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PROFESSOR: Today we're going
to continue our discussion
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of methods of integration.
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The method that I'm going to
describe today handles a
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whole class of functions
of the following form.
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You take P (x) / Q (x)
and this is known as
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a rational function.
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And all that means is that it's
a ratio off two polynomials,
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which are these functions
P ( x) and Q ( x).
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We'll handle all such functions
by a method which is known
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as partial fractions.
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And what this does is, it
splits P / Q into what you
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could call easier pieces.
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So that's going to be
some kind of algebra.
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And that's what we're going
to spend most of our
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time doing today.
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I'll start with an example.
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And all of my examples
will be illustrating
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more general methods.
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The example is to integrate
the function 1 / x - 1
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+, say, 3 / x + 2 dx.
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That's easy to do.
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It's just, we already
know the answer.
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It's ln x - 1 + 3 ln x + 3.
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Plus a constant.
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So that's done.
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So now, here's the difficulty
that is going to arise.
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The difficulty is that I can
start with this function, which
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is perfectly manageable.
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And than I can add these
two functions together.
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The way I add fractions.
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So that's getting a
common denominator.
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00:03:00 --> 00:03:06
And so that gives me x
+ 2 here + 3 ( x - 1).
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00:03:06 --> 00:03:11
And now if I combine together
all of these terms, then
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altogether I have 4x
+ 2 - 3, that's - 1.
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And if I multiply out the
denominator that's x ^2
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+ that 2 turned into a
3, that's interesting.
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Hope there aren't too many more
of those transformations.
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Is there another one here?
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STUDENT: [INAUDIBLE]
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PROFESSOR: Oh, it
happened earlier on.
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Wow that's an interesting
vibration there.
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OK.
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Thank you.
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So, I guess my 3's were
speaking to my 2's.
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Somewhere in my past.
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OK, anyway, I think
this is now correct.
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So the problem is
the following.
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This is the problem with this.
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This integral was easy.
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I'm calling it easy, we
already know how to do it.
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Over here.
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But now over here,
it's disguised.
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It's the same function,
but it's no longer clear
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how to integrate it.
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If you're faced with this
one, you say what am
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I supposed to do.
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And we have to get
around that difficulty.
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And so what we're going
to do is we're going to
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unwind this disguise.
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So we have the algebra
problem that we have.
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Oh, wow.
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There must be something
in the water.
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Impressive.
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Wow.
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OK, let's see.
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Is 2/3 = 3/2?
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Holy cow.
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Well that's good.
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Well, I'll keep you awake
today with several other
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transpositions here.
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So our algebra problem
is to detect the easy
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pieces which are inside.
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And the method that we're going
to use, the one that we'll
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emphasize anyway, is one
algebraic trick which is a
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shortcut, which is called
the cover-up method.
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But we're going to talk
about even more general
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things than that.
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But anyway, this is
where we're headed.
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Is something called
the cover-up method.
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Alright.
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So that's our intro.
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And I'll just have to
remember that 2 is not 3.
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I'll keep on repeating that.
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So now here I'm going to
describe to you how we
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unwind this disguise.
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The first step is, we write
down the function we
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want to integrate.
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Which was this.
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And now we have to undo the
first damage that we did.
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So the first step is to
factor the denominator.
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And that factors, we happen to
know the factors, so I'm not
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going to carry this out.
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But this can be a
rather difficult step.
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But we're going to
assume that it's done.
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For the purposes of
illustration here.
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So I factor the denominator.
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00:06:46 --> 00:06:51
And now, the second thing that
I'm going to do is what I'm
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going to call the setup here.
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How I'm going to set things up.
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And I'll tell you what
these things are more
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systematically in a second.
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00:07:01 --> 00:07:03
And the setup is that I
want to somehow detect
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what I did before.
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And I'm going to write
some unknowns here.
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00:07:11 --> 00:07:15
What I expect is that this will
break up into two pieces.
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One with the denominator x
- 1, and the other with
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00:07:17 --> 00:07:23
the denominator x + 2.
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00:07:23 --> 00:07:32
So now, my third step is going
to be to solve for A and B.
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00:07:32 --> 00:07:34
And then I'm done,
if I do that.
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00:07:34 --> 00:07:41
That's the complete
unwinding of this disguise.
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00:07:41 --> 00:07:43
And this is where the cover-up
method comes in handy.
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This is this method that
I'm about to describe.
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00:07:46 --> 00:07:49
Now, you can do the algebra
in a clumsy way, or you
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00:07:49 --> 00:07:50
can do it in a quick way.
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00:07:50 --> 00:07:54
And we'd like to get efficient
about the algebra involved.
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00:07:54 --> 00:07:58
And so let me show you what the
first step in the trick is.
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00:07:58 --> 00:08:12
We're going to solve for A
by multiplying by (x - 1).
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Now, notice if you multiply
by (x - 1) in that equation
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2, what you get is this.
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You got 4x - 2 / the
x - 1's cancel.
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You get this on the
left-hand side.
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00:08:23 --> 00:08:26
And on the right-hand
side you get A.
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The x - 1's cancel again.
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00:08:29 --> 00:08:31
And then we get
this extra term.
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00:08:31 --> 00:08:35
Which is B/ ( x + 2)( x - 1).
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00:08:35 --> 00:08:38
Now, the trick here, and we're
going to get even better
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00:08:38 --> 00:08:40
trick in just a second.
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00:08:40 --> 00:08:42
The trick here is that I
didn't try to clear the
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denominators completely.
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00:08:44 --> 00:08:46
I was very efficient
about the way I did it.
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00:08:46 --> 00:08:51
It just cleared one factor.
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00:08:51 --> 00:08:54
And the result here
is very useful.
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00:08:54 --> 00:09:03
Namely, if I plug in now x =
1, this term drops out too.
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00:09:03 --> 00:09:10
So what I'm going to do now is
I'm going to plug in x = 1.
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00:09:10 --> 00:09:15
And what I get on the left-hand
side here is 4 - 1 and 1 + 2,
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00:09:15 --> 00:09:17
and on the left-hand
side I get A.
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00:09:17 --> 00:09:19
That's the end.
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00:09:19 --> 00:09:21
This is my formula for A.
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00:09:21 --> 00:09:27
A happens to be equal to 1.
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00:09:27 --> 00:09:29
And that's, of course,
what I expect.
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00:09:29 --> 00:09:32
A had better be 1, because
the thing broke up into
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00:09:32 --> 00:09:37
1 / x - 1 + 3 / x + 2.
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00:09:37 --> 00:09:39
So this is the correct answer.
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00:09:39 --> 00:09:41
There was a question out
here, which I missed.
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00:09:41 --> 00:09:48
STUDENT: Aren't polynomials
defined as functions with
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00:09:48 --> 00:09:56
whole powers, or could
they be square roots?
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00:09:56 --> 00:09:58
PROFESSOR: Are polynomials
defined as functions with
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00:09:58 --> 00:10:00
whole powers, or can
they be square roots?
157
00:10:00 --> 00:10:01
That's the question.
158
00:10:01 --> 00:10:04
The answer is, they only
have whole powers.
159
00:10:04 --> 00:10:06
So for instance here I only
have the power 1 and 0.
