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PROFESSOR: Now, to start out
today we're going to finish

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up what we did last time.
 

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Which has to do with
partial fractions.

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I told you how to do partial
fractions in several special

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cases and everybody was trying
to figure out what the

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general picture was.
 

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But I'd like to lay that out.
 

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I'll still only do
it for an example.

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But it will be somehow a bigger
example so that you can see

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what the general pattern is.
 

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Partial fractions, remember,
is a method for breaking up

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so-called rational functions.
 

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Which are ratios
of polynomials.

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And it shows you that you
can always integrate them.

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That's really the theme here.
 

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And this is what's reassuring
is that it always works.

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That's really the bottom line.
 

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And that's good because there
are a lot of integrals that

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don't have formulas
and these do.

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It always works.
 

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But, maybe with lots of help.
 

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So maybe slowly.
 

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Now, there's a little bit of
bad news, and I have to be

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totally honest and tell you
what all the bad news is.

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Along with the good news.
 

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The first step, which maybe I
should be calling Step 0, I

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had a Step 1, 2 and 3 last
time, is long division.

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That's the step where you take
your polynomial divided by your

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other polynomial, and you find
the quotient plus

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some remainder.
 

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And you do that by
long division.

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And the quotient is easy to
take the antiderivative of up

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because it's just a polynomial.
 

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And the key extra property here
is that the degree of the

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numerator now over here, this
remainder, is strictly less

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than the degree of
the denominator.

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So that you can do
the next step.

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Now, the next step which I
called Step 1 last time, that's

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great imagination, it's right
after Step 0, Step 1 was to

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factor the denominator.
 

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And I'm going to illustrate by
example what the setup is here.

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I don't know maybe,
we'll do this.

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Some polynomial here,
maybe cube this one.

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So here I've factored
the denominator.

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That's what I called
Step 1 last time.

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Now, here's the first
piece of bad news.

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In reality, if somebody gave
you a multiplied out degree,

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whatever polynomial here,
you would be very hard

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pressed to factor it.
 

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A lot of them are extremely
difficult to factor.

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And so that's something
you would have to give

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to a machine to do.
 

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And it's just basically
a hard problem.

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So obviously, we're only
going to give you ones

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that you can do by hand.
 

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So very low degree examples.
 

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And that's just the way it is.
 

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So this is really a hard step
in disguise, in real life.

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Anyway, we're just going
to take it as given.

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And we have this numerator,
which is of degree less

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than the denominator.
 

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So let's count up what
its degree has to be.

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This is 4 + 2 + 6.
 

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So this is degree 4 + 2 + 6.
 

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I added that up because this is
degree 4, this is degree 2 and

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(x ^2) ^3 is the 6th power.
 

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So all together it's
this, which is 12.

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And so this thing is
of degree <= 11.

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All the way up to degree 11,
that's the possibilities

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for the numerator here.
 

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Now, the extra information
that I want to impart right

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now, is just this setup.
 

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Which I called
Step 2 last time.

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And the setup is this.
 

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Now, it's going to take
us a while to do this.

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We have this factor here.
 

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We have another factor.
 

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We have another term,
with the square.

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We have another term
with the cube.

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We have another term
with the fourth power.

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So this is what's going
to happen whenever you

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have linear factors.
 

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You'll have a collection
of terms like this.

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So you have four constants
to take care of.

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Now, with a quadratic in the
denominator, you need a linear

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function in the numerator.
 

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So that's, if you like, B0
x + C0 divided by this

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quadratic term here.
 

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And what I didn't show you last
time was how you deal with

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higher powers of
quadratic terms.

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So when you have a quadratic
term, what's going to happen

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is you're going to take
that first factor here.

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Just the way you
did in this case.

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But then you're going to
have to do the same thing

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with the next power.
 

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Now notice, just as in the
case of this top row, I

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have just a constant here.
 

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And even though I increased the
degree of the denominator I'm

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not increasing the numerator.
 

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It's staying just a constant.
 

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It's not linear up here.
 

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It's better than that.
 

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It's just a constant.
 

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And here it stayed a constant.
 

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And here I stayed a constant.
 

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Similarly here, even though I'm
increasing the degree of the

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denominator, I'm leaving the
numerator, the form of

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the numerator, alone.
 

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It's just a linear factor
and a linear factor.

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So that's the key
to this pattern.

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I don't have quite as
jazzy a song on mine.

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So this is so long that it
runs off the blackboard here.

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So let's continue
it on the next.

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We've got this B2 x +
C2 - sorry, B3 x +

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C3 / (x ^2 + 4) ^3.
 

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I guess I have room
for it over here.

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I'm going to talk about
this in just a second.

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Alright, so here's the pattern.
 

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Now, let me just do a
count of the number of

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unknowns we have here.
 

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The number of unknowns that
we have here is 1, 2, 3, 4,

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5, 6, 7, 8, 9, 10, 11, 12.
 

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That 12 is no coincidence.
 

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That's the degree
of the polynomial.

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And it's the number of
unknowns that we have.

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And it's the number of
degrees of freedom in a

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polynomial of degree 11.
 

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If you have all these free
coefficients here, you have

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the coefficient x^ 0, x ^
1, all the way up to x^ 11.

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And 0 through 11 is 12
different coefficients.

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And so this is a very
complicated system of

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equations for unknowns.
 

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This is twelve equations
for twelve unknowns.

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So I'll get rid of
this for a second.

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So we have twelve equations,
twelve unknowns.

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So that's the other bad news.
 

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Machines handle this very
well, but human beings have

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a little trouble with 12.
 

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Now, the cover-up method
works very neatly and

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picks out this term here.
 

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But that's it.
 

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So it reduces it
to an 11 by 11.

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You'll be able to evaluate
this in no time.

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But that's it.
 

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That's the only simplification
of your previous method.

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We don't have a
method for this.

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So I'm just showing what the
whole method looks like but

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really you'd have to have a
machine to implement this once

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it gets to be any size at all.
 

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Yeah, question.
 

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STUDENT: [INAUDIBLE]
 

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PROFESSOR: It's one big
equation, but it's a

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polynomial equation.
 

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So there's an equation, there's
this function r ( x) = a11

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x^2 11 + a10 x^ 10... and
these things are known.

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This is a known
expression here.

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And then when you
cross-multiply on the other

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side, what you have is, well,
it's a1 times, if you cancel

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this denominator with that,
you're going to get (x + 2) ^3

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( x ^2 + 2x + 3)( x^2 + 4)^3 +
the term for a2, etc.

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00:10:08 --> 00:10:10
It's a monster equation.
 

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And then to separate it out
into separate equations, you

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take the coefficient on x^
11th, x ^ 10, ... all

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the way down to x^ 0.
 

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And all told, that means there
are a total of 12 equations.

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11 through 0 is 12 equations.
 

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Yeah, another question.
 

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STUDENT: [INAUDIBLE]
 

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PROFESSOR: Should I write
down rest of this?

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STUDENT: [INAUDIBLE]
 

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PROFESSOR: Should you write
down all this stuff?

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Well, that's a good question.
 

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So you notice I didn't
write it down.

