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PROFESSOR: Today we're moving
on from theoretical things from

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the mean value theorem to the
introduction to what's going to

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occupy us for the whole rest of
the course, which

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is integration.
 

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So, in order to introduce that
subject, I need to introduce

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for you a new notation, which
is called differentials.

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I'm going to tell you what a
differential is, and we'll get

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used to using it over time.
 

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If you have a function which
is y = f ( x), then the

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differential of y is going to
be denoted dy, and it's by

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definition f' ( x ) dx.
 

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So here's the notation.
 

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And because y is really equal
to f, sometimes we also call

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it the differential of f.
 

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It's also called the
differential of f.

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That's the notation, and it's
the same thing as what happens

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if you formally just take this
dx, act like it's a number

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and divide it into dy.
 

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So it means the same thing
as this statement here.

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And this is more or less
the Leibniz, not Leibniz,

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interpretation of derivatives.
 

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Of a derivative as a ratio of
these so called differentials.

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It's a ratio of what are
known as infinitesimals.

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Now, this is kind of a vague
notion, this little bit here

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being an infinitesimal.
 

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It's sort of like an
infinitely small quantity.

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And Leibniz perfected the
idea of dealing with

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these intuitively.
 

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And subsequently,
mathematicians use

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them all the time.
 

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They're way more effective than
the notation that Newton used.

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You might think that notations
are a small matter, but they

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allow you to think much
faster, sometimes.

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When you have the right
names and the right

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symbols for everything.
 

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And in this case it made
it very big difference.

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Leibniz's notation was adopted
on the Continent and Newton

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dominated in Britain and, as a
result, the British fell behind

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by one or two hundred years in
the development of calculus.

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It was really a serious matter.
 

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So it's really well worth
your while to get used

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to this idea of ratios.
 

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And it comes up all over the
place, both in this class and

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also in multivariable calculus.
 

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It's used in many contexts.
 

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So first of all, just to
go a little bit easy.

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We'll illustrate it by its use
in linear approximations,

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which we've already done.
 

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The picture here, which we've
drawn a number of times, is

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that you have some function.
 

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And here's a value
of the function.

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And it's coming up like that.
 

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So here's our function.
 

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And we go forward a little
increment to a place which

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is dx further along.
 

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The idea of this notation is
that dx is going to replace

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the symbol delta x, which
is the change in x.

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And we won't think too hard
about - well, this is a small

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quantity, this is a small
quantity, we're not going to

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think too hard about
what that means.

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Now, similarly, if you see how
much we've gone up - well,

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this is kind of low, so
it's a small bit here.

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So this distance here
is, previously we

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called it delta y.
 

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But now we're just
going to call it dy.

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So dy replaces delta y.
 

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So this is the change in
level of the function.

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And we'll represent it
symbolically this way.

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Very frequently, this just
saves a little bit of notation.

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For the purposes of this, we'll
be doing the same things we did

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with delta x and delta y,
but this is the way that

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Leibniz thought of it.
 

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And he would just have
drawn it with this.

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So this distance here is dx
and this distance here is dy.

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So for an example of linear
approximation, we'll say what's

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64.1, say, to the 1/3 power,
approximately equal to?

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Now, I'm going to carry this
out in this new notation here.

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The function involved
is x ^ 1/3.

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And then it's a
differential, dy.

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Now, I want to use this
rule to get used to it.

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Because this is what we're
going to be doing all of today

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is, we're differentiatating, or
taking the differential of y.

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So that is going to be
just the derivative.

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That's 1/3 x ^ - 2/3 dx.
 

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And now I'm just going to fill
in exactly what this is.

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At x = 64, which is the natural
place close by where it's easy

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to do the evaluations, we have
y = 64 ^ 1/3, which is just 4.

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And how about dy?
 

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Well, so this is a little
bit more complicated.

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Put it over here.
 

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So dy = 1/3 ( 64) ^ - 2/3 dx.
 

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And that is (1/3 ) 1/16
dx, which is 1/48 dx.

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And now I'm going to work out
what 64 to the, whatever it is

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here, this strange fraction.
 

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I just want to be very
careful to explain to

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you one more thing.
 

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Which is that we're using x =
64, and so we're thinking of

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x + dx is going to be 64.1.
 

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So that means that dx
is going to be 1/10.

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So that's the increment
that we're interested in.

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And now I can carry out
the approximation.

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The approximation says that
64.1 ^ 1/3 is, well, it's

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approximately what I'm
going to call y + dy.

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Because really, the dy that I'm
determining here is determined

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by this linear relation.
dy = 1/48 dx.

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And so this is only
approximately true.

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Because what's really true
is that this = y + delta y.

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In our previous notation.
 

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So this is in disguise.
 

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What this is equal to.
 

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And that's the only
approximately equal to what

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the linear approximation
would give you.

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So, really, even though I wrote
dy is this increment here, what

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it really is if dx is exactly
that, is it's the amount it

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would go up if you went
straight up the tangent line.

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So I'm not going to do
that because that's

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not what people write.
 

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And that's not even
what they think.

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They're really thinking of
both dx and dy as being

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infinitesimally small.
 

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And here we're going to the
finite level and doing it.

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So this is just something you
have to live with, is a little

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ambiguity in this notation.
 

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This is the approximation.
 

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And now I can just calculate
these numbers here. y

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00:10:33 --> 00:10:36
at this value is 4.
 