160
00:10:06 --> 00:10:10
Here I have the powers 2, 1
and 0 in the denominator.
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00:10:10 --> 00:10:16
Square roots are no
good for this method.
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00:10:16 --> 00:10:17
Another question.
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00:10:17 --> 00:10:18
STUDENT: [INAUDIBLE]
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00:10:18 --> 00:10:22
PROFESSOR: Why did I say x = 1?
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00:10:22 --> 00:10:26
The reason why I said x = 1 was
that it works really fast.
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00:10:26 --> 00:10:30
You can't know this in advance,
that's part of the method.
167
00:10:30 --> 00:10:32
It just turns out to be
the best thing to do.
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00:10:32 --> 00:10:35
The fastest way of getting
at the coefficient A.
169
00:10:35 --> 00:10:38
Now the curious thing,
let me just pause for a
170
00:10:38 --> 00:10:39
second before I do it.
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00:10:39 --> 00:10:44
If I had plugged x = 1 into the
original equation, I would
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00:10:44 --> 00:10:45
have gotten nonsense.
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00:10:45 --> 00:10:48
Because I would've gotten
0 in the denominator.
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00:10:48 --> 00:10:50
And that seems like the
most horrible thing to do.
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00:10:50 --> 00:10:54
The worst possible thing
to do, is to set x = 1.
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00:10:54 --> 00:10:56
On the other hand, what
we did is a trick.
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00:10:56 --> 00:10:58
We multiplied by x - 1.
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00:10:58 --> 00:11:01
And that turned the
equation into this.
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00:11:01 --> 00:11:05
So now, in disguise,
I multiplied by 0.
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00:11:05 --> 00:11:07
But that turns out
to be legitimate.
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00:11:07 --> 00:11:11
Because really this equation
is true for all x except 1.
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And then instead of taking
x = 1, I can really
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take x tends to 1.
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00:11:15 --> 00:11:17
That's really what I need.
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00:11:17 --> 00:11:18
The limit is x goes to one.
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00:11:18 --> 00:11:20
The equation is
still valid then.
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00:11:20 --> 00:11:23
So I'm using the worst case,
the case that looks like
188
00:11:23 --> 00:11:24
it's dividing by 0.
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00:11:24 --> 00:11:26
And it's helping me because
it's cancelling out all the
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information in terms of B.
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00:11:29 --> 00:11:33
So the advantage here is
this cancellation that
192
00:11:33 --> 00:11:36
occurs in this part.
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00:11:36 --> 00:11:37
So that's the method.
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00:11:37 --> 00:11:39
We're going to shorten it
much, much more in a second.
195
00:11:39 --> 00:11:44
But let me carry it out for the
other coefficient as well.
196
00:11:44 --> 00:11:51
So the other coefficient I'm
going to solve for B, I'm
197
00:11:51 --> 00:11:57
going to multiply by x + 2.
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00:11:57 --> 00:12:02
And when I do that, I get 4x -
1 / x - 1, that's the left-hand
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00:12:02 --> 00:12:05
side, the very top
expression there.
200
00:12:05 --> 00:12:10
And then down below I get
A/ ( x - 1)( x + 2).
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00:12:10 --> 00:12:14
And then again the
x + 2's cancel.
202
00:12:14 --> 00:12:15
So I get B sitting alone.
203
00:12:15 --> 00:12:18
And now I'm going to
do the same trick.
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00:12:18 --> 00:12:21
I'm going to set x = - 2.
205
00:12:21 --> 00:12:28
That's the value which is going
to knock out this A term here.
206
00:12:28 --> 00:12:30
So that cancels this
term completely.
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00:12:30 --> 00:12:37
And what we get here all told
is - 8 - 1 / - 2 - 1 = B.
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00:12:37 --> 00:12:43
In other words, B = 3,
which was also what it
209
00:12:43 --> 00:12:44
was supposed to be.
210
00:12:44 --> 00:12:48
B was this number
3, right here.
211
00:12:48 --> 00:12:50
Which I'm now not
going to change to 2.
212
00:12:50 --> 00:12:52
Because I know that it's not 2.
213
00:12:52 --> 00:12:53
There was a question.
214
00:12:53 --> 00:12:59
STUDENT: [INAUDIBLE]
215
00:12:59 --> 00:13:01
PROFESSOR: All right.
216
00:13:01 --> 00:13:05
Now, this is the method
which is called cover-up.
217
00:13:05 --> 00:13:09
But it's really carried out
much, much faster than this.
218
00:13:09 --> 00:13:11
So I'm going to review the
method and I'm going to show
219
00:13:11 --> 00:13:14
you what it is in general.
220
00:13:14 --> 00:13:22
So the first step is to
factor the denominator, Q.
221
00:13:22 --> 00:13:24
That's what I labeled
1 over there.
222
00:13:24 --> 00:13:29
That was the factorization
of the denominator up top.
223
00:13:29 --> 00:13:36
The second step is what I'm
going to call the setup.
224
00:13:36 --> 00:13:37
That's step 2.
225
00:13:37 --> 00:13:41
And that's where I knew what
I was aiming for in advance.
226
00:13:41 --> 00:13:42
And I'm going to have to
explain to you in every
227
00:13:42 --> 00:13:45
instance exactly what
this setup should be.
228
00:13:45 --> 00:13:49
That is, what the unknowns
should be and what target,
229
00:13:49 --> 00:13:53
simplified expression,
we're aiming for.
230
00:13:53 --> 00:13:54
So that's the setup.
231
00:13:54 --> 00:14:01
And then the third step is
what I'll now call cover-up.
232
00:14:01 --> 00:14:04
Which is just a very fast way
of doing what I did on this
233
00:14:04 --> 00:14:09
last board, which is solving
for the unknown coefficients.
234
00:14:09 --> 00:14:12
So now, let me perform
it for you again.
235
00:14:12 --> 00:14:14
Over here.
236
00:14:14 --> 00:14:18
So it's 4x - 1 divided by,
so this is to eliminate
237
00:14:18 --> 00:14:19
writing here.
238
00:14:19 --> 00:14:24
Handwriting it makes
it much faster.
239
00:14:24 --> 00:14:28
So this part just factoring
the denominator, that
240
00:14:28 --> 00:14:31
was 1, that was step 1.
241
00:14:31 --> 00:14:34
And then step 2, again,
is the setup, which is
242
00:14:34 --> 00:14:39
setting it up like this.
243
00:14:39 --> 00:14:42
Alright, that's the setup.
244
00:14:42 --> 00:14:46
And now I claim that without
writing very much, I can
245
00:14:46 --> 00:14:49
figure out what A and B are.
246
00:14:49 --> 00:14:51
Just by staring at this.
247
00:14:51 --> 00:14:54
So now what I'm going to do
is I'm just going to think
248
00:14:54 --> 00:14:55
what I did over there.
249
00:14:55 --> 00:14:57
And I'm just going
to do it directly.
250
00:14:57 --> 00:15:02
So let me show you what the
method consists of visually.
251
00:15:02 --> 00:15:09
I'm going to cover up, that is,
knock out this factor, and
252
00:15:09 --> 00:15:13
focus on this number here.
253
00:15:13 --> 00:15:15
And I'm going to plug in
the thing that makes
254
00:15:15 --> 00:15:17
the 0, which is x = 1.
