.930
Now the second row, I'm going
to replace it with the first

00:04:20.930 --> 00:04:23.890
row minus the second row.

00:04:23.890 --> 00:04:25.770
So what do I get.

00:04:25.770 --> 00:04:28.840
1 minus 1 is 0.

00:04:28.840 --> 00:04:31.380
2 minus 2 is 0.

00:04:31.380 --> 00:04:35.120
1 minus 2 is minus 1.

00:04:35.120 --> 00:04:39.800
And then 1 minus minus 1 is 2.

00:04:39.800 --> 00:04:40.630
That's 1 plus 1.

00:04:40.630 --> 00:04:45.710
And then 7 minus
12 is minus 5.

00:04:45.710 --> 00:04:47.610
Now I want to get rid
of this row here.

00:04:47.610 --> 00:04:48.360
I don't want to get rid of it.

00:04:48.360 --> 00:04:49.940
I want to get rid of
this 2 right here.

00:04:49.940 --> 00:04:51.830
I want to turn it into a 0.

00:04:51.830 --> 00:04:56.350
Let's replace this row
with this row minus

00:04:56.350 --> 00:04:59.040
2 times that row.

00:04:59.040 --> 00:05:01.360
What I'm going to do is,
this row minus 2

00:05:01.360 --> 00:05:02.420
times the first row.

00:05:02.420 --> 00:05:04.820
I'm going to replace
this row with that.

00:05:04.820 --> 00:05:07.730
2 minus 2 times 1 is 0.

00:05:07.730 --> 00:05:09.050
That was the whole point.

00:05:09.050 --> 00:05:13.690
4 minus 2 times 2 is 0.

00:05:13.690 --> 00:05:19.310
0 minus 2 times 1 is minus 2.

00:05:19.310 --> 00:05:26.430
6 minus 2 times 1 is 6
minus 2, which is 4.

00:05:26.430 --> 00:05:36.835
4 minus 2 times 7, is 4 minus
14, which is minus 10.

00:05:36.835 --> 00:05:38.210
Now what can I do next.

00:05:42.230 --> 00:05:45.770
You can kind of see that this
row, well talk more about what

00:05:45.770 --> 00:05:46.590
this row means.

00:05:46.590 --> 00:05:48.860
When all of a sudden it's all
been zeroed out, there's

00:05:48.860 --> 00:05:49.730
nothing here.

00:05:49.730 --> 00:05:52.870
If I had non-zero term here,
then I'd want to zero this guy

00:05:52.870 --> 00:05:54.880
out, although it's already
zeroed out.

00:05:54.880 --> 00:05:56.850
I'm just going to move
over to this row.

00:05:56.850 --> 00:06:00.990
The first thing I want to do is
I want to make this leading

00:06:00.990 --> 00:06:02.930
coefficient here a 1.

00:06:02.930 --> 00:06:07.260
What I want to do is, I'm going
to multiply this entire

00:06:07.260 --> 00:06:09.840
row by minus 1.

00:06:09.840 --> 00:06:13.040
If I multiply this entire
row times minus 1.

00:06:13.040 --> 00:06:14.880
I don't even have to
rewrite the matrix.

00:06:14.880 --> 00:06:19.940
This becomes plus 1,
minus 2, plus 5.

00:06:19.940 --> 00:06:22.260
I think you can accept that.

00:06:22.260 --> 00:06:23.420
Now what can we do?

00:06:23.420 --> 00:06:27.290
Well, let's turn this
right here into a 0.

00:06:27.290 --> 00:06:32.820
Let me rewrite my augmented
matrix in the new form that I

00:06:32.820 --> 00:06:35.860
have. I'm going to keep the
middle row the same this time.

00:06:35.860 --> 00:06:42.940
My middle row is 0, 0, 1,
minus 2, and then it's

00:06:42.940 --> 00:06:45.710
augmented, and I
get a 5 there.

00:06:45.710 --> 00:06:50.330
What I want to do is I want to
eliminate this minus 2 here.

00:06:50.330 --> 00:06:54.420
Why don't I add this row
to 2 times that row.

00:06:54.420 --> 00:06:57.600
Then I would have minus 2, plus
2, and that'll work out.

00:06:57.600 --> 00:06:59.450
What do I get.

00:06:59.450 --> 00:07:01.900
Well, these are just
leading 0's.

00:07:01.900 --> 00:07:06.540
Then I have minus 2,
plus 2 times 1.

00:07:06.540 --> 00:07:08.680
That's just 0.

00:07:08.680 --> 00:07:17.830
4 plus 2 times minus
2, that is minus 4.

00:07:17.830 --> 00:07:22.310
That's 4 plus minus 4,
that's 0 as well.

00:07:22.310 --> 00:07:26.720
Then you have minus
10 plus 2 times 5.

00:07:26.720 --> 00:07:32.200
Well, that's just minus 10
plus 10, which is 0.

00:07:32.200 --> 00:07:34.970
That one just got zeroed out.

00:07:34.970 --> 00:07:37.400
Normally, when I just did
regular elimination, I was

00:07:37.400 --> 00:07:40.110
happy just having the situation
where I had these

00:07:40.110 --> 00:07:40.790
leading 1's.

00:07:40.790 --> 00:07:43.420
Everything below it were 0's.

00:07:43.420 --> 00:07:46.100
I wasn't too concerned about
what was above our 1's.

00:07:46.100 --> 00:07:47.320
What I want to do is,
I want to make

00:07:47.320 --> 00:07:49.730
those into a 0 as well.

00:07:49.730 --> 00:07:53.920
I want to make this
guy a 0 as well.

00:07:53.920 --> 00:07:59.170
What I can do is, I can replace
this first row with

00:07:59.170 --> 00:08:02.740
that first row minus
this second row.

00:08:02.740 --> 00:08:05.680
What is 1 minus 0?

00:08:05.680 --> 00:08:07.200
That's just 1.

00:08:07.200 --> 00:08:09.900
2 minus 0 is 2.

00:08:09.900 --> 00:08:13.770
1 minus 1 is 0.

00:08:13.770 --> 00:08:17.830
1 minus minus 2 is 3.

00:08:17.830 --> 00:08:22.540
7 minus 5 is 2.

00:08:22.540 --> 00:08:23.810
There you have it.

00:08:23.810 --> 00:08:27.970
We have our matrix in reduced
row echelon form.

00:08:27.970 --> 00:08:32.350
This is the reduced row echelon
form of our matrix,

00:08:32.350 --> 00:08:35.950
I'll write it in bold, of our
matrix A right there.

00:08:35.950 --> 00:08:39.620
You know it's in reduced row
echelon form because all of

00:08:39.620 --> 00:08:43.360
your leading 1's in each
row-- so what are my

00:08:43.360 --> 00:08:44.420
leading 1's in each row?

00:08:44.420 --> 00:08:46.880
I have this 1 and
I have that 1.

00:08:46.880 --> 00:08:51.020
They're the only non-zero
entry in their columns.

00:08:51.020 --> 00:08:53.760
These are called the
pivot entries.

00:08:53.760 --> 00:08:55.290
Let me label that for you.

00:08:55.290 --> 00:08:58.080
That's called a pivot entry.

00:08:58.080 --> 00:08:59.810
Pivot entry.

00:08:59.810 --> 00:09:01.700
They're the only non-zero
entry in

00:09:01.700 --> 00:09:03.320
their respective columns.

00:09:03.320 --> 00:09:07.080
If I have any zeroed out rows,
and I do have a zeroed out

00:09:07.080 --> 00:09:08.930
row, it's right there.

00:09:08.930 --> 00:09:12.660
This is zeroed out row.

00:09:12.660 --> 00:09:15.330
Just the style, or just the
convention, is that for

00:09:15.330 --> 00:09:20.200
reduced row echelon form, that
has to be your last row.

00:09:20.200 --> 00:09:24.250
We have the leading entries are
the only -- they're all 1.

00:09:24.250 --> 00:09:25.020
That's one case.

00:09:25.020 --> 00:09:25.980
You can't have this a 5.

00:09:25.980 --> 00:09:29.410
You'd want to divide that
equation by 5 if this was a 5.

00:09:29.410 --> 00:09:31.730
So your leading entries
in each row are a 1.

00:09:31.730 --> 00:09:35.760
That the leading entry in each
successive row is to the right

00:09:35.760 --> 00:09:38.120
of the leading entry of
the row before it.

00:09:38.120 --> 00:09:40.290
This guy right here is to
the right of that guy.

00:09:40.290 --> 00:09:43.400
This is just the style, the
convention, of reduced row

00:09:43.400 --> 00:09:44.790
echelon form.

00:09:44.790 --> 00:09:47.630
If you have any zeroed out rows,
it's in the last row.

00:09:47.630 --> 00:09:49.930
And finally, of course, and I
think I've said this multiple

00:09:49.930 --> 00:09:52.860
times, this is the only non-zero
entry in the row.

00:09:52.860 --> 00:09:54.430
What does this do for me?

00:09:54.430 --> 00:09:56.460
Now I can go back from
this world, back

00:09:56.460 --> 00:09:57.850
to my linear equations.

00:09:57.850 --> 00:10:00.860
We remember that these were the
coefficients on x1, these

00:10:00.860 --> 00:10:02.260
were the coefficients on x2.

00:10:02.260 --> 00:10:05.350
These were the coefficients on
x3, on x4, and then these were

00:10:05.350 --> 00:10:07.000
my constants out here.

00:10:07.000 --> 00:10:10.730
I can rewrite this system of
equations using my reduced row

00:10:10.730 --> 00:10:18.900
echelon form as x1,
x1 plus 2x2.

00:10:18.900 --> 00:10:20.500
There's no x3 there.

00:10:20.500 --> 00:10:26.000
So plus 3x4 is equal to 2.

00:10:26.000 --> 00:10:29.190
This equation, no x1,
no x2, I have an x3.

00:10:29.190 --> 00:10:38.310
I have x3 minus 2x4
is equal to 5.

00:10:38.310 --> 00:10:39.490
I have no other equation here.

00:10:39.490 --> 00:10:41.440
This one got completely
zeroed out.

00:10:41.440 --> 00:10:45.490
I was able to reduce this system
of equations to this

00:10:45.490 --> 00:10:47.280
system of equations.

00:10:47.280 --> 00:10:50.430
The variables that you associate
with your pivot

00:10:50.430 --> 00:10:54.160
entries, we call these
pivot variables.

00:10:54.160 --> 00:10:56.225
x1 and x3 are pivot variables.

00:11:00.700 --> 00:11:02.670
The variables that aren't
associated with the pivot

00:11:02.670 --> 00:11:05.810
entry, we call them
free variables.

00:11:05.810 --> 00:11:12.140
x2 and x4 are free variables.