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Why didn't I write it down?
 

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Because it's incredibly long.
 

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In fact, you probably,
so how many pages of

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writing would this take?
 

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This is about a
page of writing.

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So just think of your machine,
how much time you want

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to spend on this.
 

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00:11:01 --> 00:11:05
So the answer is that you
have to be realistic.

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You're a human being,
not a machine.

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00:11:07 --> 00:11:10
And so there's certain things
that you can write down and

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00:11:10 --> 00:11:12
other things you should
attempt to write down.

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So do not do this at home.
 

194
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So that's the first
down-side to this method.

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It gets more and more
complicated as time goes on.

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The second down-side, I want to
point out to you, is what's

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00:11:28 --> 00:11:35
happening with the pieces.
 

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00:11:35 --> 00:11:42
So the pieces still
need to be integrated.

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We simplified this problem,
but we didn't get rid of it.

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00:11:48 --> 00:11:50
We still have the problem
of integrating the pieces.

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Now, some of the
pieces are very easy.

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This top row here, the
antiderivatives of these,

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you can just write down.
 

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By advanced guessing.
 

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00:12:01 --> 00:12:04
I'm going to skip over to the
most complicated one over here.

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00:12:04 --> 00:12:06
For one second here.
 

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00:12:06 --> 00:12:09
And what is it that you'd have
to deal with for that one.

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00:12:09 --> 00:12:13
You'd have to deal with,
for example, so e.g., for

209
00:12:13 --> 00:12:21
example, I need to deal
with the this guy.

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I've got to get this
antiderivative here.

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00:12:26 --> 00:12:29
Now, this one you're supposed
to be able to know.

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00:12:29 --> 00:12:30
So this is why I'm
mentioning this.

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00:12:30 --> 00:12:33
Because this kind of
ingredient is something

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00:12:33 --> 00:12:34
you already covered.
 

215
00:12:34 --> 00:12:35
And what is it?
 

216
00:12:35 --> 00:12:39
Well, you do this one by
advanced guessing, although

217
00:12:39 --> 00:12:42
you it as the method
of substitution.

218
00:12:42 --> 00:12:46
You realize that it's going
to be of the form (x^2 + 4)

219
00:12:46 --> 00:12:49
^ - 2, roughly speaking.
 

220
00:12:49 --> 00:12:51
And now we're going
to fix that.

221
00:12:51 --> 00:12:54
Because if you differentiate
it you get 2x (-2), that's

222
00:12:54 --> 00:12:56
- 4 (x)(x) times this.
 

223
00:12:56 --> 00:12:58
There's an x in the
numerator here.

224
00:12:58 --> 00:13:02
So it's - 1/4 of that
will fix the factor.

225
00:13:02 --> 00:13:06
And here's the answer
for that one.

226
00:13:06 --> 00:13:10
So that's one you can do.
 

227
00:13:10 --> 00:13:19
The second piece is this guy.
 

228
00:13:19 --> 00:13:20
This is the other piece.
 

229
00:13:20 --> 00:13:25
Now, this was the piece
that came from B3.

230
00:13:25 --> 00:13:27
This is the one
that came from B3.

231
00:13:27 --> 00:13:30
And this is the one
that's coming from C3.

232
00:13:30 --> 00:13:32
This is coming from C3.
 

233
00:13:32 --> 00:13:35
We still need to get
this one out there.

234
00:13:35 --> 00:13:37
So C3 times that will be the
correct answer, once we've

235
00:13:37 --> 00:13:40
found these numbers.
 

236
00:13:40 --> 00:13:44
So how do we do this?
 

237
00:13:44 --> 00:13:45
How's this one integrated?
 

238
00:13:45 --> 00:13:49
STUDENT: Trig substitution?
 

239
00:13:49 --> 00:13:51
PROFESSOR: Trig substitution.
 

240
00:13:51 --> 00:13:57
So the trig substitution
here is x = 2 tan u.

241
00:13:57 --> 00:14:00
Or 2 tan theta.
 

242
00:14:00 --> 00:14:03
And when you do that, there are
a couple of simplifications.

243
00:14:03 --> 00:14:06
Well, I wouldn't call
this a simplification.

244
00:14:06 --> 00:14:10
This is just the
differentiation formula.

245
00:14:10 --> 00:14:14
dx = 2 sec^2 u du.
 

246
00:14:14 --> 00:14:19
And then you have to plug in,
and you're using the fact that

247
00:14:19 --> 00:14:23
when you plug in the tan^2, 4
tan ^2 + 4, you'll

248
00:14:23 --> 00:14:24
get a sec ^2.
 

249
00:14:24 --> 00:14:32
So altogether, this
thing is, 2 sec^2 u du.

250
00:14:32 --> 00:14:40
And then there's a (4 sec^2
u) ^3, in the denominator.

251
00:14:40 --> 00:14:44
So that's what happens when
you change variables here.

252
00:14:44 --> 00:14:46
And now look, this
keeps on going.

253
00:14:46 --> 00:14:49
This is not the end
of the problem.

254
00:14:49 --> 00:14:50
Because what does
that simplify to?

255
00:14:50 --> 00:14:55
That is, let's see, it's
2/64, the integral

256
00:14:55 --> 00:14:58
of sec^6 and sec^2.
 

257
00:14:58 --> 00:15:00
That's the same as cos^4.
 

258
00:15:00 --> 00:15:04
 


259
00:15:04 --> 00:15:06
And now, you did a trig
substitution but you still

260
00:15:06 --> 00:15:11
have a trig integral.
 

261
00:15:11 --> 00:15:15
The trig integral now,
there's a method for this.

262
00:15:15 --> 00:15:19
The method for this is when
it's an even power, you have to

263
00:15:19 --> 00:15:22
use the double angle formula.
 

264
00:15:22 --> 00:15:31
So that's this guy here.
 

265
00:15:31 --> 00:15:33
And you're still not done.
 

266
00:15:33 --> 00:15:35
You have to square
this thing out.

267
00:15:35 --> 00:15:37
And then you'll still
get a cos^2 2u.

268
00:15:37 --> 00:15:38
And it keeps on going.
 

269
00:15:38 --> 00:15:41
So this thing goes
on for a long time.

270
00:15:41 --> 00:15:43
But I'm not even going
to finish this, but I

271
00:15:43 --> 00:15:44
just want to show you.
 

272
00:15:44 --> 00:15:47
The point is, we're not
showing you how to do

273
00:15:47 --> 00:15:48
any complicated problem.
 

274
00:15:48 --> 00:15:50
We're just showing you all
the little ingredients.

275
00:15:50 --> 00:15:53
And you have to string them
together a long, long, long

276
00:15:53 --> 00:15:56
process to get to the final
answer of one of

277
00:15:56 --> 00:15:57
these questions.
 

278
00:15:57 --> 00:16:07
So it always works,
but maybe slowly.

279
00:16:07 --> 00:16:10
By the way, there's
even another horrible

280
00:16:10 --> 00:16:13
thing that happens.
 

281
00:16:13 --> 00:16:22
Which is, if you handle this
guy here, what's the technique.