255
00:15:17 --> 00:15:20
So I'm plugging in x = 1.
256
00:15:20 --> 00:15:21
To this left-hand side.
257
00:15:21 --> 00:15:28
And what I get is 4
- 1 / 1 + 2 = A.
258
00:15:28 --> 00:15:29
Now, that's the same
thing I did over there.
259
00:15:29 --> 00:15:33
I just did it by skipping the
intermediate algebra step,
260
00:15:33 --> 00:15:35
which is a lot of writing.
261
00:15:35 --> 00:15:38
So the cover-up method really
amounts to the following thing.
262
00:15:38 --> 00:15:40
I'm thinking of multiplying
this over here.
263
00:15:40 --> 00:15:43
It cancels this and it gets
rid of everything else.
264
00:15:43 --> 00:15:45
And it just leaves me with
A on the right-hand side.
265
00:15:45 --> 00:15:47
And I have to get rid
of it on this side.
266
00:15:47 --> 00:15:51
So in other words, by
eliminating this, I'm isolating
267
00:15:51 --> 00:15:53
a on the right-hand side.
268
00:15:53 --> 00:15:55
So the cover-up is that
I'm covering this and
269
00:15:55 --> 00:15:57
getting A out of it.
270
00:15:57 --> 00:16:01
Similarly, I can do the
same thing with B.
271
00:16:01 --> 00:16:04
It's focused on the
value x = - 2.
272
00:16:04 --> 00:16:07
And b is what I'm getting
on the right-hand side.
273
00:16:07 --> 00:16:10
And then I have to
cover up this.
274
00:16:10 --> 00:16:14
So if I cover up that, then
what's left over with x = - 2
275
00:16:14 --> 00:16:22
is again, - 8 - 1 / - 2 - 1.
276
00:16:22 --> 00:16:26
So this is the way the method
gets carried out in practice.
277
00:16:26 --> 00:16:32
Writing, essentially,
the least you can.
278
00:16:32 --> 00:16:40
Now, when you get to several
variables, it becomes just way
279
00:16:40 --> 00:16:42
more convenient to do this.
280
00:16:42 --> 00:16:45
So now, let me just review
when cover-up works.
281
00:16:45 --> 00:17:00
So this cover-up method works
if Q ( x) has distinct
282
00:17:00 --> 00:17:06
linear factors.
283
00:17:06 --> 00:17:11
And, so you need
two things here.
284
00:17:11 --> 00:17:14
It has to factor completely,
the denominator has
285
00:17:14 --> 00:17:15
to factor completely.
286
00:17:15 --> 00:17:21
And the degree of the numerator
has to be strictly less than
287
00:17:21 --> 00:17:27
the degree of the denominator.
288
00:17:27 --> 00:17:30
I'm going to give you
an example here.
289
00:17:30 --> 00:17:35
So, for instance, and this
tells you the general
290
00:17:35 --> 00:17:38
pattern of the setup also.
291
00:17:38 --> 00:17:44
Say you had x ^2 +
3x + 8, let's say.
292
00:17:44 --> 00:17:50
Over (x - 1) ( x - 2)( x + 5).
293
00:17:50 --> 00:17:52
So here I'm going to
tell you the setup.
294
00:17:52 --> 00:18:02
The setup is going to be A / (x
- 1) + B / (x - 2) + c / x + 5.
295
00:18:02 --> 00:18:04
And it will always break
up into something.
296
00:18:04 --> 00:18:07
So however many factors you
have, you'll have to put in
297
00:18:07 --> 00:18:09
a term for each of those.
298
00:18:09 --> 00:18:13
And then you can find
each number here by
299
00:18:13 --> 00:18:26
this cover-up method.
300
00:18:26 --> 00:18:29
Now we're done with that.
301
00:18:29 --> 00:18:33
And now we have to go on to
the algebraic complications.
302
00:18:33 --> 00:18:38
So would the first typical
algebraic complication be.
303
00:18:38 --> 00:18:50
It would be repeated roots
or repeated factors.
304
00:18:50 --> 00:18:54
Let me get one that
doesn't come out to be
305
00:18:54 --> 00:18:56
extremely ugly here.
306
00:18:56 --> 00:19:02
So this is what we'll
call Example 2.
307
00:19:02 --> 00:19:06
And this is going to work when
the degree, you always need
308
00:19:06 --> 00:19:08
that the degree of the
numerator is less than the
309
00:19:08 --> 00:19:12
degree of the denominator.
310
00:19:12 --> 00:19:22
And Q has now repeated
linear factors.
311
00:19:22 --> 00:19:26
So let's see which example
I wanted to show you.
312
00:19:26 --> 00:19:28
So let's just give this here.
313
00:19:28 --> 00:19:34
I'll just repeat
the denominator.
314
00:19:34 --> 00:19:39
With an extra factor like this.
315
00:19:39 --> 00:19:41
Now, the main thing you
need to know, since I've
316
00:19:41 --> 00:19:44
already performed the
factorization for you.
317
00:19:44 --> 00:19:46
Already performed Step 1.
318
00:19:46 --> 00:19:49
This is Step 1 here.
319
00:19:49 --> 00:19:51
You have to factor things
all the way, and that's
320
00:19:51 --> 00:19:53
already been done for you.
321
00:19:53 --> 00:19:56
And here's what this setup is.
322
00:19:56 --> 00:20:00
The setup is that it's
of the form A / (x -
323
00:20:00 --> 00:20:05
1) + B / (x - 1) ^2.
324
00:20:05 --> 00:20:15
We need another term for the
square here. + C / (x + 2).
325
00:20:15 --> 00:20:17
In general, if you have more
powers you just need to keep
326
00:20:17 --> 00:20:19
on putting in those powers.
327
00:20:19 --> 00:20:24
You need one for
each of the powers.
328
00:20:24 --> 00:20:26
Why does it have to be squared?
329
00:20:26 --> 00:20:27
OK.
330
00:20:27 --> 00:20:28
Good question.
331
00:20:28 --> 00:20:31
So why in the world
am I doing this?
332
00:20:31 --> 00:20:37
Let me just give you one hint
as to why I'm doing this.
333
00:20:37 --> 00:20:40
It's very, very much like
the decimal expansion of a
334
00:20:40 --> 00:20:44
number or, say, the base
2 expansion of a number.
335
00:20:44 --> 00:20:57
So, for, example the number
7/16 is 0 / 2 + 1 / 2 ^2 +
336
00:20:57 --> 00:21:02
1/2 ^3 +, is that right?
337
00:21:02 --> 00:21:07
So it's 4/16 + 1/2 ^4.
338
00:21:07 --> 00:21:09
It's this sort of thing.
339
00:21:09 --> 00:21:11
And I'm getting this
power and this power.
340
00:21:11 --> 00:21:13
If I have higher powers,
I'm going to have to
341
00:21:13 --> 00:21:14
have more and more.
342
00:21:14 --> 00:21:17
So this is what happens
when I have a 2 ^ 4.
343
00:21:17 --> 00:21:20
I have to represent
things like this.
344
00:21:20 --> 00:21:23
That's what's coming
out of this piece with
345
00:21:23 --> 00:21:24
the repetitious here.