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And dy, as I said, is 1/48 dx.
 

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And that turns out to be 4 +
1/480, because dx is 1/10.

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So that's approximately 4.002.
 

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And that's our approximation.
 

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Now, let's just compare it
to our previous notation.

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This will serve as a review
of, if you like, of

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linear approximation.
 

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00:11:35 --> 00:11:39
But what I want to emphasize
is that these things are

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supposed to be the same.
 

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Just that it's really
the same thing.

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It's just a different
notation for the same thing.

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I remind you the basic formula
for linear approximation is

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that f ( x ) is approximately
f ( a) + f' ( a )( x - a).

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And we're applying it in
the situation that a =

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64 and f(x) = x ^ 1/3.
 

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And so f ( a ), which is
f ( 64 ) is of course 4.

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00:12:27 --> 00:12:38
And f' ( a ), which is
a, (1/3)a ^ - 2/3,

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is in our case 1/16.
 

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No, 1/48.
 

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OK, that's the same
calculation as before.

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And then our relationship
becomes x ^ 1/3 is

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approximately equal to 4 +
1/48 ( x - a ), which is 64.

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So look, every single number
that I've written over here

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has a corresponding number
for this other method.

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And now if I plug in the value
we happen to want, which is the

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64.1, this would be 4 + 1/48 (
1/10 ), which is just the

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same thing we had before.
 

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So again, same answer.
 

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Same method, new notation.
 

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Well, now I get to use this
notation in a novel way.

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So again, here's the notation.
 

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This notation of differential.
 

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The way I'm going to use it is
in discussing something called

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antiderivative Again, this
is a new notation now.

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But it's also a new idea.
 

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It's one that we
haven't discussed yet.

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Namely, the notation that
I want to describe here

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is what's called the
integral of g ( x ) dx.

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And I'll denote that by a
function capital G (x).

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So it's, you start with a
function g ( x ) and you

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produce a function capital
G ( x ), which is called

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the antiderivative of G.
 

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Notice there's a differential
sitting in here.

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This symbol, this guy here,
is called an integral sign.

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Or an integral, or this whole
thing is called an integral.

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And another name for the
antiderivative of g is the

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indefinite integral of g.
 

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And I'll explain to you why
it's indefinite in just,

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very shortly here.
 

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00:16:04 --> 00:16:13
Well, so let's carry
out some examples.

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Basically what I'd like to do
is as many examples along the

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lines of all the derivatives
that we derived at the

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00:16:20 --> 00:16:21
beginning of the course.
 

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In other words, in principle
you want to be able

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to integrate as many
things as possible.

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We're going to start out with
the integral of sine x dx.

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00:16:34 --> 00:16:40
That's a function whose
derivative is sine x.

191
00:16:40 --> 00:16:44
So what function would that be?
 

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Cosine x, minus, right.
 

193
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It's - cos x.
 

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So - cos x differentiated
gives you sine x.

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So that is an
antiderivative of sine.

196
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And it satisfies this property.
 

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So this function, capital G ( x
) = - cos x, has the property

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00:17:10 --> 00:17:15
that its derivative is sine x.
 

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On the other hand, if
you differentiate a

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constant, you get 0.
 

201
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So this answer is what's
called indefinite.

202
00:17:25 --> 00:17:28
Because you can also
add any constant here.

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00:17:28 --> 00:17:33
And the same thing
will be true.

204
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So, c is constant.
 

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And as I said, the integral
is called indefinite.

206
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So that's an explanation
for this modifier in

207
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front of the integral.
 

208
00:17:46 --> 00:17:49
It's indefinite because we
actually didn't specify

209
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a single function.
 

210
00:17:50 --> 00:17:52
We don't get a single answer.
 

211
00:17:52 --> 00:17:54
Whenever you take the
antiderivative of something

212
00:17:54 --> 00:18:08
it's ambiguous up
to a constant.

213
00:18:08 --> 00:18:12
Next, let's do some other
standard functions

214
00:18:12 --> 00:18:13
from our repertoire.
 

215
00:18:13 --> 00:18:17
We have an integral
of (x ^ a)dx.

216
00:18:17 --> 00:18:20
Some power, the
integral of a power.

217
00:18:20 --> 00:18:24
And if you think about it, what
you should be differentiating

218
00:18:24 --> 00:18:27
is one power larger than that.
 

219
00:18:27 --> 00:18:33
But then you have to divide by
1 / a + 1, in order that the

220
00:18:33 --> 00:18:36
differentiatiation be correct.
 

221
00:18:36 --> 00:18:45
So this just is the fact that d
/ dx of x ^ a + 1, or maybe I

222
00:18:45 --> 00:18:46
should even say it this way.
 

223
00:18:46 --> 00:18:49
Maybe I'll do it in
differential notation. d ( x ^

224
00:18:49 --> 00:18:57
a + 1) = (a + 1) (x ^ a) dx.
 

225
00:18:57 --> 00:19:03
So if I divide that through
by a + 1, then I get

226
00:19:03 --> 00:19:06
the relation above.
 

227
00:19:06 --> 00:19:12
And because this is ambiguous
up to a constant, it could be

228
00:19:12 --> 00:19:20
any additional constant
added to that function.

229
00:19:20 --> 00:19:26
Now, the identity that I
wrote down below is correct.

230
00:19:26 --> 00:19:35
But this one is not always
correct What's the exception?