00:11:12.140 --> 00:11:16.540
Now let's solve for, essentially
you can only solve

00:11:16.540 --> 00:11:17.620
for your pivot variables.

00:11:17.620 --> 00:11:20.370
The free variables we can
set to any variable.

00:11:20.370 --> 00:11:21.960
I said that in the beginning
of this equation.

00:11:21.960 --> 00:11:24.370
We have fewer equations
than unknowns.

00:11:24.370 --> 00:11:27.940
This is going to be a not well
constrained solution.

00:11:27.940 --> 00:11:31.170
You're not going to have just
one point in R4 that solves

00:11:31.170 --> 00:11:31.550
this equation.

00:11:31.550 --> 00:11:35.020
You're going to have
multiple points.

00:11:35.020 --> 00:11:37.860
Let's solve for our pivot
variables, because that's all

00:11:37.860 --> 00:11:39.360
we can solve for.

00:11:39.360 --> 00:11:45.170
This equation tells us, right
here, it tells us x3, let me

00:11:45.170 --> 00:11:59.960
do it in a good color, x3
is equal to 5 plus 2x4.

00:11:59.960 --> 00:12:12.050
Then we get x1 is equal to
2 minus x2, 2 minus 2x2.

00:12:12.050 --> 00:12:20.560
2 minus 2x2 plus, sorry,
minus 3x4.

00:12:20.560 --> 00:12:24.930
I just subtracted these from
both sides of the equation.

00:12:24.930 --> 00:12:27.630
This right here is essentially
as far as we can go to the

00:12:27.630 --> 00:12:30.180
solution of this system
of equations.

00:12:30.180 --> 00:12:33.730
I can pick, really, any values
for my free variables.

00:12:33.730 --> 00:12:37.155
I can pick any values for my
x2's and my x4's and I can

00:12:37.155 --> 00:12:38.540
solve for x3.

00:12:38.540 --> 00:12:40.560
What I want to do right now is
write this in a slightly

00:12:40.560 --> 00:12:43.510
different form so we can
visualize a little bit better.

00:12:43.510 --> 00:12:45.680
Of course, it's always hard to
visualize things in four

00:12:45.680 --> 00:12:46.530
dimensions.

00:12:46.530 --> 00:12:49.840
So we can visualize things a
little bit better, as to the

00:12:49.840 --> 00:12:51.810
set of this solution.

00:12:51.810 --> 00:12:53.680
Let's write it this way.

00:12:53.680 --> 00:12:57.260
If I were to write it in vector
form, our solution is

00:12:57.260 --> 00:13:03.070
the vector x1, x3, x3, x4.

00:13:03.070 --> 00:13:04.320
What is it equal to?

00:13:06.760 --> 00:13:10.270
Well it's equal to-- let
me write it like this.

00:13:10.270 --> 00:13:12.370
It's equal to-- I'm just
rewriting, I'm just

00:13:12.370 --> 00:13:15.700
essentially rewriting this
solution set in vector form.

00:13:15.700 --> 00:13:19.360
So x1 is equal to 2-- let
me write a little column

00:13:19.360 --> 00:13:23.250
there-- plus x2.

00:13:23.250 --> 00:13:24.000
Let me write it this way.

00:13:24.000 --> 00:13:32.540
Plus x2 times something plus
x4 times something.

00:13:37.130 --> 00:13:45.280
x1 is equal to 2 minus 2 times
x2, or plus x2 minus 2.

00:13:45.280 --> 00:13:47.450
I put a minus 2 there.

00:13:47.450 --> 00:13:50.820
I can say plus x4
times minus 3.

00:13:50.820 --> 00:13:53.430
I can put a minus 3 there.

00:13:53.430 --> 00:13:57.300
This right here, the first
entries of these vectors

00:13:57.300 --> 00:14:00.690
literally represent that
equation right there. x1 is

00:14:00.690 --> 00:14:06.590
equal to 2 plus x2 times minus
2 plus x4 times minus 3.

00:14:06.590 --> 00:14:08.505
What does x3 equal?

00:14:14.380 --> 00:14:16.600
x3 is equal to 5.

00:14:16.600 --> 00:14:18.090
Put that 5 right there.

00:14:18.090 --> 00:14:19.930
Plus x4 times 2.

00:14:23.680 --> 00:14:24.800
x2 doesn't apply to it.

00:14:24.800 --> 00:14:25.760
We can just put a 0.

00:14:25.760 --> 00:14:28.570
0 times x2 plus 2 times x4.

00:14:28.570 --> 00:14:30.430
Now what does x2 equal?

00:14:30.430 --> 00:14:35.000
You could say, x2 is equal
to 0 plus 1 times x2

00:14:35.000 --> 00:14:37.090
plus 0 times x4.

00:14:37.090 --> 00:14:38.430
x2 is just equal to x2.

00:14:38.430 --> 00:14:39.750
It's a free variable.

00:14:39.750 --> 00:14:41.600
Similarly, what does
x4 equal to?

00:14:41.600 --> 00:14:50.020
x4 is equal to 0 plus 0 times
x2 plus 1 times x4.

00:14:50.020 --> 00:14:51.770
What does this do for us?

00:14:51.770 --> 00:14:54.250
Well, all of a sudden here,
we've expressed our solution

00:14:54.250 --> 00:14:59.170
set as essentially the linear
combination of the linear

00:14:59.170 --> 00:15:00.360
combination of three vectors.

00:15:00.360 --> 00:15:02.040
This is a vector.

00:15:02.040 --> 00:15:05.290
You can view it as
a coordinate.

00:15:05.290 --> 00:15:07.590
Either a position vector.

00:15:07.590 --> 00:15:09.140
It is a vector in R4.

00:15:09.140 --> 00:15:12.440
You can view it as a position
vector or a coordinate in R4.

00:15:12.440 --> 00:15:16.230
You could say, look, our
solution set is essentially--

00:15:16.230 --> 00:15:17.290
this is in R4.

00:15:17.290 --> 00:15:18.770
Each of these have four
components, but you can

00:15:18.770 --> 00:15:21.640
imagine it in r3.

00:15:21.640 --> 00:15:24.130
That my solution set
is equal to some

00:15:24.130 --> 00:15:27.180
vector, some vector there.

00:15:27.180 --> 00:15:28.190
That's the vector.

00:15:28.190 --> 00:15:29.800
Think of it is as a
position vector.

00:15:29.800 --> 00:15:34.210
It would be the coordinate
2, 0, 5, 0.

00:15:34.210 --> 00:15:37.590
Which obviously, this is four
dimensions right there.

00:15:37.590 --> 00:15:42.160
It's equal to multiples
of these two vectors.

00:15:42.160 --> 00:15:44.380
Let's call this vector,
right here, let's

00:15:44.380 --> 00:15:46.650
call this vector a.

00:15:46.650 --> 00:15:51.390
Let's call this vector,
right here, vector b.

00:15:51.390 --> 00:15:55.650
Our solution set is all of this
point, which is right

00:15:55.650 --> 00:15:59.440
there, or I guess we could call
it that position vector.

00:15:59.440 --> 00:16:01.590
That position vector will
look like that.

00:16:01.590 --> 00:16:04.560
Where you're starting at the
origin right there, plus

00:16:04.560 --> 00:16:06.745
multiples of these two guys.

00:16:06.745 --> 00:16:11.390
If this is vector a, let's do
vector a in a different color.

00:16:11.390 --> 00:16:13.330
Vector a looks like that.

00:16:13.330 --> 00:16:17.260
Let's say vector a looks like
that, and then vector

00:16:17.260 --> 00:16:19.620
b looks like that.

00:16:19.620 --> 00:16:22.660
This is vector b, and
this is vector a.

00:16:22.660 --> 00:16:24.690
I don't know if this is going to
be easier or harder for you

00:16:24.690 --> 00:16:26.720
to visualize, because obviously
we are dealing in

00:16:26.720 --> 00:16:29.340
four dimensions right here, and
I'm just drawing on a two

00:16:29.340 --> 00:16:30.910
dimensional surface.

00:16:30.910 --> 00:16:34.260
What you can imagine is, is that
the solution set is equal

00:16:34.260 --> 00:16:38.040
to this fixed point, this
position vector, plus linear

00:16:38.040 --> 00:16:40.490
combinations of a and b.

00:16:40.490 --> 00:16:42.530
We're dealing, of
course, in R4.

00:16:42.530 --> 00:16:43.310
Let me write that down.

00:16:43.310 --> 00:16:45.030
We're dealing in R4.

00:16:45.030 --> 00:16:47.770
But linear combinations
of a and b are

00:16:47.770 --> 00:16:50.930
going to create a plane.

00:16:50.930 --> 00:16:54.850
You can multiply a times 2,
and b times 3, or a times

00:16:54.850 --> 00:16:57.120
minus 1, and b times
minus 100.

00:16:57.120 --> 00:16:59.450
You can keep adding and
subtracting these linear

00:16:59.450 --> 00:17:01.230
combinations of a and b.

00:17:01.230 --> 00:17:05.960
They're going to construct
a plane that contains the

00:17:05.960 --> 00:17:10.710
position vector, or contains
the point 2, 0, 5, 0.

00:17:10.710 --> 00:17:16.760
The solution for these three
equations with four unknowns,

00:17:16.760 --> 00:17:18.660
is a plane in R4.

00:17:23.990 --> 00:17:26.140
I know that's really hard to
visualize, and maybe I'll do

00:17:26.140 --> 00:17:28.010
another one in three
dimensions.

00:17:28.010 --> 00:17:31.890
Hopefully this at least gives
you a decent understanding of

00:17:31.890 --> 00:17:35.570
what an augmented matrix is,
what reduced row echelon form

00:17:35.570 --> 00:17:38.470
is, and what are the valid
operations I can perform on a

00:17:38.470 --> 00:17:42.320
matrix without messing
up the system.


Matrices: Reduced row echelon form 2

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.840
I figure it never hurts getting
as much practice as

00:00:02.840 --> 00:00:07.010
possible solving systems of
linear equations, so let's

00:00:07.010 --> 00:00:07.770
solve this one.

00:00:07.770 --> 00:00:09.930
What I'm going to do is I'm
going to solve it using an

00:00:09.930 --> 00:00:13.310
augmented matrix, and I'm going
to put it in reduced row

00:00:13.310 --> 00:00:14.130
echelon form.

00:00:14.130 --> 00:00:17.940
So what's the augmented matrix
for this system of equations?

00:00:17.940 --> 00:00:19.405
Three unknowns with
three equations.

00:00:22.720 --> 00:00:24.990
I just have to do
the coefficents.

00:00:24.990 --> 00:00:29.010
So the coefficients of x
terms are just 1, 1, 1.

00:00:29.010 --> 00:00:33.590
Coefficients of the y terms
are 1, 2, and 3.