282
00:16:22 --> 00:16:25
This is another technique
that you learned, supposedly

283
00:16:25 --> 00:16:28
within the last few days.
 

284
00:16:28 --> 00:16:30
Completing the square.
 

285
00:16:30 --> 00:16:39
So this, it turns out, you
have to rewrite it this way.

286
00:16:39 --> 00:16:41
And then the evaluation is
going to be expressed in

287
00:16:41 --> 00:16:44
terms of, I'm going
to jump to the end.

288
00:16:44 --> 00:16:49
It's going to turn out to be
expressed in terms of this.

289
00:16:49 --> 00:16:53
That's what will eventually
show up in the formula.

290
00:16:53 --> 00:16:58
And not only that, but if you
deal with ones involving x as

291
00:16:58 --> 00:17:06
well, you'll also need to deal
with something like ln of

292
00:17:06 --> 00:17:09
this denominator here.
 

293
00:17:09 --> 00:17:13
So all of these things
will be involved.

294
00:17:13 --> 00:17:16
So now, the last message that
I have for you is just this.

295
00:17:16 --> 00:17:18
This thing is very complicated.
 

296
00:17:18 --> 00:17:20
We're certainly never going
to ask you to do it.

297
00:17:20 --> 00:17:23
But you should just be aware
that this level of complexity,

298
00:17:23 --> 00:17:26
we are absolutely stuck
with in this problem.

299
00:17:26 --> 00:17:32
And the reason why we're stuck
with it is that this is what

300
00:17:32 --> 00:17:36
the formulas look
like in the end.

301
00:17:36 --> 00:17:39
If the answers look like
this, the formulas have

302
00:17:39 --> 00:17:41
to be this complicated.
 

303
00:17:41 --> 00:17:43
If you differentiate this, you
get your polynomial, your

304
00:17:43 --> 00:17:44
ratio of polynomials.
 

305
00:17:44 --> 00:17:46
If you differentiate this, you
get some ratio of polynomials.

306
00:17:46 --> 00:17:48
These are the things that
come up when you take

307
00:17:48 --> 00:17:51
antiderivatives of those
rational functions.

308
00:17:51 --> 00:17:56
So we're just stuck
with these guys.

309
00:17:56 --> 00:17:58
And so don't let it
get to you too much.

310
00:17:58 --> 00:17:59
I mean, it's not so bad.
 

311
00:17:59 --> 00:18:02
In fact, there are computer
programs that will do this

312
00:18:02 --> 00:18:03
for you anytime you want.
 

313
00:18:03 --> 00:18:05
And you just have to be
not intimidated by them.

314
00:18:05 --> 00:18:10
They're like other functions.
 

315
00:18:10 --> 00:18:20
OK, that's it for the general
comments on partial fractions.

316
00:18:20 --> 00:18:24
Now we're going to change
subjects to our last technique.

317
00:18:24 --> 00:18:25
This is one more technical
thing to get you

318
00:18:25 --> 00:18:27
familiar with functions.
 

319
00:18:27 --> 00:18:32
And this technique is called
integration by parts.

320
00:18:32 --> 00:18:35
Please, just because its name
sort of sounds like partial

321
00:18:35 --> 00:18:37
fractions, don't think
it's the same thing.

322
00:18:37 --> 00:18:38
It's totally different.
 

323
00:18:38 --> 00:18:44
It's not the same.
 

324
00:18:44 --> 00:19:06
So this one is called
integration by parts.

325
00:19:06 --> 00:19:09
Now, unlike the previous case,
where I couldn't actually

326
00:19:09 --> 00:19:13
justify to you that the
linear algebra always works.

327
00:19:13 --> 00:19:14
I claimed it worked, but I
wasn't able to prove it.

328
00:19:14 --> 00:19:17
That's a complicated
theorem which I'm not

329
00:19:17 --> 00:19:19
able to do in this class.
 

330
00:19:19 --> 00:19:22
Here I can explain to you
what's going on with

331
00:19:22 --> 00:19:24
integration by parts.
 

332
00:19:24 --> 00:19:26
It's just the fundamental
theorem of calculus, if

333
00:19:26 --> 00:19:30
you like, coupled with
the product formula.

334
00:19:30 --> 00:19:33
Sort of unwound and
read in reverse.

335
00:19:33 --> 00:19:35
And here's how that works.
 

336
00:19:35 --> 00:19:38
If you take the product of two
functions and you differentiate

337
00:19:38 --> 00:19:40
them, then we know that the
product rule says that

338
00:19:40 --> 00:19:43
this is u' v + uv'.
 

339
00:19:45 --> 00:19:50
And now I'm just going to
rearrange in the following way.

340
00:19:50 --> 00:19:53
I'm going to solve for uv'.
 

341
00:19:53 --> 00:19:54
That is, this term here.
 

342
00:19:54 --> 00:19:56
So what is this term?
 

343
00:19:56 --> 00:19:58
It's this other term, (uv)'.
 

344
00:19:59 --> 00:20:04
Minus the other piece.
 

345
00:20:04 --> 00:20:08
So I just rewrote
this equation.

346
00:20:08 --> 00:20:10
And now I'm going
to integrate it.

347
00:20:10 --> 00:20:11
So here's the formula.
 

348
00:20:11 --> 00:20:16
The integral of the left-hand
side is equal to the integral

349
00:20:16 --> 00:20:17
of the right-hand side.
 

350
00:20:17 --> 00:20:19
Well when I integrate a
derivative, of I get back

351
00:20:19 --> 00:20:21
the function itself.
 

352
00:20:21 --> 00:20:27
That's the fundamental theorem.
 

353
00:20:27 --> 00:20:27
So this it.
 

354
00:20:27 --> 00:20:30
Sorry, I missed the dx,
which is important.

355
00:20:30 --> 00:20:32
I apologize.
 

356
00:20:32 --> 00:20:35
Let's put that in there.
 

357
00:20:35 --> 00:20:41
So this is the integration
by parts formula.

358
00:20:41 --> 00:20:46
I'm going to write it one more
time with the limits stuck in.

359
00:20:46 --> 00:20:49
It's also written this
way, when you have a

360
00:20:49 --> 00:21:02
definite integral.
 

361
00:21:02 --> 00:21:13
Just the same formula,
written twice.

362
00:21:13 --> 00:21:18
Alright, now I'm going to
show you how it works

363
00:21:18 --> 00:21:24
on a few examples.
 

364
00:21:24 --> 00:21:29
And I have to give you a
flavor for how it works.

365
00:21:29 --> 00:21:34
But it'll grow as we get
more and more experience.

366
00:21:34 --> 00:21:40
The first example that I'm
going to take is one that looks

367
00:21:40 --> 00:21:43
intractable on the face of it.
 

368
00:21:43 --> 00:21:49
Which is the integral
of ln x dx.

369
00:21:49 --> 00:21:52
Now, it looks like there's sort
of nothing we can do with this.

370
00:21:52 --> 00:21:55
And we don't know what
the solution is.

371
00:21:55 --> 00:21:59
However, I claim that if we fit
it into this form, we can

372
00:21:59 --> 00:22:03
figure out what the integral
is relatively easily.