231
00:19:35 --> 00:19:38
Yeah. a equals
 

232
00:19:38 --> 00:19:38
STUDENT: 0.
 

233
00:19:38 --> 00:19:42
PROFESSOR: Negative 1.
 

234
00:19:42 --> 00:19:47
So this one is OK for all a.
 

235
00:19:47 --> 00:19:49
But this one fails
because we've divided

236
00:19:49 --> 00:19:52
by 0 when a = - 1.
 

237
00:19:52 --> 00:20:04
So this is only true when
a is not equal to - 1.

238
00:20:04 --> 00:20:07
And in fact, of course,
what's happening when a = 0,

239
00:20:07 --> 00:20:11
you're getting 0 when you
differentiate the constant.

240
00:20:11 --> 00:20:15
So there's a third case
that we have to carry out.

241
00:20:15 --> 00:20:25
Which is the exceptional case,
namely the integral of dx/x.

242
00:20:25 --> 00:20:32
And this time, if we just think
back to what are - so what

243
00:20:32 --> 00:20:34
we're doing is thinking
backwards here, which a

244
00:20:34 --> 00:20:38
very important thing to do
in math at all stages.

245
00:20:38 --> 00:20:41
We got all of our formulas, now
we're reading them backwards.

246
00:20:41 --> 00:20:49
And so this one, you
may remember, is ln x.

247
00:20:49 --> 00:20:53
The reason why I want to do
this carefully and slowly now,

248
00:20:53 --> 00:20:56
is right now I also want
to write the more standard

249
00:20:56 --> 00:20:58
form which is presented.
 

250
00:20:58 --> 00:21:01
So first of all, first we
have to add a constant.

251
00:21:01 --> 00:21:04
And please don't put
the parentheses here.

252
00:21:04 --> 00:21:10
The parentheses go there.
 

253
00:21:10 --> 00:21:14
But there's another formula
hiding in the woodwork

254
00:21:14 --> 00:21:16
here behind this one.
 

255
00:21:16 --> 00:21:19
Which is that you can also
get the correct formula

256
00:21:19 --> 00:21:20
when x is negative.
 

257
00:21:20 --> 00:21:27
And that turns out to
be this one here.

258
00:21:27 --> 00:21:31
So I'm treating the case,
x positive, as being

259
00:21:31 --> 00:21:34
something that you know.
 

260
00:21:34 --> 00:21:43
But let's check the
case, x negative.

261
00:21:43 --> 00:21:46
In order to check the case
x negative, I have to

262
00:21:46 --> 00:21:51
differentiate the logarithm
of the absolute value

263
00:21:51 --> 00:21:55
of x in that case.
 

264
00:21:55 --> 00:21:58
And that's the same thing,
again, for x negative as

265
00:21:58 --> 00:22:02
the derivative of the
lograrithm of negative x.

266
00:22:02 --> 00:22:08
That's the formula,
when x is negative.

267
00:22:08 --> 00:22:12
And if you carry that out, what
you get, maybe I'll put this

268
00:22:12 --> 00:22:20
over here, is, well,
it's the chain rule.

269
00:22:20 --> 00:22:23
It's (1 / -x) d/dx(-x).
 

270
00:22:23 --> 00:22:27
 


271
00:22:27 --> 00:22:30
So see that there are
two minus signs.

272
00:22:30 --> 00:22:32
There's a - x in the
denominator and then there's

273
00:22:32 --> 00:22:35
the derivative of -
x in the numerator.

274
00:22:35 --> 00:22:38
That's just - 1.
 

275
00:22:38 --> 00:22:39
This part is - 1.
 

276
00:22:39 --> 00:22:43
So this negative 1 over
negative x, which is 1 / x.

277
00:22:43 --> 00:22:53
So the negative signs cancel.
 

278
00:22:53 --> 00:22:59
If you just keep track of this
in terms of ln negative x and

279
00:22:59 --> 00:23:05
its graph, that's a function
that looks like this.

280
00:23:05 --> 00:23:08
For x negative.
 

281
00:23:08 --> 00:23:14
And its derivative
is 1 / x, I claim.

282
00:23:14 --> 00:23:18
And if you just look at it a
little bit carefully, you see

283
00:23:18 --> 00:23:23
that the slope is
always negative.

284
00:23:23 --> 00:23:23
Right?
 

285
00:23:23 --> 00:23:26
So here the slope is negative.
 

286
00:23:26 --> 00:23:30
So it's going to be
below the axis.

287
00:23:30 --> 00:23:32
And, in fact, it's getting
steeper and steeper

288
00:23:32 --> 00:23:34
negative as we go down.
 

289
00:23:34 --> 00:23:37
And it's getting less and less
negative as we go horizontally.

290
00:23:37 --> 00:23:41
So it's going like this, which
is indeed the graph of this

291
00:23:41 --> 00:23:43
function, for x negative.
 

292
00:23:43 --> 00:23:53
Again, x negative.
 

293
00:23:53 --> 00:23:56
So that's one other
standard formula.

294
00:23:56 --> 00:24:00
And very quickly, very
often, we won't put the

295
00:24:00 --> 00:24:01
absolute value signs.
 

296
00:24:01 --> 00:24:03
We'll only consider the
case x positive here.

297
00:24:03 --> 00:24:07
But I just want you to have the
tools to do it in case we want

298
00:24:07 --> 00:24:14
to use, we want to handle,
both positive and negative x.