00:00:33.590 --> 00:00:35.920
Coefficients of the z terms
are 1, 3, and 4.

00:00:39.340 --> 00:00:41.280
And let me show that
it's augmented.

00:00:41.280 --> 00:00:47.030
And then they equal
3, 0, and minus 2.

00:00:47.030 --> 00:00:50.330
Now, I want to get this
augmented matrix into reduced

00:00:50.330 --> 00:00:51.570
row echelon form.

00:00:51.570 --> 00:00:53.840
So the first thing, I have
a leading 1 here

00:00:53.840 --> 00:00:55.240
that's a pivot entry.

00:00:55.240 --> 00:00:59.550
Let me make everything else in
that column equal to a 0.

00:00:59.550 --> 00:01:01.320
So I'm not going to change
my first row.

00:01:01.320 --> 00:01:05.740
So it will just be a 1, a 1, a
1, and then my dividing line,

00:01:05.740 --> 00:01:08.450
and then I have a 3.

00:01:08.450 --> 00:01:13.770
Now, to zero this out, let me
just replace the second row

00:01:13.770 --> 00:01:16.670
with the first row minus
the second row.

00:01:16.670 --> 00:01:19.580
So 1 minus 1 is 0.

00:01:19.580 --> 00:01:25.170
1 minus 2 is-- actually, a
better thing to do, because I

00:01:25.170 --> 00:01:28.670
eventually want this to be 1
anyway, let me replace this

00:01:28.670 --> 00:01:32.230
row with this row, with the
second row minus the first row

00:01:32.230 --> 00:01:33.620
instead of the first row
minus the second row.

00:01:33.620 --> 00:01:34.910
I can do it either way.

00:01:34.910 --> 00:01:37.220
So the second row minus
the first row.

00:01:37.220 --> 00:01:39.760
So 1 minus 1 is 0.

00:01:39.760 --> 00:01:43.230
2 minus 1 is 1.

00:01:43.230 --> 00:01:46.510
3 minus 1 is 2.

00:01:46.510 --> 00:01:50.930
And then 0 minus 3 is minus 3.

00:01:50.930 --> 00:01:53.550
Now I want to also
zero this out.

00:01:53.550 --> 00:01:56.220
So let me replace this
guy with this

00:01:56.220 --> 00:01:58.200
equation minus that equation.

00:01:58.200 --> 00:02:01.170
So 1 minus 1 is 0.

00:02:01.170 --> 00:02:04.800
3 minus 1 is 2.

00:02:04.800 --> 00:02:08.910
4 minus 1 is 3.

00:02:08.910 --> 00:02:12.860
Minus 2 minus 3 is minus 5.

00:02:12.860 --> 00:02:13.490
Fair enough.

00:02:13.490 --> 00:02:14.900
So I got my pivot entry here.

00:02:14.900 --> 00:02:16.200
I have another pivot
entry here.

00:02:16.200 --> 00:02:18.710
It's to the right of this one,
which is what I want for

00:02:18.710 --> 00:02:20.460
reduced row echelon form.

00:02:20.460 --> 00:02:23.710
Now, I need to target this
entry and that entry.

00:02:23.710 --> 00:02:25.960
I need to zero them out.

00:02:25.960 --> 00:02:28.500
So let's do it.

00:02:28.500 --> 00:02:30.800
So I'm going to keep my
second row the same.

00:02:30.800 --> 00:02:35.720
My second row is 0, 1, 2, and
then I have a minus 3, the

00:02:35.720 --> 00:02:38.650
augmented part of it.

00:02:38.650 --> 00:02:42.420
And to zero this guy out, what
I can do is I can replace the

00:02:42.420 --> 00:02:46.640
first row with the first row
minus the second row.

00:02:46.640 --> 00:02:51.770
So I get 1 minus 0 is 1.

00:02:51.770 --> 00:02:54.700
1 minus 1-- there's
a bird outside.

00:02:54.700 --> 00:02:56.720
Let me close my window.

00:02:56.720 --> 00:02:58.280
So where was I?

00:02:58.280 --> 00:03:01.410
I'm replacing the first row with
the first row minus the

00:03:01.410 --> 00:03:02.050
second row.

00:03:02.050 --> 00:03:04.800
So 1 minus 0 is 1.

00:03:04.800 --> 00:03:07.240
1 minus 1 is 0.

00:03:11.140 --> 00:03:14.780
1 minus 2 is minus 1.

00:03:14.780 --> 00:03:21.130
And then 3 minus minus 3, so
that's equal to 3 plus 3, so

00:03:21.130 --> 00:03:23.340
that's equal to 6.

00:03:23.340 --> 00:03:24.680
1 minus 0 is 1.

00:03:24.680 --> 00:03:27.510
1 minus 1 is 0, negative 1.

00:03:27.510 --> 00:03:29.930
And then 3 minus negative
3, that's 6.

00:03:29.930 --> 00:03:32.390
I always want to make sure I
don't make a careless mistake.

00:03:32.390 --> 00:03:35.230
Now, let me get rid of this
entry right here.

00:03:35.230 --> 00:03:36.590
Let me zero that out.

00:03:36.590 --> 00:03:41.780
So let me replace the third row
with the third row minus 2

00:03:41.780 --> 00:03:43.880
times the second row.

00:03:43.880 --> 00:03:45.980
So we have 0 minus--
well, 2 times 0.

00:03:45.980 --> 00:03:47.760
That's just going to be 0.

00:03:47.760 --> 00:03:52.790
2 minus 2 times 1, that's
2 minus 2, that's 0.

00:03:52.790 --> 00:03:58.980
3 minus 2 times 2, that's
3 minus 4, or minus 1.

00:03:58.980 --> 00:04:04.130
And then finally, minus 5
minus 2 times minus 3.

00:04:04.130 --> 00:04:05.380
Let me write that down.

00:04:10.120 --> 00:04:15.390
That's minus 5, minus minus 6.

00:04:15.390 --> 00:04:18.089
That's minus 5 plus
6 is equal to 1.

00:04:18.089 --> 00:04:19.680
I really wanted to make
sure I didn't make a

00:04:19.680 --> 00:04:21.220
careless mistake there.

00:04:21.220 --> 00:04:24.110
So that is equal to 1.

00:04:24.110 --> 00:04:26.540
So I'm almost done, but I'm
still not in reduced row

00:04:26.540 --> 00:04:27.760
echelon form.

00:04:27.760 --> 00:04:30.260
This has to be a positive
1 in order to get there.

00:04:30.260 --> 00:04:31.670
It can't be anything
other than a 1.

00:04:31.670 --> 00:04:33.980
That's just the style of reduced
row echelon form.

00:04:33.980 --> 00:04:36.390
And then these guys up here
have to be zeroed out.

00:04:36.390 --> 00:04:38.990
Well, the easy thing to do,
let me just multiply this

00:04:38.990 --> 00:04:41.130
equation by minus 1.

00:04:41.130 --> 00:04:44.490
So then this becomes a plus 1
and that becomes a minus 1.

00:04:44.490 --> 00:04:48.120
And then I just need to zero
out these two guys up here.

00:04:48.120 --> 00:04:49.810
So let's do it.

00:04:49.810 --> 00:04:53.980
So my equation, I'm going to
keep my third row the same.

00:04:53.980 --> 00:04:59.410
My third row is now
0, 0, 1, minus 1.

00:05:03.530 --> 00:05:07.330
And now I want to zero
this guy out.

00:05:07.330 --> 00:05:10.420
So what I can do is I could set
my first row equal to my

00:05:10.420 --> 00:05:13.970
first row plus my last row,
because if these two add up,

00:05:13.970 --> 00:05:15.730
they're going to
be equal to 0.

00:05:15.730 --> 00:05:16.580
So what do I get?

00:05:16.580 --> 00:05:19.590
1 plus 0 is 1.

00:05:19.590 --> 00:05:22.680
0 plus 0 is 0.

00:05:22.680 --> 00:05:27.050
Minus 1 plus 1 is 0.

00:05:27.050 --> 00:05:31.390
6 plus minus 1 is 5.

00:05:31.390 --> 00:05:34.300
Now, I want to zero
this row out.

00:05:34.300 --> 00:05:37.560
And to zero this row out, what
I can do is I'll replace it

00:05:37.560 --> 00:05:42.490
with the second row minus
2 times the first row.

00:05:42.490 --> 00:05:46.580
So 0 minus 2 times
0 is just 0.

00:05:46.580 --> 00:05:50.990
1 minus 2 times 0 is just 1.

00:05:50.990 --> 00:05:55.320
2 minus 2 times 1 is 0.

00:05:55.320 --> 00:05:58.180
Minus 3 minus 2 times
negative 1.

00:05:58.180 --> 00:05:59.040
Let me write that down.

00:05:59.040 --> 00:06:03.060
Minus 3 minus 2 times minus 1.

00:06:03.060 --> 00:06:04.960
I don't want to make
a careless mistake.

00:06:04.960 --> 00:06:06.070
So what is that equal to?

00:06:06.070 --> 00:06:14.390
This is equal to minus 3 minus
minus 2, or minus 3 plus 2,

00:06:14.390 --> 00:06:15.790
which is equal to minus 1.

00:06:15.790 --> 00:06:17.830
So that's equal to minus 1.

00:06:17.830 --> 00:06:21.220
And now I have my augmented
matrix in

00:06:21.220 --> 00:06:22.990
reduced row echelon form.

00:06:30.070 --> 00:06:35.290
My pivot entries are the only
entries in their columns.

00:06:35.290 --> 00:06:38.690
Each pivot entry in each
successive row is to the right

00:06:38.690 --> 00:06:40.190
of the pivot entry before it.

00:06:40.190 --> 00:06:42.040
And actually, I have
no free variables.

00:06:42.040 --> 00:06:44.000
Every column has
a pivot entry.

00:06:44.000 --> 00:06:46.780
So let's go back from the
augmented matrix world and

00:06:46.780 --> 00:06:49.550
kind of put back our
variables there.

00:06:49.550 --> 00:06:51.440
So what do we get?

00:06:51.440 --> 00:06:59.980
We get x plus 0y plus
0z is equal to 5.

00:06:59.980 --> 00:07:02.050
That's that row right there.

00:07:02.050 --> 00:07:10.230
We get 0x plus 1y plus 0z
is equal to minus 1.

00:07:10.230 --> 00:07:12.260
That's that row right there.

00:07:12.260 --> 00:07:18.000
And then finally, we have
0x plus 0y plus 1z is

00:07:18.000 --> 00:07:19.740
equal to minus 1.

00:07:19.740 --> 00:07:21.590
That's that row right there.

00:07:21.590 --> 00:07:24.510
And just like that, we've
actually solved our system of

00:07:24.510 --> 00:07:26.600
three equations with
three unknowns.