373
00:22:03 --> 00:22:07
By some little magic of
cancellation it happens.

374
00:22:07 --> 00:22:08
The idea is the following.
 

375
00:22:08 --> 00:22:13
If I consider this function to
be u, then what's going to

376
00:22:13 --> 00:22:17
appear on the other side in the
integrated form is the function

377
00:22:17 --> 00:22:22
u', which is -- so, if
you like, u = ln x.

378
00:22:22 --> 00:22:25
So u' = 1 / x.
 

379
00:22:25 --> 00:22:28
Now, 1 / x is a more manageable
function than ln x.

380
00:22:28 --> 00:22:31
What we're using is that when
we differentiate the function,

381
00:22:31 --> 00:22:33
it's getting nicer.
 

382
00:22:33 --> 00:22:36
It's getting more
tractable for us.

383
00:22:36 --> 00:22:39
In order for this to fit
into this pattern, however,

384
00:22:39 --> 00:22:44
I need a function v.
 

385
00:22:44 --> 00:22:48
So what in the world am I
going to put here for v?

386
00:22:48 --> 00:22:51
The answer is, well, dx is
almost the right answer.

387
00:22:51 --> 00:22:53
The answer turns out to be x.
 

388
00:22:53 --> 00:23:01
And the reason is that
that makes v' = 1.

389
00:23:01 --> 00:23:02
It makes v' = 1.
 

390
00:23:02 --> 00:23:05
So that means that this
is u, but it's also uv'.

391
00:23:05 --> 00:23:11
Which was what I had on
the left-hand side.

392
00:23:11 --> 00:23:12
So it's both u and uv'.
 

393
00:23:13 --> 00:23:14
So this is the setup.
 

394
00:23:14 --> 00:23:19
And now all I'm going to do is
read off what the formula says.

395
00:23:19 --> 00:23:23
What it says is, this
is equal to u v.

396
00:23:23 --> 00:23:25
So u is this and v is that.
 

397
00:23:25 --> 00:23:32
So it's x ln x, minus, so
that again, this is uv.

398
00:23:32 --> 00:23:37
Except in the other order, vu.
 

399
00:23:37 --> 00:23:38
And then I'm integrating,
and what do I have

400
00:23:38 --> 00:23:40
to integrate? u ' v.
 

401
00:23:40 --> 00:23:45
So look up there. u' v with
a minus sign here. u'

402
00:23:45 --> 00:23:47
= 1 / x, and v = x.
 

403
00:23:47 --> 00:23:49
So it's 1 / x, that's u'.
 

404
00:23:50 --> 00:23:56
And here is x, that's v, dx.
 

405
00:23:56 --> 00:23:58
Now, that one is
easy to integrate.

406
00:23:58 --> 00:24:00
Because (1/x) x = 1.
 

407
00:24:00 --> 00:24:07
And the integral of 1 dx
is x + c, if you like.

408
00:24:07 --> 00:24:10
So the antiderivative
of 1 is x.

409
00:24:10 --> 00:24:11
And so here's our answer.
 

410
00:24:11 --> 00:24:34
Our answer is that this
is x ln x - x + c.

411
00:24:34 --> 00:24:37
I'm going to do two
more slightly more

412
00:24:37 --> 00:24:39
complicated examples.
 

413
00:24:39 --> 00:24:43
And then really, the main
thing is to get yourself

414
00:24:43 --> 00:24:44
used to this method.
 

415
00:24:44 --> 00:24:47
And there's no one
way of doing that.

416
00:24:47 --> 00:24:49
Just practice makes perfect.
 

417
00:24:49 --> 00:24:53
And so we'll just do
a few more examples.

418
00:24:53 --> 00:24:55
And illustrate them.
 

419
00:24:55 --> 00:24:59
The second example that
I'm going to use is the

420
00:24:59 --> 00:25:03
integral of (ln x) ^2 dx.
 

421
00:25:03 --> 00:25:08
And this is just slightly
more recalcitrant.

422
00:25:08 --> 00:25:13
Namely, I'm going to
let u be (ln x)^2.

423
00:25:13 --> 00:25:17
 


424
00:25:17 --> 00:25:20
And again, v = to x.
 

425
00:25:20 --> 00:25:21
So that matches up here.
 

426
00:25:21 --> 00:25:23
That is, v' = 1.
 

427
00:25:23 --> 00:25:25
So this is u v'.
 

428
00:25:28 --> 00:25:30
So this thing is u v'.
 

429
00:25:31 --> 00:25:33
And then we'll just
see what happens.

430
00:25:33 --> 00:25:38
Now, the game that we get is
that when I differentiate the

431
00:25:38 --> 00:25:42
logarithm squared, I'm going
to to get something simpler.

432
00:25:42 --> 00:25:47
It's not going to win us
the whole battle, but

433
00:25:47 --> 00:25:49
it will get us started.
 

434
00:25:49 --> 00:25:50
So here we get u'.
 

435
00:25:51 --> 00:25:56
And that's 2 ln x ( 1 / x).
 

436
00:25:56 --> 00:26:00
Applying the chain rule.
 

437
00:26:00 --> 00:26:06
And so the formula is that this
is x (ln x)^2, minus the

438
00:26:06 --> 00:26:12
integral of, well it's u'
v, right, that's what I

439
00:26:12 --> 00:26:13
have to put over here.
 

440
00:26:13 --> 00:26:22
So u' = 2 ln x ( 1
/ x), and v = x.

441
00:26:22 --> 00:26:25
And so now, you notice
something interesting

442
00:26:25 --> 00:26:25
happening here.
 

443
00:26:25 --> 00:26:28
So let me just demarcate
this a little bit.

444
00:26:28 --> 00:26:34
And let you see what it
is that I'm doing here.

445
00:26:34 --> 00:26:36
So notice, this is
the same integral.

446
00:26:36 --> 00:26:38
So here we have x (ln x) ^2.
 

447
00:26:38 --> 00:26:41
We've already solve that part.
 

448
00:26:41 --> 00:26:43
But now know notice that the
1 / x and the x cancel.

449
00:26:43 --> 00:26:46
So we're back to
the previous case.

450
00:26:46 --> 00:26:49
We didn't win all the way,
but actually we reduced

451
00:26:49 --> 00:26:51
ourselves to this integral.
 

452
00:26:51 --> 00:26:56
To the integral of ln x,
which we already know.

453
00:26:56 --> 00:26:58
So here, I can copy that down.
 

454
00:26:58 --> 00:27:04
That's - 2 (x ln x -
x), and then I have to

455
00:27:04 --> 00:27:05
throw in a constant, c.
 

456
00:27:05 --> 00:27:07
And that's the end of
the problem here.

457
00:27:07 --> 00:27:10
That's it.
 

458
00:27:10 --> 00:27:26
So this piece, I got
from Example 1.

459
00:27:26 --> 00:27:35
Now, this illustrates a
principle which is a little bit

460
00:27:35 --> 00:27:40
more complicated than just the
one of integration by parts.