346
00:21:24 --> 00:21:26
Of the powers.
347
00:21:26 --> 00:21:31
This is just an analogy.
348
00:21:31 --> 00:21:31
Of what we're doing.
349
00:21:31 --> 00:21:33
Yeah, another
question over here.
350
00:21:33 --> 00:21:34
STUDENT: [INAUDIBLE]
351
00:21:34 --> 00:21:35
PROFESSOR: Yes.
352
00:21:35 --> 00:21:37
So this is an example, but it's
meant to represent the general
353
00:21:37 --> 00:21:41
case and I will also give
you a general picture.
354
00:21:41 --> 00:21:43
For sure, once you have the
second power here, you'll need
355
00:21:43 --> 00:21:46
both the first and the second
power mentioned over here.
356
00:21:46 --> 00:21:48
And since there's only a first
power over here I only have
357
00:21:48 --> 00:21:52
to mention a first
power over there.
358
00:21:52 --> 00:21:55
If this were a 3 here, there
would be one more term which
359
00:21:55 --> 00:22:00
would be the one for x -
1 ^2 in the denominator.
360
00:22:00 --> 00:22:03
That's what you just said.
361
00:22:03 --> 00:22:09
OK, now, what's different about
this setup is that the cover-up
362
00:22:09 --> 00:22:13
method, although it works,
it doesn't work so well.
363
00:22:13 --> 00:22:14
It doesn't work quite as well.
364
00:22:14 --> 00:22:33
The cover-up works for the
coefficients B and C, not A.
365
00:22:33 --> 00:22:38
We'll have a quick method
for the numbers B and C.
366
00:22:38 --> 00:22:40
To figure out what they are.
367
00:22:40 --> 00:22:43
But it will be a little
slower to get to A,
368
00:22:43 --> 00:22:47
which we will do last.
369
00:22:47 --> 00:22:56
Let me show you how it works.
370
00:22:56 --> 00:23:01
First of all, I'm going to do
the ordinary cover-up with C.
371
00:23:01 --> 00:23:05
So for C, I just want to
do the same old thing
372
00:23:05 --> 00:23:06
that I did before.
373
00:23:06 --> 00:23:10
I cover up this, and that's
going to get rid of all the
374
00:23:10 --> 00:23:12
junk except for the C term.
375
00:23:12 --> 00:23:16
So I have to plug x = - 2.
376
00:23:16 --> 00:23:21
And I get x -- sorry, I get (-2
) ^2 + 2 in the numerator.
377
00:23:21 --> 00:23:25
And I get (- 2 - 1)^2
in the denominator.
378
00:23:25 --> 00:23:28
Remember I'm covering this up.
379
00:23:28 --> 00:23:30
So that's all there is
on the left-hand side.
380
00:23:30 --> 00:23:37
And on the right-hand
side all there is C.
381
00:23:37 --> 00:23:39
Everything else got killed
off, because it was
382
00:23:39 --> 00:23:40
x - 2 times that.
383
00:23:40 --> 00:23:42
That's 0 times all
that other stuff.
384
00:23:42 --> 00:23:45
And the x - 2 over
here canceled.
385
00:23:45 --> 00:23:47
This is the shortcut that I
just described, and this is
386
00:23:47 --> 00:23:50
much faster than doing
all that arithmetic.
387
00:23:50 --> 00:23:52
And algebra.
388
00:23:52 --> 00:23:57
So all together this
is a 6/9, right?
389
00:23:57 --> 00:24:09
So it's C = 6/9, which is 2/3.
390
00:24:09 --> 00:24:13
Now, the other one which is
easy to do, I'm going to do
391
00:24:13 --> 00:24:14
by the slow method first.
392
00:24:14 --> 00:24:17
But you omit a term.
393
00:24:17 --> 00:24:23
The idea is to cover up
the other bad factor.
394
00:24:23 --> 00:24:27
Cover ups, I'll do it both
the way and the slow way.
395
00:24:27 --> 00:24:29
I'll do it the fast way
first, and then I'll
396
00:24:29 --> 00:24:30
show you the slow way.
397
00:24:30 --> 00:24:32
The fast way is to
cover this up.
398
00:24:32 --> 00:24:34
And then I have to cover
up everything else.
399
00:24:34 --> 00:24:35
That gets eliminated.
400
00:24:35 --> 00:24:38
And that includes
everything but B.
401
00:24:38 --> 00:24:40
So I get B on this side.
402
00:24:40 --> 00:24:42
And I get 1 on that side.
403
00:24:42 --> 00:24:48
So that's 1 ^2 + 2 / 1 + 2.
404
00:24:48 --> 00:24:56
So in other words, B = 1.
405
00:24:56 --> 00:24:58
That was pretty fast, so let
me show you what arithmetic
406
00:24:58 --> 00:25:00
was hiding behind that.
407
00:25:00 --> 00:25:01
What algebra was
hiding behind it.
408
00:25:01 --> 00:25:06
What I was really
doing is this.
409
00:25:06 --> 00:25:16
In multiplying through by
x - 1 ^2, so I got this.
410
00:25:16 --> 00:25:19
So this canceled here, so
this C just stands alone.
411
00:25:19 --> 00:25:26
And then I have here
C /x + 2 (x - 1) ^2.
412
00:25:26 --> 00:25:30
Notice again, I cleared out
that 1, this term from the
413
00:25:30 --> 00:25:33
denominator and sent it
over to the other side.
414
00:25:33 --> 00:25:38
Now, what's happening is
that when I set x = 1
415
00:25:38 --> 00:25:43
here, this term is dying.
416
00:25:43 --> 00:25:45
This term is going away,
because there's more powers
417
00:25:45 --> 00:25:47
in the numerator than
in the denominator.
418
00:25:47 --> 00:25:50
This is still 0.
419
00:25:50 --> 00:25:52
And this one is gone also.
420
00:25:52 --> 00:25:56
So all that's left is B.
421
00:25:56 --> 00:25:59
Now, I cannot pull that off
with a single power of x - 1.
422
00:25:59 --> 00:26:02
I can't expose the A term.
423
00:26:02 --> 00:26:04
It's the B term
that I can expose.
424
00:26:04 --> 00:26:06
Because I can multiply through
by this thing squared.
425
00:26:06 --> 00:26:10
If I multiply through by just x
- 1, what'll happen here is I
426
00:26:10 --> 00:26:12
won't have canceled
this (x - 1 )^2.
427
00:26:12 --> 00:26:13
It's useless.
428
00:26:13 --> 00:26:15
I still have a 0 in
the denominator.
429
00:26:15 --> 00:26:17
I'll have B / 0 when
I plug in x = 1.
430
00:26:17 --> 00:26:22
Which I can't use.
431
00:26:22 --> 00:26:32
Again, the cover-up method is
giving us B and C, not A.
432
00:26:32 --> 00:26:38
Now, for the last term, for A,
I'm going to just have to be
433
00:26:38 --> 00:26:40
straightforward about it.
434
00:26:40 --> 00:26:51
And so I'll just suggest for A,
plug in your favorite number.
435
00:26:51 --> 00:26:59
So plug in my favorite number.
436
00:26:59 --> 00:27:01
Which is x = 0.