299
00:24:14 --> 00:24:28
Now, let's do two
more examples.

300
00:24:28 --> 00:24:35
The integral of sec^2 x dx.
 

301
00:24:35 --> 00:24:38
These are supposed to get you
to remember all of your

302
00:24:38 --> 00:24:41
differentiatation formulas,
the standard ones.

303
00:24:41 --> 00:24:45
So this one, integral
of sec^2 dx is what?

304
00:24:45 --> 00:24:48
Tan x.
 

305
00:24:48 --> 00:24:50
And here we have + c, alright?
 

306
00:24:50 --> 00:24:54
And then the last one of,
a couple of, this type

307
00:24:54 --> 00:24:56
would be, let's see.
 

308
00:24:56 --> 00:25:04
I should do at least this one
here, square root of 1 - x ^2.

309
00:25:04 --> 00:25:05
This is another notation,
by the way, which is

310
00:25:05 --> 00:25:07
perfectly acceptable.
 

311
00:25:07 --> 00:25:10
Notice I've put the dx in the
numerator and the function

312
00:25:10 --> 00:25:13
in the denominator here.
 

313
00:25:13 --> 00:25:18
So this one turns
out to be sin-1 x.

314
00:25:18 --> 00:25:23
And, finally, let's see.
 

315
00:25:23 --> 00:25:28
About the integral
of dx / 1 + x ^2.

316
00:25:28 --> 00:25:41
That one is tan -1 x.
 

317
00:25:41 --> 00:25:43
For a little while, because
you're reading these things

318
00:25:43 --> 00:25:45
backwards and forwards,
you'll find this happens

319
00:25:45 --> 00:25:46
to you on exams.
 

320
00:25:46 --> 00:25:49
It gets slightly worse
for a little while.

321
00:25:49 --> 00:25:53
You will antidifferentiate when
you meant to differentiate.

322
00:25:53 --> 00:25:54
And you'll differentiate
when you're meant to

323
00:25:54 --> 00:25:57
antidifferentiate.
 

324
00:25:57 --> 00:26:00
Don't get too
frustrated by that.

325
00:26:00 --> 00:26:05
But eventually, you'll
get them squared away.

326
00:26:05 --> 00:26:11
And it actually helps to do
a lot of practice with

327
00:26:11 --> 00:26:15
antidifferentiations, or
integrations, as they're

328
00:26:15 --> 00:26:17
sometimes called.
 

329
00:26:17 --> 00:26:20
Because that will solidify
your remembering all of the

330
00:26:20 --> 00:26:25
differentiation formulas.
 

331
00:26:25 --> 00:26:31
So, last bit of information
that I want to emphasize before

332
00:26:31 --> 00:26:45
we go on some more complicated
examples is this It's obvious

333
00:26:45 --> 00:26:49
because the derivative
of a constant is 0.

334
00:26:49 --> 00:26:55
That the antiderivative is
ambiguous up to a constant.

335
00:26:55 --> 00:26:59
But it's very important to
realize that this is the only

336
00:26:59 --> 00:27:01
ambiguity that there is.
 

337
00:27:01 --> 00:27:04
So the last thing that I want
to tell you about is

338
00:27:04 --> 00:27:24
uniqueness of antiderivatives
up to a constant.

339
00:27:24 --> 00:27:30
The theorem is the following.
 

340
00:27:30 --> 00:27:41
The theorem is if F'
= F', then F = G.

341
00:27:41 --> 00:27:44
So F ( x ) = G ( x) + c.
 

342
00:27:44 --> 00:27:48
 


343
00:27:48 --> 00:27:54
But that means, not only that
these are antiderivatives, all

344
00:27:54 --> 00:27:56
these things with these +
c's are antiderivatives.

345
00:27:56 --> 00:28:02
But these are the only ones.
 

346
00:28:02 --> 00:28:03
Which is very reassuring.
 

347
00:28:03 --> 00:28:05
And that's a kind of
uniqueness, although its

348
00:28:05 --> 00:28:09
uniqueness up to a constant,
it's acceptable to us.

349
00:28:09 --> 00:28:12
Now, the proof of
this is very quick.

350
00:28:12 --> 00:28:18
But this is a fundamental fact.
 

351
00:28:18 --> 00:28:19
The proof is the following.
 

352
00:28:19 --> 00:28:29
If F' = G', then if you take
the difference between the two

353
00:28:29 --> 00:28:40
functions, its derivative,
which of course is F' - G' = 0.

354
00:28:40 --> 00:28:55
Hence, F( x) - G
(x) is a constant.

355
00:28:55 --> 00:28:58
Now, this is a key fact.
 

356
00:28:58 --> 00:28:59
Very important fact.
 

357
00:28:59 --> 00:29:03
We deduced it last time from
the mean value theorem.

358
00:29:03 --> 00:29:05
It's not a small matter.
 

359
00:29:05 --> 00:29:06
It's a very, very
important thing.

360
00:29:06 --> 00:29:08
It's the basis for calculus.
 

361
00:29:08 --> 00:29:11
It's the reason why
calculus make sense.

362
00:29:11 --> 00:29:14
If we didn't have the fact that
the derivative is 0 implied

363
00:29:14 --> 00:29:18
that the function was
constant, we would be done.

364
00:29:18 --> 00:29:23
We would have, calculus would
be just useless for us.