461
00:27:40 --> 00:27:44
Which is a sort of a general
principle which I'll call my

462
00:27:44 --> 00:27:48
Example 3, which is
something which is called a

463
00:27:48 --> 00:27:56
reduction formula.
 

464
00:27:56 --> 00:28:00
A reduction formula is a case
where we apply some rule and

465
00:28:00 --> 00:28:03
we figure out one of these
integrals in terms

466
00:28:03 --> 00:28:05
of something else.
 

467
00:28:05 --> 00:28:07
Which is a little bit simpler.
 

468
00:28:07 --> 00:28:10
And eventually we'll get down
to the end, but it may take us

469
00:28:10 --> 00:28:12
n steps from the beginning.
 

470
00:28:12 --> 00:28:18
So the example is
l(n x^ n) dx. .

471
00:28:18 --> 00:28:21
And the claim is that if I do
what I did in Example 2, to

472
00:28:21 --> 00:28:26
this case, I'll get a simpler
one which will involve

473
00:28:26 --> 00:28:28
the n - 1st power.
 

474
00:28:28 --> 00:28:30
And that way I can get
all the way back down

475
00:28:30 --> 00:28:32
to the final answer.
 

476
00:28:32 --> 00:28:34
So here's what happens.
 

477
00:28:34 --> 00:28:36
We take u as ln x^ n.
 

478
00:28:37 --> 00:28:40
This is the same discussion
as before, v = x.

479
00:28:40 --> 00:28:47
And then u' is n l(n
x) ^ n - 1( 1 / x).

480
00:28:47 --> 00:28:50
And v' is 1.
 

481
00:28:50 --> 00:28:52
And so the setup is similar.
 

482
00:28:52 --> 00:28:59
We have here x ( ln x)^
n minus the integral.

483
00:28:59 --> 00:29:05
And there's n times, it turns
out to be (ln x)^ n - 1.

484
00:29:05 --> 00:29:26
And then there's a 1 / x
and an x, which cancel.

485
00:29:26 --> 00:29:32
So I'm going to explain this
also abstractly a little bit

486
00:29:32 --> 00:29:35
just to show you what's
happening here.

487
00:29:35 --> 00:29:44
If you use the notation Fn (x)
is the integral of (ln x)^n dx,

488
00:29:44 --> 00:29:46
and we're going to forget
the constant here.

489
00:29:46 --> 00:29:52
Then the relationship that we
have here is that Fn (x) = n

490
00:29:52 --> 00:29:56
ln, I'm sorry, x (ln x)^ n.
 

491
00:29:56 --> 00:29:59
That's the first
term over here.

492
00:29:59 --> 00:30:03
Minus n times the
preceding one.

493
00:30:03 --> 00:30:07
This one here.
 

494
00:30:07 --> 00:30:11
And the idea is that
eventually we can get down.

495
00:30:11 --> 00:30:14
If we start with the nth one,
we have a formula that

496
00:30:14 --> 00:30:17
includes, so the reduction
is to the n - 1st.

497
00:30:17 --> 00:30:21
Then we can reduce to
the n - 2nd, and so on.

498
00:30:21 --> 00:30:23
Until we reduce to
the 1, the first 1.

499
00:30:23 --> 00:30:29
And then in fact we can even
go down to the 0th one.

500
00:30:29 --> 00:30:32
So this is the idea of
a reduction formula.

501
00:30:32 --> 00:30:36
And let me illustrate it
exactly in the context

502
00:30:36 --> 00:30:38
of Examples 1 and 2.
 

503
00:30:38 --> 00:30:44
So the first step would be
to evaluate the first one.

504
00:30:44 --> 00:30:48
Which is, if you
like, (ln x)^ 0 dx.

505
00:30:48 --> 00:30:52
That's very easy, that's x.
 

506
00:30:52 --> 00:31:01
And then F1 ( x) =
x ln x - F0 (x).

507
00:31:01 --> 00:31:03
Now, that's applying this rule.
 

508
00:31:03 --> 00:31:06
So let me just put
it in a box here.

509
00:31:06 --> 00:31:09
This is the method
of induction.

510
00:31:09 --> 00:31:13
Here's the rule.
 

511
00:31:13 --> 00:31:21
And I'm applying it for n = 1.
 

512
00:31:21 --> 00:31:23
I plugged in n = 1 here.
 

513
00:31:23 --> 00:31:32
So here, I have x ln
x ^ 1 - 1 ( F0 ( x).

514
00:31:32 --> 00:31:39
And that's what I put right
here, on the right-hand side.

515
00:31:39 --> 00:31:41
And that's going to generate
for me the formula that I

516
00:31:41 --> 00:31:44
want, which is x ln x - x.
 

517
00:31:44 --> 00:31:49
That's the answer to
this problem over here.

518
00:31:49 --> 00:31:51
This was Example 1.
 

519
00:31:51 --> 00:31:53
Notice I dropped the
constants because I can

520
00:31:53 --> 00:31:54
add them in at the end.
 

521
00:31:54 --> 00:31:57
So I'll put in
parentheses here, (+ c).

522
00:31:57 --> 00:32:01
That's what would happen at
the end of the problem.

523
00:32:01 --> 00:32:08
The next step, so that
was Example 1, and now

524
00:32:08 --> 00:32:12
Example 2 works more
or less the same way.

525
00:32:12 --> 00:32:14
I'm just summarizing what
I did on that blackboard

526
00:32:14 --> 00:32:16
right up here.
 

527
00:32:16 --> 00:32:21
The same thing, but in much
more compact notation.

528
00:32:21 --> 00:32:25
If I take F2 ( x), that's
going to be equal to x

529
00:32:25 --> 00:32:31
(ln x)^2 - 2 F1 ( x).
 

530
00:32:31 --> 00:32:41
Again, this is box for n = 2.
 

531
00:32:41 --> 00:32:46
And if I plug it in, what I'm
getting here is x (ln x) ^2

532
00:32:46 --> 00:32:49
- twice this stuff here.
 

533
00:32:49 --> 00:32:55
Which is right
here. x ln x - x.

534
00:32:55 --> 00:32:58
If you like, + c.
 

535
00:32:58 --> 00:33:07
So I'll leave the c off.
 

536
00:33:07 --> 00:33:12
So this is how reduction
formulas work in general.

537
00:33:12 --> 00:33:22
I'm going to give you one more
example of a reduction formula.

538
00:33:22 --> 00:33:30
So I guess we have to
call this Example 4.

539
00:33:30 --> 00:33:34
Let's be fancy, let's
make it the sine.

540
00:33:34 --> 00:33:35
No no, no, let's
be fancier still.

541
00:33:35 --> 00:33:38
Let's make it e^ x.
 

542
00:33:38 --> 00:33:48
So this would also work
for cosine x and sine x.

543
00:33:48 --> 00:33:50
The same sort of thing.
 

544
00:33:50 --> 00:33:53
And I should mention that
on your homework, you have

545
00:33:53 --> 00:33:54
to do it for cosine of x.
 

546
00:33:54 --> 00:33:56
I decided to change my mind
on the spur of the moment.