437
00:27:01 --> 00:27:04
And you won't be able to
plug in x = 0 if you've
438
00:27:04 --> 00:27:05
already used it.
439
00:27:05 --> 00:27:08
Here the two numbers
we've already used are
440
00:27:08 --> 00:27:13
x = 1 and x = - 2.
441
00:27:13 --> 00:27:18
But we haven't used x =
0 yet, so that's good.
442
00:27:18 --> 00:27:21
I'm going to plug in now
x = 0 into the equation.
443
00:27:21 --> 00:27:22
What do I get?
444
00:27:22 --> 00:27:35
I get 0 ^2 + 2 / (- 1) ^2 *
2 is equal to, let's see.
445
00:27:35 --> 00:27:36
A is the thing that
I don't know.
446
00:27:36 --> 00:27:44
So it's A / - 1 +, B /
x - 1 ^2 so B = 1, so
447
00:27:44 --> 00:27:48
that's 1 / (- 1) ^2.
448
00:27:48 --> 00:27:51
And then C was 2/3.
449
00:27:51 --> 00:27:55
2/3 / x + 2.
450
00:27:55 --> 00:28:00
So that's 0 + 2.
451
00:28:00 --> 00:28:02
Don't give up at this point.
452
00:28:02 --> 00:28:03
This is a lot of algebra.
453
00:28:03 --> 00:28:06
You really have to plug
in all these numbers.
454
00:28:06 --> 00:28:08
You make one arithmetic mistake
and you're always going
455
00:28:08 --> 00:28:09
to get the wrong answer.
456
00:28:09 --> 00:28:16
This is very
arithmetically intensive.
457
00:28:16 --> 00:28:18
However, it does
simplify at this point.
458
00:28:18 --> 00:28:22
We have 2 / 2, that's 1.
459
00:28:22 --> 00:28:27
Is equal to - A + 1 + 1/3.
460
00:28:27 --> 00:28:29
So let's see.
461
00:28:29 --> 00:28:34
A on the other side,
this becomes A = 1/3.
462
00:28:34 --> 00:28:35
And that's it.
463
00:28:35 --> 00:28:36
This is the end.
464
00:28:36 --> 00:28:40
We've we've simplified
our function.
465
00:28:40 --> 00:28:48
And now it's easy to integrate.
466
00:28:48 --> 00:28:48
Question.
467
00:28:48 --> 00:28:49
Another question.
468
00:28:49 --> 00:29:02
STUDENT: [INAUDIBLE]
469
00:29:02 --> 00:29:04
PROFESSOR: So the question
is, if x = 0 has already
470
00:29:04 --> 00:29:05
been used, what do I do?
471
00:29:05 --> 00:29:10
And the answer is,
pick something else.
472
00:29:10 --> 00:29:11
And you said pick
a random number.
473
00:29:11 --> 00:29:13
And that's right, except that
if you really picked a random
474
00:29:13 --> 00:29:19
number it would be 4.12567843,
which would be difficult.
475
00:29:19 --> 00:29:21
What you want to pick is the
easiest possible number
476
00:29:21 --> 00:29:24
you can think of.
477
00:29:24 --> 00:29:24
Yeah.
478
00:29:24 --> 00:29:32
STUDENT: [INAUDIBLE]
479
00:29:32 --> 00:29:37
PROFESSOR: If you had, as in
this sort of situation here.
480
00:29:37 --> 00:29:39
More powers.
481
00:29:39 --> 00:29:41
Wouldn't you have
more variables.
482
00:29:41 --> 00:29:42
Very good question.
483
00:29:42 --> 00:29:44
That's absolutely right.
484
00:29:44 --> 00:29:48
This was a 3 by 3 system in
disguise, for these three
485
00:29:48 --> 00:29:50
unknowns, A, B and C.
486
00:29:50 --> 00:29:52
What we started with in
the previous problem
487
00:29:52 --> 00:29:53
was two variables.
488
00:29:53 --> 00:29:56
It's over here, the
variables A and B.
489
00:29:56 --> 00:29:59
And as the degree of the
denominator goes up, the
490
00:29:59 --> 00:30:02
number of variables goes up.
491
00:30:02 --> 00:30:05
It gets more and more
and more complicated.
492
00:30:05 --> 00:30:06
More and more
arithmetically intensive.
493
00:30:06 --> 00:30:09
STUDENT: [INAUDIBLE]
494
00:30:09 --> 00:30:09
PROFESSOR: Well, so.
495
00:30:09 --> 00:30:11
The question is, how would
you solve it if you
496
00:30:11 --> 00:30:12
have two unknowns.
497
00:30:12 --> 00:30:16
That's exactly the point here.
498
00:30:16 --> 00:30:19
This is a system of
simultaneous equations
499
00:30:19 --> 00:30:19
for unknowns.
500
00:30:19 --> 00:30:23
And we have little tricks for
isolating single variables.
501
00:30:23 --> 00:30:25
Otherwise we're stuck with
solving the whole system.
502
00:30:25 --> 00:30:28
And you'd have to solve the
whole system by elimination,
503
00:30:28 --> 00:30:35
various other tricks.
504
00:30:35 --> 00:30:38
I'll say a little more
about that later.
505
00:30:38 --> 00:30:47
Now, I have to get one step
more complicated with
506
00:30:47 --> 00:30:53
my next example.
507
00:30:53 --> 00:31:02
My next example is going to
have a quadratic factor.
508
00:31:02 --> 00:31:05
So still I'm sticking to the
degree of the polynomial and
509
00:31:05 --> 00:31:08
the numerator is less than
the degree of the polynomial
510
00:31:08 --> 00:31:09
in the denominator.
511
00:31:09 --> 00:31:21
And I'm going to take the case
where Q has a quadratic factor.
512
00:31:21 --> 00:31:26
Let me just again illustrate
this by example.
513
00:31:26 --> 00:31:30
I have here (x - 1) (x^2 + 1).
514
00:31:30 --> 00:31:34
I'll make it about as
easy as they come.
515
00:31:34 --> 00:31:38
Now, the setup will be
slightly different here.
516
00:31:38 --> 00:31:40
Here's the setup.
517
00:31:40 --> 00:31:42
It's already factored.
518
00:31:42 --> 00:31:44
I've already done as
much as I can do.
519
00:31:44 --> 00:31:48
I can't factor this x^2 + 1
into linear factors unless you
520
00:31:48 --> 00:31:49
know about complex numbers.
521
00:31:49 --> 00:31:52
If you know about complex
numbers this method
522
00:31:52 --> 00:31:53
becomes much easier.
523
00:31:53 --> 00:31:55
And it comes back to
the cover-up method.
524
00:31:55 --> 00:31:58
Which is the way that the
cover-up method was originally
525
00:31:58 --> 00:32:01
conceived by heavy side.
526
00:32:01 --> 00:32:04
But you won't get to
that until 18.03.
527
00:32:04 --> 00:32:05
So we'll wait.
528
00:32:05 --> 00:32:08
This, by the way, is a method
which is used for integration.
529
00:32:08 --> 00:32:11
But it was invented to do
something with Laplace
530
00:32:11 --> 00:32:14
transforms and inversion
of certain kinds of
531
00:32:14 --> 00:32:15
differential equations.
532
00:32:15 --> 00:32:17
By heavy side.