00:07:26.600 --> 00:07:27.570
That's the solution
right there.

00:07:27.570 --> 00:07:29.570
I just wrote it in this way
just so you can see the

00:07:29.570 --> 00:07:31.610
corresponding, but obviously,
I could have written them

00:07:31.610 --> 00:07:32.875
closer to their equal sign.

00:07:32.875 --> 00:07:36.260
So hopefully, you found
that vaguely useful.


Integrals: Trig Substitution 3 (long problem)

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:03.270
Let's say we have the
indefinite integral of the

00:00:03.270 --> 00:00:06.890
square root of 6x minus
x squared minus 5.

00:00:06.890 --> 00:00:09.290
And obviously this is not
some simple integral.

00:00:09.290 --> 00:00:11.350
I don't have just, you know,
this expression and its

00:00:11.350 --> 00:00:14.540
derivative lying around, so
u-substitution won't work.

00:00:14.540 --> 00:00:17.410
And so you can guess from just
the title of this video that

00:00:17.410 --> 00:00:19.060
we're going to have to
do something fancier.

00:00:19.060 --> 00:00:21.950
And we'll probably have to do
some type of trig substitution.

00:00:21.950 --> 00:00:25.360
But this immediately doesn't
look kind of amenable

00:00:25.360 --> 00:00:26.250
to trig substitution.

00:00:26.250 --> 00:00:30.190
I like to do trig substitution
when I see kind of a 1 minus x

00:00:30.190 --> 00:00:33.280
squared under a radical sign,
or maybe an x squared minus

00:00:33.280 --> 00:00:37.590
1 under a radical sign, or
maybe a x squared plus 1.

00:00:37.590 --> 00:00:40.030
These are the type of things
that get my brain thinking in

00:00:40.030 --> 00:00:41.340
terms of trig substitution.

00:00:41.340 --> 00:00:43.650
but that doesn't quite
look like that just yet.

00:00:43.650 --> 00:00:44.600
I have a radical sign.

00:00:44.600 --> 00:00:47.455
I have some x squared, but it
doesn't look like this form.

00:00:47.455 --> 00:00:50.810
So let's if we can get
it to be in this form.

00:00:50.810 --> 00:00:55.000
Let me delete these guys
right there real fast.

00:00:55.000 --> 00:01:00.280
So let's see if we can maybe
complete the square down here.

00:01:00.280 --> 00:01:00.920
So let's see.

00:01:00.920 --> 00:01:04.029
If this is equal to,
let me rewrite this.

00:01:04.029 --> 00:01:06.110
And if this completing the
square doesn't look familiar

00:01:06.110 --> 00:01:08.610
to you, I have a whole
bunch of videos on that.

00:01:08.610 --> 00:01:15.740
Let me rewrite this as equal to
minus 5 minus-- I need more

00:01:15.740 --> 00:01:20.530
space up here-- minus
5 minus x squared.

00:01:20.530 --> 00:01:22.520
Now there's a plus 6x, but
I have a minus out here.

00:01:22.520 --> 00:01:24.070
So minus 6 x, right?

00:01:24.070 --> 00:01:26.730
A minus and a minus
will becomes plus 6x.

00:01:26.730 --> 00:01:30.090
And then, I want to make
this into a perfect square.

00:01:30.090 --> 00:01:34.430
So what number when I add it
to itself will be minus 6?

00:01:34.430 --> 00:01:36.550
Well, it's minus 3
and minus 3 squared.

00:01:36.550 --> 00:01:38.290
So you take half of this
number, you get minus

00:01:38.290 --> 00:01:39.760
3, and you square it.

00:01:39.760 --> 00:01:42.200
Then you put a 9 there.

00:01:42.200 --> 00:01:44.220
Now, I can't just
arbitrarily add nines.

00:01:44.220 --> 00:01:45.610
Or actually, I didn't
add a 9 here.

00:01:45.610 --> 00:01:46.050
What did I do?

00:01:46.050 --> 00:01:47.080
I subtracted a 9.

00:01:47.080 --> 00:01:50.050
Because I threw a 9 there, but
it's really a minus 9, because

00:01:50.050 --> 00:01:52.060
of this minus sign out there.

00:01:52.060 --> 00:01:55.030
So in order to make this
neutral to my 9 that I

00:01:55.030 --> 00:01:57.180
just threw in there,
this is a minus 9.

00:01:57.180 --> 00:01:58.210
I have to add a 9.

00:01:58.210 --> 00:01:59.240
So let me add a 9.

00:01:59.240 --> 00:02:02.280
So plus 9, right there.

00:02:02.280 --> 00:02:04.920
If this doesn't make complete
sense, what I just did, and

00:02:04.920 --> 00:02:08.350
obviously you have the dx right
there, multiply this out.

00:02:08.350 --> 00:02:11.560
You get minus x squared plus
6x, which are these two terms

00:02:11.560 --> 00:02:14.155
right there, minus 9, and then
you'll have this plus 9, and

00:02:14.155 --> 00:02:16.790
these two will cancel out, and
you'll just get exactly back

00:02:16.790 --> 00:02:17.750
to what we had before.

00:02:17.750 --> 00:02:20.110
Because I want you to realize,
I didn't change the equation.

00:02:20.110 --> 00:02:22.000
This is a minus 9
because of this.

00:02:22.000 --> 00:02:24.570
So I added a 9, so I
really added 0 to it.

00:02:24.570 --> 00:02:28.750
But what this does, it gets
it into a form that I like.

00:02:28.750 --> 00:02:32.020
Obviously this, right here,
just becomes a 4, and then this

00:02:32.020 --> 00:02:34.110
term right here becomes, what?

00:02:34.110 --> 00:02:38.290
That is x minus 3 squared.

00:02:38.290 --> 00:02:39.680
x minus 3 squared.

00:02:39.680 --> 00:02:43.880
So my indefinite integral now
becomes the integral, I'm just

00:02:43.880 --> 00:02:48.290
doing a little bit of algebra,
the integral of the square root

00:02:48.290 --> 00:02:57.990
of 4 minus x minus
3 squared dx.

00:02:57.990 --> 00:03:00.160
Now this is starting to look
like a form that I like, but

00:03:00.160 --> 00:03:01.510
I like to have a 1 here.

00:03:01.510 --> 00:03:03.220
So let's factor a 4 out.

00:03:03.220 --> 00:03:07.410
So this is equal to, I'll
switch colors, that's equal to

00:03:07.410 --> 00:03:12.720
the integral of the radical,
and we'll have the 4, times 1

00:03:12.720 --> 00:03:18.520
minus x minus 3 squared over 4.

00:03:18.520 --> 00:03:21.100
I just took a 4 out of
both of these terms.

00:03:21.100 --> 00:03:23.270
If I multiply this out, I'll
just get back to that,

00:03:23.270 --> 00:03:26.030
right there, dx.

00:03:26.030 --> 00:03:28.790
And now this is starting to
look like a form that I like.

00:03:28.790 --> 00:03:30.150
Let me simplify it even more.

00:03:30.150 --> 00:03:34.630
So this is equal to the
integral, if I take the 4 out,

00:03:34.630 --> 00:03:39.790
it becomes 2 times the square
root of 1 minus, and I can

00:03:39.790 --> 00:03:45.800
rewrite this as x minus 3,
let me write this way.

00:03:45.800 --> 00:03:51.310
1 minus x minus 3
over 2 squared dx.

00:03:51.310 --> 00:03:52.680
And where did I
get that 2 from?

00:03:52.680 --> 00:03:55.010
Well, if I just square both
of these, I get x minus 3

00:03:55.010 --> 00:03:56.330
squared over 2 squared.

00:03:56.330 --> 00:03:58.740
Which is x minus 3 over 4.

00:03:58.740 --> 00:04:01.630
So I have done no
calculus so far.

00:04:01.630 --> 00:04:05.370
I just algebraically rewrote
this indefinite integral as

00:04:05.370 --> 00:04:06.420
this indefinite integral.

00:04:06.420 --> 00:04:07.690
They are equivalent.

00:04:07.690 --> 00:04:10.070
But this, all of a sudden,
looks like a form

00:04:10.070 --> 00:04:11.290
that I recognize.

00:04:11.290 --> 00:04:14.580
I showed you in the last video
that cosine squared of theta

00:04:14.580 --> 00:04:17.290
is just equal to 1 minus
sine squared of theta.

00:04:17.290 --> 00:04:18.490
You could actually do
it the other way.

00:04:18.490 --> 00:04:20.260
You could do sine squared
is equal to 1 minus

00:04:20.260 --> 00:04:20.890
cosine squared.

00:04:20.890 --> 00:04:21.970
No difference.

00:04:21.970 --> 00:04:23.360
But they both will work out.

00:04:23.360 --> 00:04:27.120
But this looks an
awful lot like this.

00:04:27.120 --> 00:04:30.200
In fact, it would look exactly
like this if I say that that is

00:04:30.200 --> 00:04:33.670
equal to sine squared of theta.

00:04:33.670 --> 00:04:35.170
So let me make that
substitution.

00:04:35.170 --> 00:04:46.180
Let me write that x minus
3 over 2 squared is equal

00:04:46.180 --> 00:04:51.620
to sine squared of theta.

00:04:51.620 --> 00:04:53.840
Now, if we take the square root
of both sides of that equation,

00:04:53.840 --> 00:05:01.230
I get x minus 3 over 2 is
equal to sine of theta.

00:05:01.230 --> 00:05:03.410
Now we're eventually, you know
where this is going to go.

00:05:03.410 --> 00:05:07.130
We're eventually going to have
to substitute back for theta.

00:05:07.130 --> 00:05:09.050
So let's solve for
theta in terms of x.

00:05:09.050 --> 00:05:11.980
So theta in terms of x, we
could just say, just take the

00:05:11.980 --> 00:05:13.530
arc sine of both sides of this.

00:05:13.530 --> 00:05:16.900
You get theta is equal to,
right, the arc sine of

00:05:16.900 --> 00:05:18.030
the sine is just theta.

00:05:18.030 --> 00:05:28.380
Theta is equal to the arc
sine of x minus 3 over 2.

00:05:28.380 --> 00:05:29.340
Fair enough.

00:05:29.340 --> 00:05:30.910
Now, to actually do the
substitution, though, we're

00:05:30.910 --> 00:05:32.960
going to have figure out what
dx is, we're going to have to

00:05:32.960 --> 00:05:35.290
solve for x in terms of theta.

00:05:35.290 --> 00:05:36.360
So let me do that.

00:05:36.360 --> 00:05:40.160
So we get, if we multiply both
sides of the equation by 2, we

00:05:40.160 --> 00:05:46.340
get x minus 3 is equal to 2
times the sine of theta,

00:05:46.340 --> 00:05:51.195
or that x is equal to 2
sine of theta plus 3.