547
00:33:56 --> 00:33:58
I'm not going to do it for
cosine because you have

548
00:33:58 --> 00:34:00
to work it out on your
homework for cosine.

549
00:34:00 --> 00:34:03
In a later homework you'll
even do this case.

550
00:34:03 --> 00:34:05
So it's fine.
 

551
00:34:05 --> 00:34:07
You need the practice.
 

552
00:34:07 --> 00:34:10
OK, so how am I going
to do it this time.

553
00:34:10 --> 00:34:13
This is again, a
reduction formula.

554
00:34:13 --> 00:34:19
And the trick here is to pick
u to be this function here.

555
00:34:19 --> 00:34:20
And the reason is
the following.

556
00:34:20 --> 00:34:23
So it's very important to pick
which function is the u and

557
00:34:23 --> 00:34:25
which function is the v.
 

558
00:34:25 --> 00:34:27
That's the only decision you
have to make if you're going to

559
00:34:27 --> 00:34:30
apply integration by parts.
 

560
00:34:30 --> 00:34:33
When I pick this function as
the u, the advantage that I

561
00:34:33 --> 00:34:38
have is that u' is simpler.
 

562
00:34:38 --> 00:34:39
How is it simpler?
 

563
00:34:39 --> 00:34:42
It's simpler because
it's one degree down.

564
00:34:42 --> 00:34:45
So that's making
progress for us.

565
00:34:45 --> 00:34:49
On the other hand, this
function here is going to

566
00:34:49 --> 00:34:52
be what I'll use for v.
 

567
00:34:52 --> 00:34:55
And if I differentiated that,
if I did it the other way

568
00:34:55 --> 00:34:57
around and I differentiated
that, I would just get the

569
00:34:57 --> 00:34:58
same level of complexity.
 

570
00:34:58 --> 00:35:01
Differentiating e^x just
gives you back e ^ x.

571
00:35:01 --> 00:35:02
So that's boring.
 

572
00:35:02 --> 00:35:05
It doesn't make any
progress in this process.

573
00:35:05 --> 00:35:11
And so I'm going to instead
let v = e ^ x and,

574
00:35:11 --> 00:35:12
sorry this is v '.
 

575
00:35:12 --> 00:35:14
Make it v ' = e ^ x.
 

576
00:35:14 --> 00:35:15
And then v = e ^x.
 

577
00:35:15 --> 00:35:17
At least it isn't any
worse when I went

578
00:35:17 --> 00:35:20
backwards like that.
 

579
00:35:20 --> 00:35:28
So now, I have u and v ', and
now I get (x ^ n)( e ^ x).

580
00:35:28 --> 00:35:30
This again is u, and this is v.
 

581
00:35:30 --> 00:35:34
So it happens that v = t v ' so
it's a little confusing here.

582
00:35:34 --> 00:35:37
But this is the one
we're calling v '.

583
00:35:37 --> 00:35:38
And here's v.
 

584
00:35:38 --> 00:35:43
And now minus the integral
and I have here nx ^ n - 1.

585
00:35:43 --> 00:35:45
And I have here e ^ x.
 

586
00:35:45 --> 00:35:52
So this is u ' and
this is v dx.

587
00:35:52 --> 00:35:55
So this recurrence is
a new recurrence.

588
00:35:55 --> 00:35:57
And let me summarize it here.
 

589
00:35:57 --> 00:36:00
It's saying that Gn ( x)
should be the integral

590
00:36:00 --> 00:36:05
of (x ^ n)( e ^ x) dx.
 

591
00:36:05 --> 00:36:06
Again, I'm dropping the c.
 

592
00:36:06 --> 00:36:17
And then the reduction formula
is that Gn (x) = this

593
00:36:17 --> 00:36:25
expression here, (x ^ n)(
e ^ x) - n Gn - 1 (x).

594
00:36:25 --> 00:36:32
So here's our
reduction formula.

595
00:36:32 --> 00:36:38
And to illustrate this, if I
take G0 (x), if you think

596
00:36:38 --> 00:36:40
about it for a second that's
just, there's nothing here.

597
00:36:40 --> 00:36:44
The antiderivative of e ^ x,
that's going to be e ^ x,

598
00:36:44 --> 00:36:48
that getting started at
the real basement here.

599
00:36:48 --> 00:36:52
Again, as always, 0 is
my favorite number.

600
00:36:52 --> 00:36:52
Not 1.
 

601
00:36:52 --> 00:36:55
I always start with the
easiest one, if possible.

602
00:36:55 --> 00:37:00
And now G1, applying this
formula, is going to be

603
00:37:00 --> 00:37:06
equal to x e ^ x - G0 ( x).
 

604
00:37:06 --> 00:37:11
Which is just right, because
n is 1 and n - 1 is 0.

605
00:37:11 --> 00:37:17
And so that's just
x e ^ x - e^ x.

606
00:37:17 --> 00:37:20
So this is a very, very
fancy way of saying

607
00:37:20 --> 00:37:22
the following fact.
 

608
00:37:22 --> 00:37:32
I'll put it over on
this other board.

609
00:37:32 --> 00:37:44
Which is that the integral of
x e^ x dx = x e^ x - x + c.

610
00:37:44 --> 00:37:45
Yeah, question.
 

611
00:37:45 --> 00:37:50
STUDENT: [INAUDIBLE]
 

612
00:37:50 --> 00:37:53
PROFESSOR: The question
is, why is this true.

613
00:37:53 --> 00:37:54
Why is this statement true.
 

614
00:37:54 --> 00:37:56
Why is G 0 = e^ x.
 

615
00:37:56 --> 00:37:58
I did that in my head.
 

616
00:37:58 --> 00:38:02
What I did was, I first wrote
down the formula for G0.

617
00:38:02 --> 00:38:11
Which was G0 is equal to
the integral of e^ x dx.

618
00:38:11 --> 00:38:12
Because there's an x to
the 0 power in there,

619
00:38:12 --> 00:38:15
which is just 1.
 

620
00:38:15 --> 00:38:17
And then I know the
antiderivative of e ^ x.

621
00:38:17 --> 00:38:23
It's e ^x.
 

622
00:38:23 --> 00:38:31
STUDENT: [INAUDIBLE]
 

623
00:38:31 --> 00:38:33
PROFESSOR: How do you know
when this method will work?

624
00:38:33 --> 00:38:37
The answer is only
by experience.

625
00:38:37 --> 00:38:40
You must get practice
doing this.

626
00:38:40 --> 00:38:42
If you look in your
textbook, you'll see

627
00:38:42 --> 00:38:44
hints as to what to do.
 

628
00:38:44 --> 00:38:47
The other hint that I want to
say is that if you find that

629
00:38:47 --> 00:38:50
you have one factor in your
expression which when you

630
00:38:50 --> 00:38:52
differentiate it,
it gets easier.

631
00:38:52 --> 00:38:56
And when you antidifferentiate
the other half, it doesn't get

632
00:38:56 --> 00:38:58
any worse, then that's
when this method has

633
00:38:58 --> 00:39:01
a chance of helping.
 

634
00:39:01 --> 00:39:04
And there is there's
no general thing.