533
00:32:17 --> 00:32:21
And so it came much
later than integration.
534
00:32:21 --> 00:32:26
But anyway, it's a very
convenient method.
535
00:32:26 --> 00:32:30
So here's the set
up with this one.
536
00:32:30 --> 00:32:34
Again, we want a term for
this (x - 1) factor.
537
00:32:34 --> 00:32:36
And now we're going to also
have a term with the
538
00:32:36 --> 00:32:39
denominator x squared plus 1.
539
00:32:39 --> 00:32:40
But this is the difference.
540
00:32:40 --> 00:32:44
It's now going to be a
first degree polynomial.
541
00:32:44 --> 00:32:53
One degree down from
the quadratic here.
542
00:32:53 --> 00:32:55
So this is what I keep
on calling the setup,
543
00:32:55 --> 00:32:57
this is number 2.
544
00:32:57 --> 00:32:59
You have to know that in
advance based on the pattern
545
00:32:59 --> 00:33:03
that you see on the
left-hand side.
546
00:33:03 --> 00:33:03
Yes.
547
00:33:03 --> 00:33:13
STUDENT: [INAUDIBLE]
548
00:33:13 --> 00:33:15
PROFESSOR: The question is, if
the degree of the numerator.
549
00:33:15 --> 00:33:19
So in this case, if this were
cubed, and this is matching
550
00:33:19 --> 00:33:23
with the denominator, which
is total of degree 3.
551
00:33:23 --> 00:33:26
The answer is that
this does not work.
552
00:33:26 --> 00:33:29
STUDENT: [INAUDIBLE]
553
00:33:29 --> 00:33:31
PROFESSOR: It definitely
doesn't work.
554
00:33:31 --> 00:33:32
And we're going to have
to do something totally
555
00:33:32 --> 00:33:34
different to handle it.
556
00:33:34 --> 00:33:37
Which turns out, fortunately,
to be much easier than this.
557
00:33:37 --> 00:33:41
But we'll deal with
that at the end.
558
00:33:41 --> 00:33:43
Keep this in mind.
559
00:33:43 --> 00:33:45
This is an easy way to make
a mistake if you start with
560
00:33:45 --> 00:33:47
a higher degree numerator.
561
00:33:47 --> 00:33:51
You'll never get
the right answer.
562
00:33:51 --> 00:33:54
So now, so I have my setup now.
563
00:33:54 --> 00:33:56
And now what can I do?
564
00:33:56 --> 00:34:03
Well, I claim that I can
still do cover-up for A.
565
00:34:03 --> 00:34:05
It's the same idea.
566
00:34:05 --> 00:34:07
I cover this guy up.
567
00:34:07 --> 00:34:09
And if I really multiply
by it it would knock
568
00:34:09 --> 00:34:11
everything out but A.
569
00:34:11 --> 00:34:14
So I cover this up
and I plug in x = 1.
570
00:34:14 --> 00:34:20
So I get here 1 ^2
/ 1 ^2 + 1 = A.
571
00:34:20 --> 00:34:25
In other words, A = 1/2.
572
00:34:25 --> 00:34:28
Again cover-up is pretty
fast, as you can see.
573
00:34:28 --> 00:34:32
It's not too bad.
574
00:34:32 --> 00:34:41
Now, at this next stage,
I want to find B and C.
575
00:34:41 --> 00:34:45
And the best idea
is the slow way.
576
00:34:45 --> 00:34:48
Here, it's not too terrible.
577
00:34:48 --> 00:34:50
But it's just what
we're going to do.
578
00:34:50 --> 00:34:54
Which is to clear the
denominators completely.
579
00:34:54 --> 00:35:05
So for B and C, just
clear the denominator.
580
00:35:05 --> 00:35:08
That means multiply through
by that whole business.
581
00:35:08 --> 00:35:09
Now, when you do that on
the left-hand side you're
582
00:35:09 --> 00:35:10
going to get x ^2.
583
00:35:10 --> 00:35:12
Because I got rid of
the whole denominator.
584
00:35:12 --> 00:35:16
On the right-hand side when
I bring this up, the x -
585
00:35:16 --> 00:35:18
1 will cancel with this.
586
00:35:18 --> 00:35:23
So the a term will
be A ( x ^2 + 1).
587
00:35:23 --> 00:35:29
And the Bx + C term will have
a remaining factor of x - 1.
588
00:35:29 --> 00:35:33
Because the x ^2
+ 1 will cancel.
589
00:35:33 --> 00:35:38
Again, the arithmetic here
is not too terrible.
590
00:35:38 --> 00:35:41
Now I'm going to
do the following.
591
00:35:41 --> 00:35:46
I'm going to look
at the x ^2 term.
592
00:35:46 --> 00:35:49
On the left-hand side and
the right-hand side.
593
00:35:49 --> 00:35:51
And that will give me one
equation for B and C.
594
00:35:51 --> 00:35:54
And then I'm going to do the
same thing with another term.
595
00:35:54 --> 00:35:57
The x^2 term on the left-hand
side, the coefficient is 1.
596
00:35:57 --> 00:35:59
It's 1 ( x ^2).
597
00:35:59 --> 00:36:06
On the other side, it's A.
remember I actually have A.
598
00:36:06 --> 00:36:08
So I'm going to put
it in, it's 1/2.
599
00:36:08 --> 00:36:11
So this is the A term.
600
00:36:11 --> 00:36:14
And so I get 1/2 ( x ^2).
601
00:36:14 --> 00:36:18
And then the only other x^2 is
when this Bx multiplies this x.
602
00:36:18 --> 00:36:23
So Bx * x is Bx ^2, so this
is the other coefficient
603
00:36:23 --> 00:36:25
on x ^2 is B.
604
00:36:25 --> 00:36:36
And that forces B to be 1/2.
605
00:36:36 --> 00:36:41
And last of all, I'm going
to do the x^ 0 power term.
606
00:36:41 --> 00:36:44
Or, otherwise known as
the constant term.
607
00:36:44 --> 00:36:48
And on the left-hand side,
the constant term is 0.
608
00:36:48 --> 00:36:51
There is no constant term.
609
00:36:51 --> 00:36:57
On the right-hand side there's
a constant term, 1/2 * 1.
610
00:36:57 --> 00:36:58
That's 1/2 here.
611
00:36:58 --> 00:37:01
And then there's another
constant term, which is this
612
00:37:01 --> 00:37:03
constant times this - 1.
613
00:37:03 --> 00:37:09
Which is - C.
614
00:37:09 --> 00:37:18
And so the conclusion
here is that C = 1/2.
615
00:37:18 --> 00:37:18
Another question.
616
00:37:18 --> 00:37:19
Yeah.
617
00:37:19 --> 00:37:33
STUDENT: [INAUDIBLE]
618
00:37:33 --> 00:37:37
PROFESSOR: There's also an
x^ 0 power hidden in here.
619
00:37:37 --> 00:37:40
Sorry, an x ^ 1 , that's what
you were asking about, sorry.
620
00:37:40 --> 00:37:42
There's also an x ^ 1 .
621
00:37:42 --> 00:37:45
The only reason why I didn't go
to the x^ 1 is that it turns
622
00:37:45 --> 00:37:49
out with these two
I didn't need it.