365
00:29:23 --> 00:29:24
The point is, the rate of
change is supposed to

366
00:29:24 --> 00:29:29
determine the function up
to this starting value.

367
00:29:29 --> 00:29:32
So this conclusion
is very important.

368
00:29:32 --> 00:29:35
And we already checked it
last time, this conclusion.

369
00:29:35 --> 00:29:39
And now just by algebra, I
can rearrange this to say

370
00:29:39 --> 00:30:03
that f ( x) = G ( x) + c.
 

371
00:30:03 --> 00:30:07
Now, maybe I should leave
differentials up here.

372
00:30:07 --> 00:30:20
Because I want to illustrate.
 

373
00:30:20 --> 00:30:22
So let's go on to some
trickier, slightly

374
00:30:22 --> 00:30:29
trickier, integrals.
 

375
00:30:29 --> 00:30:35
Here's an example.
 

376
00:30:35 --> 00:30:51
The integral of, say, x^3
( x ^ 4 + 2) ^ 5 dx.

377
00:30:51 --> 00:30:54
This is a function which you
actually do know how to

378
00:30:54 --> 00:30:59
integrate, because we already
have a formula for all powers.

379
00:30:59 --> 00:31:03
Namely, the integral of
x ^ a is equal to this.

380
00:31:03 --> 00:31:06
And even if it were a negative
power, we could do it.

381
00:31:06 --> 00:31:08
So it's OK.
 

382
00:31:08 --> 00:31:11
On the other hand, to
expand the 5th power

383
00:31:11 --> 00:31:14
here is quite a mess.
 

384
00:31:14 --> 00:31:18
And this is just a
very, very bad idea.

385
00:31:18 --> 00:31:21
There's another trick for doing
this that evaluates this

386
00:31:21 --> 00:31:23
much more efficiently.
 

387
00:31:23 --> 00:31:27
And it's the only device
that we're going to learn

388
00:31:27 --> 00:31:31
now for integrating.
 

389
00:31:31 --> 00:31:36
Integration actually is much
harder than differentiation.

390
00:31:36 --> 00:31:37
Symbolically.
 

391
00:31:37 --> 00:31:39
It's quite difficult.
 

392
00:31:39 --> 00:31:42
And occasionally impossible.
 

393
00:31:42 --> 00:31:45
And so we have to go
about it gently.

394
00:31:45 --> 00:31:47
But for the purposes of this
unit, we're only going

395
00:31:47 --> 00:31:50
to use one method.
 

396
00:31:50 --> 00:31:50
Which is very good.
 

397
00:31:50 --> 00:31:53
That means whenever you see an
integral, either you'll be able

398
00:31:53 --> 00:31:56
to divine immediately what the
answer is, or you'll

399
00:31:56 --> 00:31:57
use this method.
 

400
00:31:57 --> 00:31:59
So this is it.
 

401
00:31:59 --> 00:32:09
The trick is called the
method of substitution.

402
00:32:09 --> 00:32:17
And it is tailor-made for
notion of differentials.

403
00:32:17 --> 00:32:36
So tailor-made. for
differential notation.

404
00:32:36 --> 00:32:37
The idea is the following.
 

405
00:32:37 --> 00:32:40
I'm going to to define
a new function.

406
00:32:40 --> 00:32:43
And it's the messiest
function that I see here.

407
00:32:43 --> 00:32:50
It's u = x ^ 4 + 2.
 

408
00:32:50 --> 00:32:56
And then, I'm going to take
its differential and what I

409
00:32:56 --> 00:32:59
discover, if I look at its
formula, is and the rule for

410
00:32:59 --> 00:33:02
differentials, which
is right here.

411
00:33:02 --> 00:33:06
Its formula is what?
 

412
00:33:06 --> 00:33:10
4x^3 dx.
 

413
00:33:10 --> 00:33:14
Now, lo and behold with these
two quantities, I can

414
00:33:14 --> 00:33:17
substitute, I can plug
in to this integral.

415
00:33:17 --> 00:33:21
And I will simplify
it considerably.

416
00:33:21 --> 00:33:23
So how does that happen?
 

417
00:33:23 --> 00:33:35
Well, this integral is the same
thing as, well, really I should

418
00:33:35 --> 00:33:36
combine it the other way.
 

419
00:33:36 --> 00:33:41
So let me move this over.
 

420
00:33:41 --> 00:33:43
So there are two pieces here.
 

421
00:33:43 --> 00:33:46
And this one is u ^ 5.
 

422
00:33:46 --> 00:33:54
And this one is 1/4 du.
 

423
00:33:54 --> 00:34:01
Now, that makes it the
integral of (u ^ 5 du) / 4.

424
00:34:01 --> 00:34:04
And that's relatively
easy to integrate.

425
00:34:04 --> 00:34:05
That is just a power.
 

426
00:34:05 --> 00:34:06
So let's see.
 

427
00:34:06 --> 00:34:11
It's just 1/20 u to
the - not 1/20.

428
00:34:11 --> 00:34:15
The antiderivative
of u ^ 5 is u ^ 6.

429
00:34:15 --> 00:34:25
With the 1/6, so it's
1/24 u ^ 6 + c.

430
00:34:25 --> 00:34:29
Now, that's not the
answer to the question.

431
00:34:29 --> 00:34:32
It's almost the answer
to the question.