635
00:39:04 --> 00:39:08
The thing is, though, if you it
with x^ n (e^ x), x ^ n cosine

636
00:39:08 --> 00:39:11
x, especially on sine x, those
are examples where it works.

637
00:39:11 --> 00:39:15
This power of the ln.
 

638
00:39:15 --> 00:39:19
I'll give you er one
more example here.

639
00:39:19 --> 00:39:26
So this was G1 ( x), right.
 

640
00:39:26 --> 00:39:28
I'll give you one more
example in a second.

641
00:39:28 --> 00:39:29
Yeah.
 

642
00:39:29 --> 00:39:33
STUDENT: [INAUDIBLE]
 

643
00:39:33 --> 00:39:35
PROFESSOR: Thank you.
 

644
00:39:35 --> 00:39:38
There's a mistake here.
 

645
00:39:38 --> 00:39:39
That's bad.
 

646
00:39:39 --> 00:39:40
I was thinking in the
back of my head of the

647
00:39:40 --> 00:39:45
following formula.
 

648
00:39:45 --> 00:39:51
Which is another one
which we've just done.

649
00:39:51 --> 00:39:53
So these are the types of
formulas that you can get out

650
00:39:53 --> 00:39:57
of integration by parts.
 

651
00:39:57 --> 00:40:00
There's also another way of
getting these, which I'm not

652
00:40:00 --> 00:40:02
going to say anything about.
 

653
00:40:02 --> 00:40:04
Which is called
advance guessing.

654
00:40:04 --> 00:40:06
You guess in advance what the
form is, you differentiate

655
00:40:06 --> 00:40:08
it and you check.
 

656
00:40:08 --> 00:40:14
That does work too, with
many of these cases.

657
00:40:14 --> 00:40:21
I want to give you
an illustration.

658
00:40:21 --> 00:40:30
Just because you these
formulas are somewhat dry.

659
00:40:30 --> 00:40:34
So I want to give you just
at least one application.

660
00:40:34 --> 00:40:42
We're almost done with the
idea of these formulas.

661
00:40:42 --> 00:40:46
And we're going to get back now
to being able to handle lots

662
00:40:46 --> 00:40:47
more integrals than
we could before.

663
00:40:47 --> 00:40:50
And what's satisfying is that
now we can get numbers out

664
00:40:50 --> 00:40:54
instead of being stuck and
hamstrung with only

665
00:40:54 --> 00:40:55
a few techniques.
 

666
00:40:55 --> 00:40:57
Now we have all of the
techniques of integration

667
00:40:57 --> 00:40:59
that anybody has.
 

668
00:40:59 --> 00:41:02
And so we can do pretty
much anything we want

669
00:41:02 --> 00:41:04
that's possible to do.
 

670
00:41:04 --> 00:41:14
So here's, if you like, an
application that illustrates

671
00:41:14 --> 00:41:18
how integration by
parts can be helpful.

672
00:41:18 --> 00:41:26
And we're going to find the
volume of an exponential

673
00:41:26 --> 00:41:34
wine glass here.
 

674
00:41:34 --> 00:41:38
Again, don't try
this at home, but.

675
00:41:38 --> 00:41:40
So let's see.
 

676
00:41:40 --> 00:41:44
It's going to be this
beautiful guy here.

677
00:41:44 --> 00:41:46
I think.
 

678
00:41:46 --> 00:41:49
OK, so what's it going to be.
 

679
00:41:49 --> 00:41:52
This graph is going
to be y = e^ x.

680
00:41:52 --> 00:42:04
Then we're going to rotate
it around the y axis.

681
00:42:04 --> 00:42:10
And this level here
is the height y = 1.

682
00:42:10 --> 00:42:12
And the top, let's
say, is y = e.

683
00:42:12 --> 00:42:22
So that the horizontal here,
coming down, is x = 1.

684
00:42:22 --> 00:42:35
Now, there are two ways
to set up this problem.

685
00:42:35 --> 00:42:40
And so there are two methods.
 

686
00:42:40 --> 00:42:44
And this is also a good review
because, of course, we did

687
00:42:44 --> 00:42:46
this in the last unit.
 

688
00:42:46 --> 00:42:58
The two methods are horizontal
and vertical slices.

689
00:42:58 --> 00:43:00
Those are the two
ways we can do this.

690
00:43:00 --> 00:43:03
Now, if we do it with,
so let's start out with

691
00:43:03 --> 00:43:09
the horizontal ones.
 

692
00:43:09 --> 00:43:12
That's this shape here.
 

693
00:43:12 --> 00:43:15
And we're going like that.
 

694
00:43:15 --> 00:43:19
And the horizontal slices
mean that this little bit

695
00:43:19 --> 00:43:22
here is a thickness dy.
 

696
00:43:22 --> 00:43:24
And then we're going
to wrap that around.

697
00:43:24 --> 00:43:30
So this is going
to become a disk.

698
00:43:30 --> 00:43:34
This is the method of disks.
 

699
00:43:34 --> 00:43:35
And what's this distance here?
 

700
00:43:35 --> 00:43:37
Well, this place is x.
 

701
00:43:37 --> 00:43:40
And so the disk
has area pi x ^2.

702
00:43:40 --> 00:43:43
And we're going to add
up the thickness of the

703
00:43:43 --> 00:43:45
disks and we're going to
integrate from 1 to e.

704
00:43:45 --> 00:43:51
So here's our volume.
 

705
00:43:51 --> 00:43:54
And now we have one last little
item of business before we

706
00:43:54 --> 00:43:56
can evaluate this integral.
 

707
00:43:56 --> 00:43:58
And that is that we need to
know the relationship here on

708
00:43:58 --> 00:44:01
the curve, that y = e ^ x.
 

709
00:44:01 --> 00:44:07
So that means x = ln y.
 

710
00:44:07 --> 00:44:10
And in order to evaluate this
integral, we have to evaluate x

711
00:44:10 --> 00:44:13
correctly as a function of y.
 

712
00:44:13 --> 00:44:26
So that's the integral from 1
to e of (ln y)^2, times pi, dy.

713
00:44:26 --> 00:44:28
So now you see that this
is an integral that we

714
00:44:28 --> 00:44:30
did calculate already.
 

715
00:44:30 --> 00:44:34
And in fact, it's
sitting right here.

716
00:44:34 --> 00:44:37
Except with the variable x
instead of the variable y.

717
00:44:37 --> 00:44:44
So the answer, which we already
had, is this F2 ( y) here.

718
00:44:44 --> 00:44:47
So maybe I'll write
it that way.

719
00:44:47 --> 00:44:52
So this is F2 (y)
between 1 and e.

720
00:44:52 --> 00:45:00
And now let's figure
out what it is.

721
00:45:00 --> 00:45:02
It's written over there.
 

722
00:45:02 --> 00:45:15
It's y (ln y) ^2
- 2(y ln y - y).

723
00:45:15 --> 00:45:24
The whole thing
evaluated at 1e.

724
00:45:24 --> 00:45:29
And that is, if I plug
in e here, I get e.