00:05:51.195 --> 00:05:54.010
Now, if we take the derivative
of both sides with respect to

00:05:54.010 --> 00:06:00.030
theta, we get dx d theta, is
equal to 2 cosine of theta,

00:06:00.030 --> 00:06:01.790
derivative of this is just 0.

00:06:01.790 --> 00:06:05.460
Or we can multiply both sides
by d theta, and we get dx is

00:06:05.460 --> 00:06:09.530
equal to 2 cosine
of theta d theta.

00:06:09.530 --> 00:06:12.060
And we're ready to substitute
back into our original

00:06:12.060 --> 00:06:13.950
indefinite integral.

00:06:13.950 --> 00:06:19.050
So this thing will now be
rewritten as the integral of 2

00:06:19.050 --> 00:06:25.680
times the radical, if I can get
some space, of 1 minus,

00:06:25.680 --> 00:06:28.050
I'm replacing this with
sine squared of theta.

00:06:31.660 --> 00:06:32.930
And all that times dx.

00:06:32.930 --> 00:06:36.360
Well, I just said that dx is
equal to this, right here.

00:06:36.360 --> 00:06:43.020
So dx is equal to 2
cosine of theta d theta.

00:06:43.020 --> 00:06:45.460
What does this simplify to?

00:06:45.460 --> 00:06:50.730
This action right here, this is
just cosine squared of theta.

00:06:50.730 --> 00:06:51.880
And we're going to take
the square root of

00:06:51.880 --> 00:06:53.390
cosine squared of theta.

00:06:53.390 --> 00:06:56.950
So this, the square root of
cosine squared of data, this

00:06:56.950 --> 00:06:59.960
whole term right here, right?

00:06:59.960 --> 00:07:04.010
That becomes the square root of
cosine squared of theta, which

00:07:04.010 --> 00:07:06.780
is just the same thing
as cosine of theta.

00:07:06.780 --> 00:07:12.900
So our integral becomes, so our
integral is equal to, 2 times

00:07:12.900 --> 00:07:16.120
the square root of cosine
squared of theta, so that's

00:07:16.120 --> 00:07:22.720
just 2 times cosine of theta,
times 2 times cosine of theta.

00:07:22.720 --> 00:07:24.490
That's that one, right there.

00:07:24.490 --> 00:07:30.180
This is this, and all of
this radical sine, that's

00:07:30.180 --> 00:07:32.040
this, right there.

00:07:32.040 --> 00:07:34.370
1 minus sine squared was cosine
squared, take the radical,

00:07:34.370 --> 00:07:35.800
you get cosine squared.

00:07:35.800 --> 00:07:40.060
And then everything
times d theta.

00:07:40.060 --> 00:07:46.390
Now this is obviously equal
to 4 times cosine squared

00:07:46.390 --> 00:07:48.890
of theta d theta.

00:07:48.890 --> 00:07:52.310
Which, by itself, is still not
an easy integral to solve.

00:07:52.310 --> 00:07:55.160
I don't know, you know, I
can't do U-substitution or

00:07:55.160 --> 00:07:56.240
anything like that, there.

00:07:56.240 --> 00:07:57.480
So what do we do?

00:07:57.480 --> 00:08:02.700
Well, we resort to our
good old trig identities.

00:08:02.700 --> 00:08:05.110
Now, I don't know if you have
this one memorized, although it

00:08:05.110 --> 00:08:08.140
tends to be in the inside cover
of most calculus books,

00:08:08.140 --> 00:08:10.840
or inside cover of
most trig books.

00:08:10.840 --> 00:08:19.040
But cosine squared of theta
can be rewritten as 1/2 times

00:08:19.040 --> 00:08:22.330
1 plus cosine of 2 theta.

00:08:22.330 --> 00:08:24.390
And I've proven this
in multiple videos.

00:08:24.390 --> 00:08:26.330
So let's just make
this substitution.

00:08:26.330 --> 00:08:28.890
Let me just replace this
thing with that thing.

00:08:28.890 --> 00:08:35.010
So this integral becomes,
it equals, 4 times cosine

00:08:35.010 --> 00:08:37.920
squared of theta, but cosine
squared of theta is this.

00:08:37.920 --> 00:08:44.725
4 times 1/2 times 1 plus
cosine of 2 theta d theta.

00:08:48.520 --> 00:08:50.910
Now, this looks
easier to deal with.

00:08:50.910 --> 00:08:51.350
So what is it?

00:08:51.350 --> 00:08:53.810
4 times 1/2, that's 2.

00:08:53.810 --> 00:09:00.250
So my integral becomes the
integral of 2 times 1, so it's

00:09:00.250 --> 00:09:09.200
2, plus 2 times 2 cosine of 2
theta, all of that d theta.

00:09:09.200 --> 00:09:12.030
Now, this antiderivative is
pretty straightforward.

00:09:12.030 --> 00:09:13.240
What is this, right here?

00:09:13.240 --> 00:09:20.150
This is the derivative
with respect of theta of

00:09:20.150 --> 00:09:23.580
sine of 2 theta, right?

00:09:23.580 --> 00:09:24.250
This whole thing.

00:09:24.250 --> 00:09:25.340
What's the derivative
sine of 2 theta?

00:09:25.340 --> 00:09:27.900
Take the derivative of the
inside, that's 2, times the

00:09:27.900 --> 00:09:30.900
derivative of the outside,
cosine of 2 theta.

00:09:30.900 --> 00:09:34.110
And this, of course, is the
derivative of 2 theta.

00:09:34.110 --> 00:09:36.650
So this is equal to, the
antiderivative of 2 with

00:09:36.650 --> 00:09:41.420
respect to theta, is just 2
theta, plus the antiderivative

00:09:41.420 --> 00:09:51.740
of this, which is just sine of
2 theta, and then

00:09:51.740 --> 00:09:54.080
we have a plus c.

00:09:54.080 --> 00:09:57.330
And of course, we can't forget
that we defined theta, our

00:09:57.330 --> 00:09:59.830
original antiderivative
was in terms of x.

00:09:59.830 --> 00:10:01.230
So we can't just leave
it in terms of theta.

00:10:01.230 --> 00:10:02.840
We're going to have to
do a back substitution.

00:10:02.840 --> 00:10:05.020
So let's just remember,
theta was equal to arc

00:10:05.020 --> 00:10:07.080
sine of x minus 3 over 2.

00:10:07.080 --> 00:10:08.620
Let me write that
on the side here.

00:10:17.750 --> 00:10:20.230
Now, if I immediately
substitute this theta straight

00:10:20.230 --> 00:10:23.430
into this, I'm going to get a
sine of 2 times arc sine of x

00:10:23.430 --> 00:10:25.710
minus 3 over 2, which
would be correct.

00:10:25.710 --> 00:10:28.990
And I would have a 2 times arc
sine of x minus 3 over 2.

00:10:28.990 --> 00:10:30.880
That would all be fine,
and we would be done.

00:10:30.880 --> 00:10:32.060
But that's not satisfying.

00:10:32.060 --> 00:10:33.480
It's not a nice clean answer.

00:10:33.480 --> 00:10:37.090
So let's see if we can simplify
this, so it's only in

00:10:37.090 --> 00:10:39.100
terms of sine of theta.

00:10:39.100 --> 00:10:43.240
So when you take the sine of
the arc sine, then it just

00:10:43.240 --> 00:10:46.150
simplifies to x minus 3 over 2.

00:10:46.150 --> 00:10:49.210
Let me make that clear.

00:10:49.210 --> 00:10:51.730
So if I can write all of this
in terms of sines of theta,

00:10:51.730 --> 00:10:57.990
because the sine of theta is
equal to the sine of the arc

00:10:57.990 --> 00:11:06.230
sine of x minus 3 over 2.

00:11:06.230 --> 00:11:09.180
Which is just equal
to x minus 3 over 2.

00:11:09.180 --> 00:11:12.250
So if I can write this in terms
of sines of theta, then I can

00:11:12.250 --> 00:11:13.390
just make this substitution.

00:11:13.390 --> 00:11:15.520
Sine of theta equals
that, and everything

00:11:15.520 --> 00:11:17.700
simplifies a good bit.

00:11:17.700 --> 00:11:19.610
So let's see if we can do that.

00:11:19.610 --> 00:11:25.140
Well, you may or may not know
the other identity, and I've

00:11:25.140 --> 00:11:30.200
proven this as well, that sine
of 2 theta, that's the same

00:11:30.200 --> 00:11:42.485
thing as sine of theta plus
theta, which is equal to sine

00:11:42.485 --> 00:11:48.940
of theta cosine of theta plus,
the thetas get swapped around,

00:11:48.940 --> 00:11:53.320
sine of theta plus cosine of
theta, which is equal to, this

00:11:53.320 --> 00:11:56.740
is just the same thing written
twice, 2 sine of theta

00:11:56.740 --> 00:11:57.760
cosine of theta.

00:11:57.760 --> 00:12:00.540
Some people have this memorized
ahead of time, and if you have

00:12:00.540 --> 00:12:02.600
to take an exam on trig
substitution, it doesn't hurt

00:12:02.600 --> 00:12:04.310
to have this memorized
ahead of time.

00:12:04.310 --> 00:12:06.180
But let's rewrite
this like this.

00:12:06.180 --> 00:12:09.320
So our indefinite integral
in terms of theta, or our

00:12:09.320 --> 00:12:12.870
antiderivative, became 2 theta,
plus, instead of sine of 2

00:12:12.870 --> 00:12:18.490
theta, we could write, 2 sine
of theta cosine of theta, and

00:12:18.490 --> 00:12:20.500
of course we have a plus c.

00:12:20.500 --> 00:12:22.380
Now, I want to write everything
in terms of sines of theta, but

00:12:22.380 --> 00:12:23.960
I have a cosine if theta there.

00:12:23.960 --> 00:12:25.370
So what can we do?

00:12:25.370 --> 00:12:30.440
Well, we know we know that
cosine squared of theta is

00:12:30.440 --> 00:12:35.410
equal to 1 minus sine squared
of theta, or that cosine of

00:12:35.410 --> 00:12:39.530
theta is equal to the square
root of 1 minus sine

00:12:39.530 --> 00:12:40.990
squared of theta.

00:12:40.990 --> 00:12:43.800
Which seems like we're adding
complexity to it, but the neat

00:12:43.800 --> 00:12:45.800
thing is, it's in terms
of sine of theta.

00:12:45.800 --> 00:12:46.480
So let's do that.

00:12:46.480 --> 00:12:48.160
Let's make the substitution.

00:12:48.160 --> 00:12:53.150
So our antiderivative, this is
the same thing as 2 data plus 2

00:12:53.150 --> 00:12:57.920
sine of theta times cosine of
theta, which is equal to this,

00:12:57.920 --> 00:13:03.400
times the square root of 1
minus sine squared of theta,

00:13:03.400 --> 00:13:05.510
and all of that plus c.