432
00:34:32 --> 00:34:33
Why isn't it the answer?
 

433
00:34:33 --> 00:34:35
It isn't the answer
because now the answer's

434
00:34:35 --> 00:34:37
expressed in terms of u.
 

435
00:34:37 --> 00:34:41
Whereas the problem was posed
in terms of this variable x.

436
00:34:41 --> 00:34:45
So we must change back
to our variable here.

437
00:34:45 --> 00:34:47
And we do that just
by writing it in.

438
00:34:47 --> 00:34:56
So it's 1/24 (x ^
4 + 2) ^ 6 + c.

439
00:34:56 --> 00:35:02
And this is the end
of the problem.

440
00:35:02 --> 00:35:02
Yeah, question.
 

441
00:35:02 --> 00:35:16
STUDENT: [INAUDIBLE]
 

442
00:35:16 --> 00:35:19
PROFESSOR: The question is,
can you see it directly?

443
00:35:19 --> 00:35:20
Yeah.
 

444
00:35:20 --> 00:35:23
And we're going to talk about
that in just one second.

445
00:35:23 --> 00:35:30
OK.
 

446
00:35:30 --> 00:35:35
Now, I'm going to do
one more example and

447
00:35:35 --> 00:35:44
illustrate this method.
 

448
00:35:44 --> 00:35:45
Here's another example.
 

449
00:35:45 --> 00:35:51
The integral of x dx /
squre root of 1 + x ^2.

450
00:35:51 --> 00:35:56
Now, here's another example.
 

451
00:35:56 --> 00:36:03
Now, the method of substitution
leads us to the idea u = 1

452
00:36:03 --> 00:36:11
+ x ^2. du = 2x dx, etc.
 

453
00:36:11 --> 00:36:14
It takes about as long as
this other problem did.

454
00:36:14 --> 00:36:15
To figure out what's going on.
 

455
00:36:15 --> 00:36:17
It's a very similar
sort of thing.

456
00:36:17 --> 00:36:20
You end up integrating
u ^ - 1/2.

457
00:36:20 --> 00:36:28
It needs to the integral
of u ^ - 1/2 du.

458
00:36:28 --> 00:36:31
Is everybody seeing
where this..?

459
00:36:31 --> 00:36:37
However, there is a
slightly better method.

460
00:36:37 --> 00:36:46
So recommended method.
 

461
00:36:46 --> 00:36:59
And I call this method
advanced guessing.

462
00:36:59 --> 00:37:01
What advanced guessing means
is that you've done enough

463
00:37:01 --> 00:37:04
of these problems that you
can see two steps ahead.

464
00:37:04 --> 00:37:08
And you know what's
going to happen.

465
00:37:08 --> 00:37:11
So the advanced guessing leads
you to believe that here you

466
00:37:11 --> 00:37:14
had a power - 1/2, here you
have the differential

467
00:37:14 --> 00:37:14
of the thing.
 

468
00:37:14 --> 00:37:16
So it's going to
work out somehow.

469
00:37:16 --> 00:37:19
And the advanced guessing
allows you to guess that the

470
00:37:19 --> 00:37:26
answer should be something
like this. (1 + x ^2) ^ 1/2.

471
00:37:26 --> 00:37:27
So this is your advanced guess.
 

472
00:37:27 --> 00:37:31
And now you just differentiate
it, and see whether it works.

473
00:37:31 --> 00:37:32
Well, here it is.
 

474
00:37:32 --> 00:37:38
It's 1/2 (1 + x ^2) ^ -
1/2( 2x), that's the

475
00:37:38 --> 00:37:39
chain rule here.
 

476
00:37:39 --> 00:37:44
Which, sure enough, gives you
x / square root of 1 + x ^2.

477
00:37:44 --> 00:37:45
So we're done.
 

478
00:37:45 --> 00:37:56
And so the answer is square
root of (1 + x^2) + c.

479
00:37:56 --> 00:38:02
Let me illustrate this further
with another example.

480
00:38:02 --> 00:38:06
I strongly recommend that
you do this, but you

481
00:38:06 --> 00:38:09
have to get used to it.
 

482
00:38:09 --> 00:38:18
So here's another
example. e ^ 6x dx.

483
00:38:18 --> 00:38:26
My advanced guess is e ^ 6x.
 

484
00:38:26 --> 00:38:29
And if I check, when
I differentiate

485
00:38:29 --> 00:38:33
it, I get 6e ^ 6x.
 

486
00:38:33 --> 00:38:35
That's the derivative.
 

487
00:38:35 --> 00:38:38
And so I know that the
answer, so now I know

488
00:38:38 --> 00:38:39
what the answer is.
 

489
00:38:39 --> 00:38:46
It's 1/6 e ^ 6x + c.
 

490
00:38:46 --> 00:38:57
Now, OK, you could, it's also
OK, but slow, to use a

491
00:38:57 --> 00:39:02
substitution, to use u = 6x.
 

492
00:39:02 --> 00:39:07
Then you're going to
get du = 6dx ...

493
00:39:07 --> 00:39:23
It's going to work, it's
just a waste of time.

494
00:39:23 --> 00:39:26
Well, I'm going to give you
a couple more examples.

495
00:39:26 --> 00:39:41
So how about this one.
x ( e^ - x^2) dx.

496
00:39:41 --> 00:39:45
What's the guess?
 

497
00:39:45 --> 00:39:51
Anybody have a guess?
 