623
00:37:49 --> 00:37:51
The other thing is that by
experience, I know that the
624
00:37:51 --> 00:37:53
extreme ends of the
multiplication are
625
00:37:53 --> 00:37:54
the easiest ends.
626
00:37:54 --> 00:37:57
And the middle terms have
tons of cross terms.
627
00:37:57 --> 00:37:59
And so I don't like the middle
term as much because it always
628
00:37:59 --> 00:38:01
involves more arithmetic.
629
00:38:01 --> 00:38:05
So I stick to the lowest and
the highest terms if I can.
630
00:38:05 --> 00:38:07
So that was really
a sneaky thing.
631
00:38:07 --> 00:38:10
I did that without
saying anything.
632
00:38:10 --> 00:38:10
Yes.
633
00:38:10 --> 00:38:14
STUDENT: [INAUDIBLE]
634
00:38:14 --> 00:38:15
PROFESSOR: Another
good question.
635
00:38:15 --> 00:38:17
Could I just set equals 0?
636
00:38:17 --> 00:38:17
Absolutely.
637
00:38:17 --> 00:38:22
In fact, that's equivalent to
picking out the x^ 0 term.
638
00:38:22 --> 00:38:24
And you could plug in numbers.
639
00:38:24 --> 00:38:25
If you wanted.
640
00:38:25 --> 00:38:27
That's another way of doing
this besides doing that.
641
00:38:27 --> 00:38:30
So you can also
plug in numbers.
642
00:38:30 --> 00:38:38
Can plug in numbers. x = 0.
643
00:38:38 --> 00:38:44
Actually, not x = 1,
right? - 1, 2, etc.
644
00:38:44 --> 00:38:46
Not 1 just because
we've already used it.
645
00:38:46 --> 00:38:48
We won't get interesting
information out.
646
00:38:48 --> 00:38:48
Yes.
647
00:38:48 --> 00:38:56
STUDENT: [INAUDIBLE]
648
00:38:56 --> 00:38:58
PROFESSOR: So the question
is, could I have done
649
00:38:58 --> 00:38:59
it this other way.
650
00:38:59 --> 00:39:02
With the polynomial,
with this other one.
651
00:39:02 --> 00:39:04
Yes, absolutely.
652
00:39:04 --> 00:39:07
So in other words what I've
taught you now is two choices
653
00:39:07 --> 00:39:09
which are equally reasonable.
654
00:39:09 --> 00:39:12
The one that I picked was the
one that was the fastest for
655
00:39:12 --> 00:39:15
this guy and the one that was
fastest for this one, but I
656
00:39:15 --> 00:39:19
could've done the
other way around.
657
00:39:19 --> 00:39:22
There are a lot of ways of
solving simultaneous equations.
658
00:39:22 --> 00:39:23
Yeah, another question.
659
00:39:23 --> 00:39:24
STUDENT: [INAUDIBLE]
660
00:39:24 --> 00:39:25
PROFESSOR: The question
is the following.
661
00:39:25 --> 00:39:28
So now everybody can
understand the question.
662
00:39:28 --> 00:39:33
If this, instead of being x
^2 + 1, this were x^3 + 1.
663
00:39:33 --> 00:39:36
So that's an important
case to understand.
664
00:39:36 --> 00:39:39
That's a case in which
this denominator is
665
00:39:39 --> 00:39:41
not fully factored.
666
00:39:41 --> 00:39:46
So it's an x^3 + 1, you would
have to factor out an x + 1.
667
00:39:46 --> 00:39:49
So that would be a situation
like this, you have an x^3 + 1,
668
00:39:49 --> 00:39:57
but that's (x + 1)( x^2 + x
+ 1), this kind of thing.
669
00:39:57 --> 00:40:01
If that's the right, there must
be a minus sign in here maybe.
670
00:40:01 --> 00:40:03
OK, something like this.
671
00:40:03 --> 00:40:07
Right?
672
00:40:07 --> 00:40:08
Isn't that what it is?
673
00:40:08 --> 00:40:12
STUDENT: [INAUDIBLE]
674
00:40:12 --> 00:40:12
PROFESSOR: I think it's right.
675
00:40:12 --> 00:40:15
But anyway, the point is
that you have to factor it.
676
00:40:15 --> 00:40:17
And then you have a
linear and a quadratic.
677
00:40:17 --> 00:40:21
So you're always going to be
faced eventually with linear
678
00:40:21 --> 00:40:23
factors and quadratic factors.
679
00:40:23 --> 00:40:25
If you have a cubic,
that means you haven't
680
00:40:25 --> 00:40:28
factored sufficiently.
681
00:40:28 --> 00:40:32
So you're still back in Step 1.
682
00:40:32 --> 00:40:32
STUDENT: [INAUDIBLE]
683
00:40:32 --> 00:40:34
PROFESSOR: In the x^3 + 1 case?
684
00:40:34 --> 00:40:36
STUDENT: [INAUDIBLE]
685
00:40:36 --> 00:40:39
PROFESSOR: In the x^3 + 1 case,
we are out of luck until we've
686
00:40:39 --> 00:40:41
completed the factorization.
687
00:40:41 --> 00:40:44
Once we've completed the
factorization, we're going to
688
00:40:44 --> 00:40:48
have to deal with these two
factors as denominators.
689
00:40:48 --> 00:40:54
So it'll be this plus something
over x + 1 + a Bx + C type of
690
00:40:54 --> 00:40:58
thing over this thing here.
691
00:40:58 --> 00:41:01
That's what's eventually
going to happen.
692
00:41:01 --> 00:41:03
But hold on to that idea.
693
00:41:03 --> 00:41:14
Let me carry out one
more example here.
694
00:41:14 --> 00:41:17
So I've figured out what
all the values are.
695
00:41:17 --> 00:41:21
But I think it's also worth it
to remember now that we also
696
00:41:21 --> 00:41:29
have to carry out
the integration.
697
00:41:29 --> 00:41:36
What I've just shown you is
that the integral of x ^2 dx
698
00:41:36 --> 00:41:42
over (x - 1)( x ^2 + 1) is
equal to, and I've split
699
00:41:42 --> 00:41:43
up into these pieces.
700
00:41:43 --> 00:41:45
So what are the pieces?
701
00:41:45 --> 00:42:01
The pieces are, 1/2, x -
1 + 1/2 x / x ^2 + 1.
702
00:42:01 --> 00:42:02
This is the A term.
703
00:42:02 --> 00:42:04
This is the B term.
704
00:42:04 --> 00:42:12
And then there's the C term.
705
00:42:12 --> 00:42:16
So we'd better remember that we
know how to antidifferentiate
706
00:42:16 --> 00:42:18
these things.
707
00:42:18 --> 00:42:20
In other words, I want
to finish the problem.
708
00:42:20 --> 00:42:22
The others were pretty easy, so
I didn't bother to finish my
709
00:42:22 --> 00:42:26
sentence, but here I want to be
careful and have you realize
710
00:42:26 --> 00:42:28
that there's something
a little more to do.
711
00:42:28 --> 00:42:31
First of all we have the, the
first one is no problem.
712
00:42:31 --> 00:42:35
That's this.
713
00:42:35 --> 00:42:39
The second one actually
is not too bad either.