00:13:05.510 --> 00:13:08.030
Now we're in the home stretch.

00:13:08.030 --> 00:13:09.930
This problem was probably
harder than you thought

00:13:09.930 --> 00:13:10.980
it was going to be.

00:13:10.980 --> 00:13:15.430
We know that sine theta is
equal to x minus 3 over 2.

00:13:15.430 --> 00:13:18.190
So let's make that
reverse substitution.

00:13:18.190 --> 00:13:20.690
So we have 2 times theta.

00:13:20.690 --> 00:13:24.290
This first term is just 2
times theta, right there.

00:13:24.290 --> 00:13:28.400
So that's 2 times, we can't
escape the arc sine.

00:13:28.400 --> 00:13:30.170
If we just have a theta, we
have to say that theta is

00:13:30.170 --> 00:13:37.270
equal to arc sine of
x minus 3 over 2.

00:13:37.270 --> 00:13:40.280
And then we have plus,
let me switch colors,

00:13:40.280 --> 00:13:42.500
plus 2 sine of theta.

00:13:42.500 --> 00:13:46.710
Well, that's plus 2 times sine
of theta is x minus 3 over 2.

00:13:46.710 --> 00:13:51.970
So 2 times x minus 3 over 2,
and then all of that times

00:13:51.970 --> 00:13:59.460
the square root of 1 minus
sine of theta squared.

00:13:59.460 --> 00:14:00.630
What's sine of theta?

00:14:00.630 --> 00:14:05.706
It's x minus 3 over 2 squared.

00:14:05.706 --> 00:14:07.510
And of course we have a plus c.

00:14:07.510 --> 00:14:10.040
Let's see if we can simplify
this even more, now that

00:14:10.040 --> 00:14:11.350
we're at the home stretch.

00:14:11.350 --> 00:14:19.410
So this is equal to 2 arc sine
of x minus 3 over 2, plus these

00:14:19.410 --> 00:14:24.170
two terms, this 2 and this 2
cancel out, plus x minus 3,

00:14:24.170 --> 00:14:29.280
times the square root of, what
happens if we multiply

00:14:29.280 --> 00:14:35.090
everything here by, let's see,
if we take a, so this is equal

00:14:35.090 --> 00:14:45.180
to 1 minus x minus 3 over 4. x
minus 3 squared over 4.

00:14:45.180 --> 00:14:47.780
This simplification, I should
write that in quotes, is taking

00:14:47.780 --> 00:14:49.280
me longer than I thought.

00:14:49.280 --> 00:14:51.850
But let's see if we can
simplify this even more.

00:14:51.850 --> 00:14:54.760
If we multiply, let me just
focus on this term right here,

00:14:54.760 --> 00:14:59.180
if we multiply the outside,
or say, let's multiply

00:14:59.180 --> 00:15:01.920
and divide this by 2.

00:15:01.920 --> 00:15:04.460
So I'll write this as--
so let's just multiply

00:15:04.460 --> 00:15:08.490
this times 2 over 2.

00:15:08.490 --> 00:15:10.910
You might say, Sal, why
are you doing that?

00:15:10.910 --> 00:15:13.940
Because I can rewrite this, let
me write my whole thing here.

00:15:13.940 --> 00:15:21.000
So I have 2 arc sine of x
minus 3 over 2, and then

00:15:21.000 --> 00:15:23.530
I have, I could take this
denominator 2 right here.

00:15:23.530 --> 00:15:26.830
So I say, plus x
minus 3 over 2.

00:15:26.830 --> 00:15:29.130
That 2 is that 2 right there.

00:15:29.130 --> 00:15:32.010
And then I could write
this 2 right here as

00:15:32.010 --> 00:15:33.470
a square root of 4.

00:15:33.470 --> 00:15:37.330
Times the square root
of 4, times the square

00:15:37.330 --> 00:15:39.100
root of all of this.

00:15:39.100 --> 00:15:44.300
1 minus x minus 3
squared over 4.

00:15:44.300 --> 00:15:45.350
I think you see
where I'm going.

00:15:45.350 --> 00:15:47.090
I'm kind of reversing
everything that I did at the

00:15:47.090 --> 00:15:48.260
beginning of this problem.

00:15:48.260 --> 00:15:50.210
And maybe I'm getting a little
fixated on making this as

00:15:50.210 --> 00:15:53.820
simple as possible, but I'm
so close, so let me finish.

00:15:53.820 --> 00:15:59.390
So I get 2 times the arc sine
of x minus 3 over 2, which I'm

00:15:59.390 --> 00:16:04.410
tired of writing, plus x minus
3 over 2, and if we bring this

00:16:04.410 --> 00:16:06.780
4 in, right, the square root of
4 times the square root of that

00:16:06.780 --> 00:16:11.380
is equal to the square root
of 4 times these things.

00:16:11.380 --> 00:16:16.700
So it's 4 minus x minus
3 squared, all of

00:16:16.700 --> 00:16:18.280
that that plus c.

00:16:18.280 --> 00:16:19.790
And we're at the home stretch.

00:16:19.790 --> 00:16:27.740
This is equal to 2 times the
arc sine of x minus 3 over 2

00:16:27.740 --> 00:16:34.650
plus x minus 3 over 2 times the
radical of 4 minus, let's

00:16:34.650 --> 00:16:41.260
expand this, x squared
minus 6x plus 9.

00:16:41.260 --> 00:16:47.960
And then this expression right
here simplifies to minus minus,

00:16:47.960 --> 00:16:53.270
it's 6x minus x squared, and
then you have a 4

00:16:53.270 --> 00:16:55.050
minus 9 minus 5.

00:16:55.050 --> 00:16:56.180
Which is our original
antiderivative.

00:16:58.940 --> 00:17:01.000
So finally, we're
at the very end.

00:17:01.000 --> 00:17:08.020
We get the antiderivative is 2
arc sine of x minus 3 over 2

00:17:08.020 --> 00:17:16.430
plus x minus 3 over 2 times,
time the radical of 6x

00:17:16.430 --> 00:17:19.340
minus x squared minus 5.

00:17:19.340 --> 00:17:24.840
That right there is the
antiderivative of what this

00:17:24.840 --> 00:17:28.170
thing that we had at the very
top of my little chalkboard,

00:17:28.170 --> 00:17:29.820
which is right there.

00:17:29.820 --> 00:17:33.620
So that is equal to the
antiderivative of the

00:17:33.620 --> 00:17:39.050
square root of 6x minus
x squared minus 5 dx.

00:17:39.050 --> 00:17:42.120
And I can imagine that you're
probably as tired as I am.

00:17:42.120 --> 00:17:43.550
My hand actually hurts.

00:17:43.550 --> 00:17:45.880
But hopefully you find that
to be vaguely satisfying.

00:17:45.880 --> 00:17:47.960
Sometimes I get complaints
that I only do easy problems.

00:17:47.960 --> 00:17:51.630
Well, this was quite a hairy
and not-so-easy problem.


Integrals: Trig Substitution 2

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:05.290
Let's say we have the
indefinite integral of 1

00:00:05.290 --> 00:00:15.300
over 36 plus x squared d x.

00:00:15.300 --> 00:00:17.910
Now, as you can imagine, this
is not an easy integral to

00:00:17.910 --> 00:00:19.830
solve without trigonometry.

00:00:19.830 --> 00:00:22.130
I can't do u substitution, I
don't have the derivative of

00:00:22.130 --> 00:00:23.480
this thing sitting someplace.

00:00:23.480 --> 00:00:25.340
This would be easy if I
had a 2x sitting there.

00:00:25.340 --> 00:00:27.910
Than I would say, oh the
derivative of this is 2x,

00:00:27.910 --> 00:00:30.470
I could do u substitution
and I'd be set.

00:00:30.470 --> 00:00:32.880
But there is no 2x there,
so how do I do it?

00:00:32.880 --> 00:00:35.640
Well, I resort to our
trigonometric identities.

00:00:35.640 --> 00:00:38.100
Let's see what trig
identity we can get here.

00:00:38.100 --> 00:00:40.840
The first thing I always do,
this is just the way my brain

00:00:40.840 --> 00:00:43.920
works, I always like it-- I can
see this is a constant plus

00:00:43.920 --> 00:00:46.350
something squared, which
tells me I should use a

00:00:46.350 --> 00:00:47.300
trigonometric identity.

00:00:47.300 --> 00:00:50.600
But I always like it in terms
of 1 plus something squared.

00:00:50.600 --> 00:00:54.210
I'm just going to rewrite my
integral as being equal to,

00:00:54.210 --> 00:00:55.720
let me write the dx
in the numerator.

00:00:55.720 --> 00:00:57.870
This is just times dx.

00:00:57.870 --> 00:00:59.430
Let me write a nicer
integral than that.

00:00:59.430 --> 00:01:07.370
This is equal to the integral
of d x over 36 times 1

00:01:07.370 --> 00:01:11.800
plus x squared over 36.

00:01:11.800 --> 00:01:14.130
1 plus x squared over 36,
that's another way to

00:01:14.130 --> 00:01:15.420
write my integral.

00:01:15.420 --> 00:01:19.110
Let's see if any of our trig
identities can somehow be

00:01:19.110 --> 00:01:22.400
substituted in here for
that that would somehow

00:01:22.400 --> 00:01:24.560
simplify the problem.

00:01:24.560 --> 00:01:28.190
So the one that springs to
mind, and if you don't know

00:01:28.190 --> 00:01:30.430
this already, I'll write it
down right here, is 1 plus

00:01:30.430 --> 00:01:31.630
tangent squared of theta.

00:01:35.390 --> 00:01:36.900
Let's prove this one.

00:01:36.900 --> 00:01:39.790
Tangent squared of theta, this
is equal to 1 plus just the

00:01:39.790 --> 00:01:45.210
definition of tangent sine
squared of theta over

00:01:45.210 --> 00:01:47.160
cosine squared of theta.

00:01:47.160 --> 00:01:50.160
Now 1 is just cosine squared
over cosine squared.

00:01:50.160 --> 00:01:57.250
So I can rewrite this as equal
to cosine squared of theta over

00:01:57.250 --> 00:02:02.990
cosine squared of theta, that's
1, plus sine squared theta over

00:02:02.990 --> 00:02:04.860
cosine squared of theta,
now that we have a

00:02:04.860 --> 00:02:06.320
common denominator.

00:02:06.320 --> 00:02:08.490
Now what's cosine squared
plus sine squared?

00:02:08.490 --> 00:02:09.970
Definition of the unit circle.

00:02:09.970 --> 00:02:14.210
That equals 1 over cosine
squared of theta.