725
00:45:29 --> 00:45:32
Except there's a factor
of pi there, sorry.

726
00:45:32 --> 00:45:36
Missed the pi factor.
 

727
00:45:36 --> 00:45:38
So there's an e here.
 

728
00:45:38 --> 00:45:43
And then I subtract off,
well, at 1 this is e - e.

729
00:45:43 --> 00:45:44
So it cancels.
 

730
00:45:44 --> 00:45:45
There's nothing left.
 

731
00:45:45 --> 00:45:50
And then at 1, I get ln 1 is
0, ln 1 is 0, there's only

732
00:45:50 --> 00:45:53
one term left, which is 2.
 

733
00:45:53 --> 00:45:55
So it's - 2.
 

734
00:45:55 --> 00:46:03
That's the answer.
 

735
00:46:03 --> 00:46:11
Now we get to compare that
with what happens if we

736
00:46:11 --> 00:46:15
do it the other way.
 

737
00:46:15 --> 00:46:19
So what's the vertical?
 

738
00:46:19 --> 00:46:31
So by vertical slicing,
we get shells.

739
00:46:31 --> 00:46:38
And that starts, that's
in the x variable.

740
00:46:38 --> 00:46:43
It starts at 0 and ends
at 1 and it's dx.

741
00:46:43 --> 00:46:46
And what are the shells?
 

742
00:46:46 --> 00:46:51
Well, the shells are, if I can
draw the picture again, they

743
00:46:51 --> 00:46:55
start, the top value is e.
 

744
00:46:55 --> 00:47:02
And the bottom value is, I need
a little bit of room for this.

745
00:47:02 --> 00:47:06
The bottom value is y.
 

746
00:47:06 --> 00:47:12
And then we have 2 pi x
is the circumference, as

747
00:47:12 --> 00:47:15
we sweep it around dx.
 

748
00:47:15 --> 00:47:18
So here's our new volume.
 

749
00:47:18 --> 00:47:23
Expressed in this
different way.

750
00:47:23 --> 00:47:26
So now I'm going to
plug in what this is.

751
00:47:26 --> 00:47:30
It's the integral from
0 to 1 of e - e ^ x.

752
00:47:30 --> 00:47:32
That's the formula for y.
 

753
00:47:32 --> 00:47:36
2 pi x dx.
 

754
00:47:36 --> 00:47:39
And what you see is that
you get the integral from

755
00:47:39 --> 00:47:45
0 to 1 of 2 pi e x dx.
 

756
00:47:45 --> 00:47:46
That's easy, right?
 

757
00:47:46 --> 00:47:51
That's just 2 pi e ( 1/2).
 

758
00:47:51 --> 00:47:54
This one is just the
area of a triangle.

759
00:47:54 --> 00:47:56
If I factor out the 2 pi e.
 

760
00:47:56 --> 00:48:03
And then the other piece is the
integral of 2 pi x e^ x dx.

761
00:48:03 --> 00:48:08
From 0 to 1.
 

762
00:48:08 --> 00:48:11
STUDENT: [INAUDIBLE]
 

763
00:48:11 --> 00:48:14
PROFESSOR: Are you asking me
whether I need an x ^2 here?

764
00:48:14 --> 00:48:15
I just evaluated the integral.
 

765
00:48:15 --> 00:48:17
I just did it geometrically.
 

766
00:48:17 --> 00:48:19
I said, this is the
area of a triangle.

767
00:48:19 --> 00:48:22
I didn't antidifferentiate
and evaluate it, I just

768
00:48:22 --> 00:48:23
told you the number.
 

769
00:48:23 --> 00:48:27
Because it's a
definite integral.

770
00:48:27 --> 00:48:33
So now, this one here, I can
read off from right up here.

771
00:48:33 --> 00:48:37
Above it, this is G1.
 

772
00:48:37 --> 00:48:42
So this is equal to,
let's check it out here.

773
00:48:42 --> 00:48:52
So this is pi e, right, - 2 pi
G1 ( x), evaluated at 0 and 1.

774
00:48:52 --> 00:48:54
So let's make sure that it's
the same as what we had before.

775
00:48:54 --> 00:48:59
It's pi e - 2 pi
times here's g1.

776
00:48:59 --> 00:49:03
So it's x e ^ x - e^ x.
 

777
00:49:03 --> 00:49:05
So at x = 1, that cancels.
 

778
00:49:05 --> 00:49:08
But at the bottom
end, it's e^ 0.

779
00:49:08 --> 00:49:12
So it's - 1 here.
 

780
00:49:12 --> 00:49:13
Is that right?
 

781
00:49:13 --> 00:49:13
Yep.
 

782
00:49:13 --> 00:49:17
So it's pi e - 2.
 

783
00:49:17 --> 00:49:21
It's the same.
 

784
00:49:21 --> 00:49:22
Question.
 

785
00:49:22 --> 00:49:28
STUDENT: [INAUDIBLE]
 

786
00:49:28 --> 00:49:33
PROFESSOR: From here to
here, is that the question?

787
00:49:33 --> 00:49:39
STUDENT: [INAUDIBLE]
 

788
00:49:39 --> 00:49:43
PROFESSOR: So the step here is
just the distributive law.

789
00:49:43 --> 00:49:46
This is e 2 pi x,
that's this term.

790
00:49:46 --> 00:49:49
And the other terms, the
minus sign is outside.

791
00:49:49 --> 00:49:51
The 2 pi I factored out.
 

792
00:49:51 --> 00:49:56
And the x and the e ^x stayed
inside the integral sign.

793
00:49:56 --> 00:49:59
Thank you.
 

794
00:49:59 --> 00:50:01
The correction is that
there was a missing

795
00:50:01 --> 00:50:03
minus sign, last time.
 

796
00:50:03 --> 00:50:13
When I integrated from 0 to 1 x
e^ x dx, I had a x e^ x - e^x.

797
00:50:13 --> 00:50:15
Evaluated at 0 and 1.
 

798
00:50:15 --> 00:50:18
And that's equal to + 1.
 

799
00:50:18 --> 00:50:21
I was missing this minus sign.
 

800
00:50:21 --> 00:50:30
The place where it came in was
in this wineglass example.

801
00:50:30 --> 00:50:39
We had the integral of
2 pi x e - e ^x dx.

802
00:50:39 --> 00:50:48
And that was 2 pi e integral of
x dx, from 0 to 1, - 2 pi,

803
00:50:48 --> 00:50:52
integral from 0 to
1 of x e^x dx.

804
00:50:52 --> 00:50:58
And then I worked this
out and it was pi e.

805
00:50:58 --> 00:51:03
And then this one was - 2 pi,
and what I wrote down was - 1.

806
00:51:03 --> 00:51:05
But there should have been
an extra minus sign there.

807
00:51:05 --> 00:51:08
So it's this.
 

808
00:51:08 --> 00:51:11
The final answer was
correct, but this minus

809
00:51:11 --> 00:51:13
sign was missing.
 

810
00:51:13 --> 00:51:16
Right there.
 

811
00:51:16 --> 00:51:20
So just, right there.
 

812
00:51:20 --> 00:51:23