714
00:42:39 --> 00:42:45
This is, by the advanced
guessing method, my favorite
715
00:42:45 --> 00:42:48
method, something like the
logarithm, because that's
716
00:42:48 --> 00:42:50
what's going to appear
in the denominator.
717
00:42:50 --> 00:42:51
And then, if you differentiate
this, you're going
718
00:42:51 --> 00:42:53
to get 2x over this.
719
00:42:53 --> 00:42:55
But here we have 1/2.
720
00:42:55 --> 00:42:59
So altogether it's 1/4 of this.
721
00:42:59 --> 00:43:02
So I fixed the
coefficient here.
722
00:43:02 --> 00:43:06
And then the last one, you have
to think back to some level of
723
00:43:06 --> 00:43:10
memorization here and remember
that this is 1/2
724
00:43:10 --> 00:43:15
the arc tangent.
725
00:43:15 --> 00:43:20
STUDENT: [INAUDIBLE]
726
00:43:20 --> 00:43:22
PROFESSOR: Why did I go to 1/4?
727
00:43:22 --> 00:43:26
Because in disguise, for this
guy, I was thinking d / dx of
728
00:43:26 --> 00:43:35
ln (x ^2 + 1) = 2x / x^2 + 1.
729
00:43:35 --> 00:43:39
Because it's the derivative
of this divided by itself.
730
00:43:39 --> 00:43:46
This is the derivative
of ln u is u ' / u.
731
00:43:46 --> 00:43:50
Ln u' = u' / u.
732
00:43:50 --> 00:43:54
That was what I applied.
733
00:43:54 --> 00:43:58
And what I had was 1/2, so I
need a total of 1/4 to cancel.
734
00:43:58 --> 00:44:06
So 2/4 is 1/2.
735
00:44:06 --> 00:44:09
Now I've got to get you out
of one more deep hole.
736
00:44:09 --> 00:44:12
And I'm going to save the
general pattern for next time.
737
00:44:12 --> 00:44:27
But I do want to
clarify one thing.
738
00:44:27 --> 00:44:30
So let's handle this thing.
739
00:44:30 --> 00:44:34
What if the degree of P
is bigger than or equal
740
00:44:34 --> 00:44:39
to the degree of Q.
741
00:44:39 --> 00:44:42
That's the thing that
I claimed was easier.
742
00:44:42 --> 00:44:45
And I'm going to describe
to you how it's done.
743
00:44:45 --> 00:44:49
Now, in analogy, with
numbers you might call
744
00:44:49 --> 00:44:55
this an improper fraction.
745
00:44:55 --> 00:44:59
That's the thing that should
echo in your mind when
746
00:44:59 --> 00:45:01
you're thinking about this.
747
00:45:01 --> 00:45:04
And I'm just going to
do it by example here.
748
00:45:04 --> 00:45:08
Let's see., I cooked up an
example so that I don't
749
00:45:08 --> 00:45:11
make an arithmetistic
mistake along the way.
750
00:45:11 --> 00:45:16
So there are two or three
steps that I need to explain.
751
00:45:16 --> 00:45:18
So here's an example.
752
00:45:18 --> 00:45:21
The denominator's degree 2,
the numerator is degree 3.
753
00:45:21 --> 00:45:24
It well exceeds, so
there's a problem here.
754
00:45:24 --> 00:45:27
Our method is not
going to work.
755
00:45:27 --> 00:45:32
And the first step that
I want to carry out
756
00:45:32 --> 00:45:36
is to reverse Step 1.
757
00:45:36 --> 00:45:38
That is, I don't want the
factorization for what
758
00:45:38 --> 00:45:39
I'm going to do next.
759
00:45:39 --> 00:45:42
I want it multiplied out.
760
00:45:42 --> 00:45:48
That means I have to multiply
through, so I get x ^2 + x - 2.
761
00:45:48 --> 00:45:52
I'm back to the
starting place here.
762
00:45:52 --> 00:45:57
And now, the next thing that
I'm going to do is, I'm
763
00:45:57 --> 00:46:01
going to use long division.
764
00:46:01 --> 00:46:04
That's how you convert
an improper fraction
765
00:46:04 --> 00:46:07
to a proper fraction.
766
00:46:07 --> 00:46:12
This is something you were
supposed to learn in, I don't
767
00:46:12 --> 00:46:16
know, Grade 4, I know.
768
00:46:16 --> 00:46:23
Grade 3, Grade 4, Grade
5, Grade 6, etc.
769
00:46:23 --> 00:46:27
So here's how it works in
the case of polynomials.
770
00:46:27 --> 00:46:31
It's kind of amusing.
771
00:46:31 --> 00:46:37
So we're dividing this
polynomial into that one.
772
00:46:37 --> 00:46:41
And so the quotient to
first order here is x.
773
00:46:41 --> 00:46:44
That is, that's going to
match the top terms.
774
00:46:44 --> 00:46:48
So I get x^3 + x ^2 - 2x.
775
00:46:48 --> 00:46:50
That's the product.
776
00:46:50 --> 00:46:51
And now I subtract.
777
00:46:51 --> 00:46:53
And it cancels.
778
00:46:53 --> 00:46:58
So we get here - x ^2 + 2x.
779
00:46:58 --> 00:47:01
That's the difference.
780
00:47:01 --> 00:47:04
And now I need to divide
this next term in.
781
00:47:04 --> 00:47:08
And I need a - 1.
782
00:47:08 --> 00:47:14
So - 1 times this
is - x ^2 - x + 2.
783
00:47:14 --> 00:47:16
And I subtract.
784
00:47:16 --> 00:47:17
And the x^2's cancel.
785
00:47:17 --> 00:47:24
And here I get + 3x - 2.
786
00:47:24 --> 00:47:27
Now, this thing has a name.
787
00:47:27 --> 00:47:30
This is called the quotient.
788
00:47:30 --> 00:47:33
And this thing also has a name.
789
00:47:33 --> 00:47:39
This is called the remainder.
790
00:47:39 --> 00:47:43
And now I'll show you how
it works by sticking it
791
00:47:43 --> 00:47:44
into the answer here.
792
00:47:44 --> 00:47:47
The quotient is x - 1.
793
00:47:47 --> 00:47:51
And the remainder is,
let's get down there.
794
00:47:51 --> 00:47:58
3x - 2 / x ^2 + x - 2.
795
00:47:58 --> 00:48:03
So the punchline here
is that this thing is
796
00:48:03 --> 00:48:05
easy to integrate.
797
00:48:05 --> 00:48:08
This is easy.
798
00:48:08 --> 00:48:17
And this one, you can use, now
you can use cover-up, The
799
00:48:17 --> 00:48:18
method that we had before.
800
00:48:18 --> 00:48:21
Because the degree of the
numerator is now below the
801
00:48:21 --> 00:48:22
degree of the denominator.
802
00:48:22 --> 00:48:25
It's now first degree and
this is second degree.
803
00:48:25 --> 00:48:27
What you can't do
is use cover-up to
804
00:48:27 --> 00:48:29
start out with here.
805
00:48:29 --> 00:48:32
That will give you
the wrong answer.
806
00:48:32 --> 00:48:37
So that's the end for today,
and see you next time.
807
00:48:37 --> 00:48:37