00:02:14.210 --> 00:02:17.600
Or we could say that that
equals 1 over cosine squared.

00:02:17.600 --> 00:02:19.540
One over cosine is secant.

00:02:19.540 --> 00:02:24.090
So this is equal to the
secant squared of theta.

00:02:24.090 --> 00:02:28.270
If we make the substitution, if
we say let's make this thing

00:02:28.270 --> 00:02:32.340
right here equal to tangent of
theta, or tangent

00:02:32.340 --> 00:02:33.590
squared of theta.

00:02:33.590 --> 00:02:37.310
Then this expression will be 1
plus tangent squared of theta.

00:02:37.310 --> 00:02:38.700
Which is equal to
secant squared.

00:02:38.700 --> 00:02:42.860
Maybe that'll help simplify
this equation a bit.

00:02:42.860 --> 00:02:50.080
We're going to say that x
squared over 36 is equal to

00:02:50.080 --> 00:02:52.710
tangent squared of theta.

00:02:52.710 --> 00:02:55.440
Let's take the square root of
both sides of this equation and

00:02:55.440 --> 00:03:04.090
you get x over 6 is equal to
the tangent of theta, or that x

00:03:04.090 --> 00:03:08.540
is equal to 6 tangent of theta.

00:03:08.540 --> 00:03:10.950
If we take the derivative of
both sides of this with respect

00:03:10.950 --> 00:03:16.140
to theta we get d x d theta is
equal to-- what's the

00:03:16.140 --> 00:03:18.610
derivative of the
tangent of theta?

00:03:18.610 --> 00:03:21.120
I could show it to you just
by going from these basic

00:03:21.120 --> 00:03:23.070
principles right here.

00:03:23.070 --> 00:03:26.670
Actually let me do it
for you just in case.

00:03:26.670 --> 00:03:29.000
So the derivative of tangent
theta-- never hurts to do it

00:03:29.000 --> 00:03:31.140
on the side, let me
do it right here.

00:03:31.140 --> 00:03:34.470
It's going to be 6 times the
derivative with respect to

00:03:34.470 --> 00:03:36.490
theta of tangent of theta.

00:03:36.490 --> 00:03:39.050
Which we need to figure,
so let's figure it out.

00:03:39.050 --> 00:03:43.040
The derivative of tangent of
theta, that's the same thing

00:03:43.040 --> 00:03:48.030
as d d theta of sine of
theta over cosine of theta.

00:03:48.030 --> 00:03:50.270
That's just the
derivative of tangent.

00:03:50.270 --> 00:03:54.380
Or this is just the same thing
as the derivative with respect

00:03:54.380 --> 00:03:57.520
to theta, let me scroll to
the right a little bit.

00:03:57.520 --> 00:03:59.650
Because I never remember the
quotient rule, I've told you in

00:03:59.650 --> 00:04:04.090
the past that it's somewhat
lame, of sine of theta times

00:04:04.090 --> 00:04:09.560
cosine of theta to
the minus 1 power.

00:04:09.560 --> 00:04:10.860
What is this equal to?

00:04:10.860 --> 00:04:14.090
We could say it's equal to,
well the derivative of the

00:04:14.090 --> 00:04:17.520
first expression or the first
function we could say, which

00:04:17.520 --> 00:04:19.050
is just cosine of theta.

00:04:19.050 --> 00:04:21.760
This is equal to cosine of
theta, that's just the

00:04:21.760 --> 00:04:25.050
derivative of sine of theta
times our second expression.

00:04:25.050 --> 00:04:29.660
Times cosine of theta
to the minus 1.

00:04:29.660 --> 00:04:32.810
I've put these parentheses, and
put the minus 1 out there

00:04:32.810 --> 00:04:34.610
because I didn't want to put
the minus 1 here and make you

00:04:34.610 --> 00:04:37.590
think that I'm talking about an
inverse cosine or an arccosine.

00:04:37.590 --> 00:04:41.680
So that's the derivative of
sine times cosine and now I

00:04:41.680 --> 00:04:45.690
want to take plus the
derivative of cosine.

00:04:48.720 --> 00:04:50.770
Not just cosine, the derivative
if cosine to the minus 1.

00:04:50.770 --> 00:04:57.710
So that is minus 1 times cosine
to the minus 2 power of theta.

00:04:57.710 --> 00:05:01.466
That's the derivative of
the outside times the

00:05:01.466 --> 00:05:03.110
derivative of the inside.

00:05:03.110 --> 00:05:05.060
Let me scroll over more.

00:05:05.060 --> 00:05:06.620
So that's the derivative
of the outside.

00:05:06.620 --> 00:05:09.160
If the cosine theta was just an
x, you would say x to the minus

00:05:09.160 --> 00:05:12.360
1 derivative is minus
1 x to the minus 2.

00:05:12.360 --> 00:05:14.560
Now times the derivative
of the inside.

00:05:14.560 --> 00:05:16.230
Of cosine of theta with
respect to theta.

00:05:16.230 --> 00:05:21.180
So that's times minus
sine of theta.

00:05:21.180 --> 00:05:26.350
I'm going to multiply all of
that times sine of theta.

00:05:26.350 --> 00:05:28.690
The derivative of this thing,
which is the stuff in green,

00:05:28.690 --> 00:05:30.550
times the first expression.

00:05:30.550 --> 00:05:32.520
So what does this equal?

00:05:32.520 --> 00:05:34.890
These cosine of theta
divided by cosine of

00:05:34.890 --> 00:05:36.960
theta, that is equal to 1.

00:05:36.960 --> 00:05:40.400
And then I have a minus 1 and
I have a minus sine of theta.

00:05:40.400 --> 00:05:42.670
That's plus plus.

00:05:42.670 --> 00:05:43.190
What do I have?

00:05:43.190 --> 00:05:45.910
I have sine squared, sine of
theta time sine of theta

00:05:45.910 --> 00:05:47.980
over cosine squared.

00:05:47.980 --> 00:05:54.430
So plus sine squares of theta
over cosine squared of theta.

00:05:54.430 --> 00:05:58.570
Which is equal to 1 plus
tangent squared of theta.

00:05:58.570 --> 00:06:00.290
What's 1 plus tangent
squared of theta?

00:06:00.290 --> 00:06:00.980
I just showed you that.

00:06:00.980 --> 00:06:04.720
That's equal to secant
squared of theta.

00:06:04.720 --> 00:06:07.100
So the derivative of tangent
of theta is equal to

00:06:07.100 --> 00:06:09.410
secant squared of theta.

00:06:09.410 --> 00:06:11.940
All that work to get us
fairly something-- it's nice

00:06:11.940 --> 00:06:13.490
when it comes out simple.

00:06:13.490 --> 00:06:17.440
So d x d theta, this is
just equal to secant

00:06:17.440 --> 00:06:19.780
squared of theta.

00:06:19.780 --> 00:06:23.440
If we want to figure out what d
x is equal to, d x is equal to

00:06:23.440 --> 00:06:25.510
just both sides times d theta.

00:06:25.510 --> 00:06:32.680
So it's 6 times secant
squared theta d theta.

00:06:32.680 --> 00:06:34.290
That's our d x.

00:06:34.290 --> 00:06:37.330
Of course, in the future
we're going to have to back

00:06:37.330 --> 00:06:39.780
substitute, so we want
to solve for theta.

00:06:39.780 --> 00:06:41.000
That's fairly straightforward.

00:06:41.000 --> 00:06:43.600
Just take the arctangent of
both sides of this equation.

00:06:43.600 --> 00:06:50.270
You get that the arctangent of
x over 6 is equal to the theta.

00:06:50.270 --> 00:06:52.600
We'll save this for later.

00:06:52.600 --> 00:06:54.640
So what is our
integral reduced to?

00:06:54.640 --> 00:06:58.020
Our integral now becomes
the integral of d x?

00:06:58.020 --> 00:06:59.210
What's d x?

00:06:59.210 --> 00:07:06.430
It is 6 secant squared
theta d theta.

00:07:06.430 --> 00:07:11.860
All of that over this
denominator, which is 36

00:07:11.860 --> 00:07:19.380
times 1 plus tangent
squared of theta.

00:07:19.380 --> 00:07:23.600
We know that this right there
is secant squared of theta.

00:07:23.600 --> 00:07:25.420
I've shown you that
multiple times.

00:07:25.420 --> 00:07:27.470
So this is secant squared of
theta in the denominator.

00:07:27.470 --> 00:07:31.370
We have a secant squared on the
numerator, they cancel out.

00:07:31.370 --> 00:07:32.990
So those cancel out.

00:07:32.990 --> 00:07:37.400
So are integral reduces to,
lucky for us, 6/36 which

00:07:37.400 --> 00:07:40.910
is just 1/6 d theta.

00:07:40.910 --> 00:07:45.760
Which is equal to
1/6 theta plus c.

00:07:45.760 --> 00:07:48.650
Now we back substitute
using this result.

00:07:48.650 --> 00:07:52.240
Theta is equal to
arctangent x over 6.

00:07:52.240 --> 00:07:55.710
The anti-derivative 1 over
36 plus x squared is

00:07:55.710 --> 00:07:58.220
equal to 1/6 times theta.

00:07:58.220 --> 00:08:06.210
Theta's just equal to the
arctangent x over 6 plus c.

00:08:06.210 --> 00:08:07.000
And we're done.

00:08:07.000 --> 00:08:09.770
So that one wasn't too bad.


Integrals: Trig Substitution 1

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:01.120
Let's

00:00:01.120 --> 00:00:06.600
say I have the indefinite
integral 1 over the square

00:00:06.600 --> 00:00:11.840
root of 3 minus 2x squared.

00:00:11.840 --> 00:00:13.730
Of course I have a dx there.

00:00:13.730 --> 00:00:16.600
So right when I look at
that, there's no obvious

00:00:16.600 --> 00:00:19.240
traditional method of
taking this antiderivative.

00:00:19.240 --> 00:00:22.510
I don't have the derivative of
this sitting someplace else in

00:00:22.510 --> 00:00:25.830
the integral, so I can't do
traditional u-substitution.

00:00:25.830 --> 00:00:29.560
But what I can do is I could
say, well, this almost looks

00:00:29.560 --> 00:00:32.680
like some trig identities that
I'm familiar with, so maybe I

00:00:32.680 --> 00:00:35.610
can substitute with
trig functions.

00:00:35.610 --> 00:00:37.740
So let's see if I can find
a trig identity that

00:00:37.740 --> 00:00:39.280
looks similar to this.

00:00:39.280 --> 00:00:41.830
Well, our most basic
trigonometric identity-- this

00:00:41.830 --> 00:00:45.100
comes from the unit circle
definition-- is that the sine

00:00:45.100 --> 00:00:50.080
squared of theta plus
the cosine squared of

00:00:50.080 --> 00:00:53.040
theta is equal to 1.