498
00:39:51 --> 00:39:52
Well, you could also correct.
 

499
00:39:52 --> 00:39:54
So I don't want you to
bother - yeah, go ahead.

500
00:39:54 --> 00:39:57
STUDENT: [INAUDIBLE]
 

501
00:39:57 --> 00:39:59
PROFESSOR: Yeah, so you're
already one step ahead of me.

502
00:39:59 --> 00:40:02
Because this is too easy.
 

503
00:40:02 --> 00:40:04
When they get more complicated,
you just want to make

504
00:40:04 --> 00:40:05
this guess here.
 

505
00:40:05 --> 00:40:09
So various people have said
1/2, and they understand that

506
00:40:09 --> 00:40:10
there's 1/2 going here.
 

507
00:40:10 --> 00:40:13
But let me just show
you what happens, OK?

508
00:40:13 --> 00:40:19
If you make this guess and you
differentiate it, what you get

509
00:40:19 --> 00:40:25
here is e^ - x ^2 times the
derivative of negative 2x, so

510
00:40:25 --> 00:40:30
that's - 2x. - x^2,
so it's - 2x.

511
00:40:30 --> 00:40:37
So now you see that you're off
by a factor of not 2, but - 2.

512
00:40:37 --> 00:40:39
So a number of you
were saying that.

513
00:40:39 --> 00:40:46
So the answer is -
1/2 e^ - x ^2 + c.

514
00:40:46 --> 00:40:50
And I can guarantee you, having
watched this on various

515
00:40:50 --> 00:40:55
problems, that people who
don't write this out make

516
00:40:55 --> 00:40:57
arithmetic mistakes.
 

517
00:40:57 --> 00:41:00
In other words, there is a
limit to how much people

518
00:41:00 --> 00:41:02
can think ahead and
guess correctly.

519
00:41:02 --> 00:41:05
Another way of doing it, by the
way, is simply to write this

520
00:41:05 --> 00:41:08
thing in and then fix the
coefficient by doing the

521
00:41:08 --> 00:41:10
differentiation here.
 

522
00:41:10 --> 00:41:14
That's perfectly OK as well.
 

523
00:41:14 --> 00:41:18
Alright, one more example.
 

524
00:41:18 --> 00:41:30
We're going to integrate
sin x cos x dx.

525
00:41:30 --> 00:41:33
So what's a good
guess for this one?

526
00:41:33 --> 00:41:36
STUDENT: [INAUDIBLE]
 

527
00:41:36 --> 00:41:38
PROFESSOR: Someone
suggesting sine ^2 x.

528
00:41:38 --> 00:41:41
So let's try that.
 

529
00:41:41 --> 00:41:45
Over 2 - well, we'll get the
coefficient in just a second.

530
00:41:45 --> 00:41:47
So sine ^2 x, if I
differentiate I get

531
00:41:47 --> 00:41:50
2 sine x cosine x.
 

532
00:41:50 --> 00:41:53
So that's off by a factor of 2.
 

533
00:41:53 --> 00:42:04
So the answer is 1/2 sine ^2 x.
 

534
00:42:04 --> 00:42:14
But now I want to point out
to you that there's another

535
00:42:14 --> 00:42:17
way of doing this problem.
 

536
00:42:17 --> 00:42:31
It's also true that if you
differentiate cosine ^2 x,

537
00:42:31 --> 00:42:38
you get 2 cos x ( - sine x).
 

538
00:42:38 --> 00:42:49
So another answer is that
the integral of sin x cos

539
00:42:49 --> 00:43:01
x dx = - 1/2 cos^2 x + c.
 

540
00:43:01 --> 00:43:03
So what is going on here?
 

541
00:43:03 --> 00:43:06
What's the problem with this?
 

542
00:43:06 --> 00:43:11
STUDENT: [INAUDIBLE]
 

543
00:43:11 --> 00:43:11
PROFESSOR: Pardon me?
 

544
00:43:11 --> 00:43:15
STUDENT: [INAUDIBLE]
 

545
00:43:15 --> 00:43:18
PROFESSOR: Integrals
aren't unique.

546
00:43:18 --> 00:43:21
That's part of the - but
somehow these two answers

547
00:43:21 --> 00:43:22
still have to be the same.
 

548
00:43:22 --> 00:43:32
STUDENT: [INAUDIBLE]
 

549
00:43:32 --> 00:43:36
PROFESSOR: OK.
 

550
00:43:36 --> 00:43:36
What do you think?
 

551
00:43:36 --> 00:43:38
STUDENT: If you add them
together, you just get c.

552
00:43:38 --> 00:43:40
PROFESSOR: If you add
them together you get c.

553
00:43:40 --> 00:43:44
Well, actually,
that's almost right.

554
00:43:44 --> 00:43:45
That's not what you
want to do, though.

555
00:43:45 --> 00:43:47
You don't want to add them.
 

556
00:43:47 --> 00:43:50
You want to subtract them.
 

557
00:43:50 --> 00:43:53
So let's see what happens
when you subtract them.

558
00:43:53 --> 00:43:56
I'm going to ignore the
c, for the time being.

559
00:43:56 --> 00:44:05
I get sin^2 x, 1/2 sin^2
x - (-1/2 cos^2 x).

560
00:44:05 --> 00:44:08
So the difference between
them, we hope to be 0.