00:00:53.040 --> 00:00:56.380
And then if we subtract cosine
squared of theta from both

00:00:56.380 --> 00:00:59.510
sides, we get-- or if we
subtract sine squared of theta

00:00:59.510 --> 00:01:03.220
from both sides, we could do
either-- we could get cosine

00:01:03.220 --> 00:01:08.310
squared of theta is equal to 1
minus sine squared of theta.

00:01:08.310 --> 00:01:09.460
We could do either way.

00:01:09.460 --> 00:01:14.790
But this, all of a sudden, this
thing right here, starts to

00:01:14.790 --> 00:01:16.230
look a little bit like this.

00:01:16.230 --> 00:01:18.790
Maybe I can do a little bit of
algebraic manipulation to make

00:01:18.790 --> 00:01:20.580
this look a lot like that.

00:01:20.580 --> 00:01:22.540
So the first thing, I would
like to have a 1 here-- at

00:01:22.540 --> 00:01:25.090
least, that's how my brain
works-- so let's factor out a

00:01:25.090 --> 00:01:27.420
3 out of this denominator.

00:01:27.420 --> 00:01:33.630
So this is the same thing as
the integral of 1 over the

00:01:33.630 --> 00:01:37.920
square root of-- let me factor
out a 3 out of this expression.

00:01:37.920 --> 00:01:44.690
3 times 1 minus 2/3x squared.

00:01:44.690 --> 00:01:45.900
I did nothing fancy here.

00:01:45.900 --> 00:01:47.890
I just factored the 3
out of this expression,

00:01:47.890 --> 00:01:49.250
that's all I did.

00:01:49.250 --> 00:01:53.130
But the neat thing now is,
this expression looks a

00:01:53.130 --> 00:01:54.650
lot like that expression.

00:01:54.650 --> 00:01:58.310
In fact, if I substitute, if I
say that this thing right here,

00:01:58.310 --> 00:02:02.490
this 2/3x squared, if I set it
equal to sine squared theta, I

00:02:02.490 --> 00:02:04.250
will be able to use
this identity.

00:02:04.250 --> 00:02:05.450
So let's do that.

00:02:05.450 --> 00:02:12.130
Let's set 2/3x squared, let's
set that equal to sine

00:02:12.130 --> 00:02:15.590
squared of theta.

00:02:15.590 --> 00:02:18.210
So if we take the square root
of both sides of this equation,

00:02:18.210 --> 00:02:24.730
I get the square root of 2 over
the square root of 3 times x is

00:02:24.730 --> 00:02:28.910
equal to the sine of theta.

00:02:28.910 --> 00:02:31.600
If I want to solve for
x, what do I get?

00:02:31.600 --> 00:02:32.870
And, well, we're going to
have to solve for both

00:02:32.870 --> 00:02:34.600
x and for theta, so
let's do it both ways.

00:02:34.600 --> 00:02:37.390
First, let's solve for theta.

00:02:37.390 --> 00:02:42.430
If we solve for theta, you get
that theta is equal to the

00:02:42.430 --> 00:02:47.500
arcsine, or the inverse sine,
of square root of 2 over

00:02:47.500 --> 00:02:49.820
square root of 3x.

00:02:49.820 --> 00:02:51.300
That's if you solve for theta.

00:02:51.300 --> 00:02:54.380
Now, if you solve for x, you
just multiply both sides of

00:02:54.380 --> 00:02:58.940
this equation times the inverse
of this and you get x is equal

00:02:58.940 --> 00:03:01.650
to-- divide both sides of the
equation by this or multiply it

00:03:01.650 --> 00:03:05.180
by the inverse-- is equal to
the square root of 3 over the

00:03:05.180 --> 00:03:10.230
square root of 2 times
the sine of theta.

00:03:10.230 --> 00:03:13.060
And we were going to substitute
this with sine squared of

00:03:13.060 --> 00:03:15.410
theta, but we can't leave
this dx out there.

00:03:15.410 --> 00:03:18.000
We have to take the integral
with respect to d theta.

00:03:18.000 --> 00:03:20.530
So what's dx with
respect to d theta?

00:03:20.530 --> 00:03:25.560
So the derivative of x with
respect to theta is equal to

00:03:25.560 --> 00:03:28.690
square root of 3 over
square root of 2.

00:03:28.690 --> 00:03:32.170
Derivative of this with respect
to theta is just cosine of

00:03:32.170 --> 00:03:35.340
theta, and if we want to write
this in terms of dx, we could

00:03:35.340 --> 00:03:40.490
just write that dx is equal to
square root of 3 over the

00:03:40.490 --> 00:03:46.370
square root of 2 cosine
of theta d theta.

00:03:46.370 --> 00:03:47.870
Now we're ready to substitute.

00:03:47.870 --> 00:03:52.660
So we can rewrite this
expression up here-- I'll do it

00:03:52.660 --> 00:03:56.360
in this reddish color-- I was
using that, let me do

00:03:56.360 --> 00:03:57.770
it in the blue color.

00:03:57.770 --> 00:03:59.650
We can rewrite this
expression up here now.

00:03:59.650 --> 00:04:03.610
It's an indefinite integral
of-- dx is on the

00:04:03.610 --> 00:04:04.600
numerator, right?

00:04:04.600 --> 00:04:06.330
Instead of writing this 1
times dx, I could have

00:04:06.330 --> 00:04:08.050
just written a dx up here.

00:04:08.050 --> 00:04:09.750
That could be a dx
just like that.

00:04:09.750 --> 00:04:11.800
You're just multiplying
it times dx.

00:04:11.800 --> 00:04:12.990
So what's dx?

00:04:12.990 --> 00:04:14.880
dx is this business.

00:04:14.880 --> 00:04:16.070
I'll do it in yellow.

00:04:16.070 --> 00:04:17.950
dx is this right here.

00:04:17.950 --> 00:04:21.560
So it's the square root of
3 over the square root of

00:04:21.560 --> 00:04:26.250
2 cosine theta d theta.

00:04:26.250 --> 00:04:28.160
That's what dx was.

00:04:28.160 --> 00:04:33.250
Now, the denominator in my
equation, I have the square

00:04:33.250 --> 00:04:39.580
root of 3 times--
now it's 1 minus.

00:04:39.580 --> 00:04:43.053
Now I said 2/3x squared is
equal to sine squared of theta.

00:04:47.640 --> 00:04:49.750
Now how can I simplify this?

00:04:49.750 --> 00:04:52.960
Well, what's 1 minus
sine squared of theta?

00:04:52.960 --> 00:04:55.480
That's cosine squared of theta.

00:04:55.480 --> 00:04:59.670
So this thing right here is
cosine squared of theta.

00:04:59.670 --> 00:05:05.010
So my indefinite integral
becomes the square root of 3

00:05:05.010 --> 00:05:11.030
over the square root of 2
cosine theta d theta, all of

00:05:11.030 --> 00:05:18.340
that over the square root of
3 times the cosine

00:05:18.340 --> 00:05:19.570
squared of theta.

00:05:19.570 --> 00:05:21.900
That just became cosine
squared of theta.

00:05:21.900 --> 00:05:23.890
So let's just take the square
root of this bottom part.

00:05:23.890 --> 00:05:27.380
So this is going to be equal
to-- I'll do an arbitrary

00:05:27.380 --> 00:05:32.670
change of colors-- square root
of 3 over the square root of 2

00:05:32.670 --> 00:05:37.890
cosine of theta d theta, all of
that over-- what's the

00:05:37.890 --> 00:05:38.420
square root of this?

00:05:38.420 --> 00:05:41.230
It's equal to the square root
of 3 times the square root

00:05:41.230 --> 00:05:45.730
of cosine squared, so
times cosine of theta.

00:05:45.730 --> 00:05:47.200
Now, this simplifies
things a good bit.

00:05:47.200 --> 00:05:50.110
I have a cosine of theta
divided by a cosine of theta,

00:05:50.110 --> 00:05:53.150
those cancel out, so we'll just
get 1, and then I have a square

00:05:53.150 --> 00:05:56.150
root of 3 up here divided by a
square root of 3, so those two

00:05:56.150 --> 00:05:59.860
guys are going to cancel out,
so my integral simplifies

00:05:59.860 --> 00:06:05.730
nicely to 1 over square
root of 2 d theta.

00:06:05.730 --> 00:06:08.530
Or even better, I could write
this-- this is just a constant

00:06:08.530 --> 00:06:12.460
term, I could take it out of my
integral-- it equals 1 over the

00:06:12.460 --> 00:06:17.140
square root of 2 times my
integral of just d theta.

00:06:17.140 --> 00:06:18.800
And this is super easy.

00:06:18.800 --> 00:06:22.570
This is equal to 1 over
the square root of 2

00:06:22.570 --> 00:06:25.590
times theta plus c.

00:06:25.590 --> 00:06:26.900
Plus some constant.

00:06:26.900 --> 00:06:28.760
I mean, you could say that the
integral of this is theta plus

00:06:28.760 --> 00:06:30.830
c and then you'd multiply the
constant times this, but it's

00:06:30.830 --> 00:06:32.950
still going to be some
arbitrary constant.

00:06:32.950 --> 00:06:35.320
I think you know how to take
the antiderivative of this.

00:06:35.320 --> 00:06:36.630
But are we done?

00:06:36.630 --> 00:06:37.210
Well, no.

00:06:37.210 --> 00:06:40.420
We want to know our indefinite
integral in terms of x.

00:06:40.420 --> 00:06:42.280
So now we have to
reverse substitute.

00:06:42.280 --> 00:06:43.960
So what is theta?

00:06:43.960 --> 00:06:46.900
We figured that out here. theta
is equal to arcsine the square

00:06:46.900 --> 00:06:48.810
root of 2 over the
square root of 3x.

00:06:48.810 --> 00:06:53.840
So our original indefinite
integral, which was all of this

00:06:53.840 --> 00:06:58.180
silliness up here, now that I
reverse substitute for theta or

00:06:58.180 --> 00:07:02.140
put x back in there, it's 1
over the square root

00:07:02.140 --> 00:07:03.890
of 2 times theta.

00:07:03.890 --> 00:07:12.980
theta is just this, is just
arcsine of square root of 2

00:07:12.980 --> 00:07:17.530
over square root of 3 x, and
then I have this constant

00:07:17.530 --> 00:07:19.600
out here, plus c.

00:07:19.600 --> 00:07:26.230
So this right here is
the antiderivative of

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1 over the square root
of 3 minus 2x squared.

00:07:29.800 --> 00:07:30.840
So hopefully you
found that helpful.

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I'm going to do a couple of
more videos where we go through

00:07:32.690 --> 00:07:34.350
a bunch of these examples,
just so that you get

00:07:34.350 --> 00:07:36.220
familiar with them.