561
00:44:08 --> 00:44:10
But actually of
course it's not 0.

562
00:44:10 --> 00:44:18
What it is, is it's 1/2 (sin^2
+ cos^2) which is 1/2.

563
00:44:18 --> 00:44:24
It's not 0, it's a constant.
 

564
00:44:24 --> 00:44:26
So what's really going on
here is that these two

565
00:44:26 --> 00:44:29
formulas are the same.
 

566
00:44:29 --> 00:44:31
But you have to understand
how to interpret them.

567
00:44:31 --> 00:44:34
The two constants, here's
a constant up here.

568
00:44:34 --> 00:44:37
There's a constant, c1
associated to this one.

569
00:44:37 --> 00:44:43
There's a different constant,
c2 associated to this one.

570
00:44:43 --> 00:44:46
And this family of functions
for all possible c1s and all

571
00:44:46 --> 00:44:49
possible c2s, is the same
family of functions.

572
00:44:49 --> 00:44:52
Now, what's the relationship
between c1 and c2?

573
00:44:52 --> 00:44:57
Well, if you do the
subtraction, c1 - c2 has

574
00:44:57 --> 00:44:59
to be equal to 1/2.
 

575
00:44:59 --> 00:45:06
They're both constants,
but they differ by 1/2.

576
00:45:06 --> 00:45:09
So this explains, when you're
dealing with families of

577
00:45:09 --> 00:45:10
things, they don't have
to look the same.

578
00:45:10 --> 00:45:13
And there are lots of trig
functions which look

579
00:45:13 --> 00:45:16
a little different.
 

580
00:45:16 --> 00:45:18
So there can be several
formulas that actually

581
00:45:18 --> 00:45:19
are the same.
 

582
00:45:19 --> 00:45:21
And it's hard to check that
they're actually the same.

583
00:45:21 --> 00:45:28
You need some trig
identities to do it.

584
00:45:28 --> 00:45:55
Let's do one more example here.
 

585
00:45:55 --> 00:46:06
Here's another one.
 

586
00:46:06 --> 00:46:14
Now, you may be thinking, and
a lot of people are, thinking

587
00:46:14 --> 00:46:22
ugh, it's got a ln in it.
 

588
00:46:22 --> 00:46:25
If you're experienced, you
actually can read off the

589
00:46:25 --> 00:46:26
answer just the way there were
several people who were

590
00:46:26 --> 00:46:29
shouting out the answers when
we were doing the rest

591
00:46:29 --> 00:46:31
of these problems.
 

592
00:46:31 --> 00:46:32
But, you do need to relax.
 

593
00:46:32 --> 00:46:36
Because in this case, now this
is definitely not true in

594
00:46:36 --> 00:46:37
general when we do integrals.
 

595
00:46:37 --> 00:46:39
But, for now, when we
do integrals, they'll

596
00:46:39 --> 00:46:40
all be manageable.
 

597
00:46:40 --> 00:46:42
And there's only one method.
 

598
00:46:42 --> 00:46:47
Which is substitution.
 

599
00:46:47 --> 00:46:50
And in the substitution
method, you want to go

600
00:46:50 --> 00:46:52
for the trickiest part.
 

601
00:46:52 --> 00:46:55
And substitute for that.
 

602
00:46:55 --> 00:46:59
So the substitution that I
proposed to you is that this

603
00:46:59 --> 00:47:02
should be, you should be ln x.
 

604
00:47:02 --> 00:47:06
And the advantage that that
has is that its differential

605
00:47:06 --> 00:47:08
is simpler then itself.
 

606
00:47:08 --> 00:47:15
So du = dx /x.
 

607
00:47:15 --> 00:47:17
Remember, we use
that in logarithmic

608
00:47:17 --> 00:47:21
differentiation, too.
 

609
00:47:21 --> 00:47:28
So now we can express this
using this substitution.

610
00:47:28 --> 00:47:32
And what we get is, the
integral of, so I'll divide

611
00:47:32 --> 00:47:33
the two parts here.
 

612
00:47:33 --> 00:47:36
It's 1 / ln x, and
then it's dx / x.

613
00:47:36 --> 00:47:43
And this part is 1 / u,
and this part is du.

614
00:47:43 --> 00:47:49
So it's the integral of du / u.
 

615
00:47:49 --> 00:47:58
And that is ln u + c.
 

616
00:47:58 --> 00:48:11
Which altogether, if I put back
in what u is, is ln (ln x) + c.

617
00:48:11 --> 00:48:14
And now we see some
uglier things.

618
00:48:14 --> 00:48:16
In fact, technically
speaking, we could take

619
00:48:16 --> 00:48:18
the absolute value here.
 

620
00:48:18 --> 00:48:28
And then this would be
absolute values there.

621
00:48:28 --> 00:48:33
So this is the type of example
where I really would recommend

622
00:48:33 --> 00:48:39
that you actually use the
substitution, at least for now.

623
00:48:39 --> 00:48:42
Alright, tomorrow we're
going to be doing

624
00:48:42 --> 00:48:43
differential equations.
 

625
00:48:43 --> 00:48:45
And we're going to
review for the test.

626
00:48:45 --> 00:48:47
I'm going to give you a handout
telling you just exactly what's

627
00:48:47 --> 00:48:48
going to be on the test.
 

628
00:48:48 --> 00:48:52
So, see you tomorrow.
 

629
00:48:52 --> 00:48:52