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volunteer
or P and Q are polynomials in X and PFU obviously can't, can't be have a root, can't have a root.
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volunteer
that will um
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volunteer
affect the expression.
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volunteer
Actually, that's not right. I can't say it can't have fruits it's just that it, we don't consider
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volunteer
the times when uh P(x) has the root. We just don't include that in our domain.
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volunteer
cause we know that they'll be undefined.
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volunteer
Um, and this is assuming you can't just
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volunteer
factor out and then get rid of the term.
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volunteer
It just won't be, it just be irrational at that point.
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volunteer
So you can factor out and get rid of the roots.
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student
oh okay
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volunteer
Anyway.
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volunteer
Um, so if the degree of PF X
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volunteer
is less than q of x.
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volunteer
the rational fraction is called proper.
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volunteer
Right, because just like we have our um
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volunteer
fractions by itself, when we have a fraction that is, that has a numerator that is smaller than the denominator, we call that a proper fraction.
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volunteer
right? Because you can't divide anymore.
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student
right
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volunteer
Whereas if the numerator is greater than the the denominator, then you have an improper fraction because then you can
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volunteer
pull it out like, you know, 4/3, something like that, and you can, it's like 1 and 1/3, um
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volunteer
you get the point
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volunteer
So the improper, so it's otherwise, OK, the improper or rational facts could be reduced to the proper rational right.
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volunteer
Function by long division process. Thus, if POX is improper, then P over Q is equal tot of X + R(x) over Q of X.
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volunteer
which is um realizing now important.
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volunteer
I was trying to do a problem and it gave me this and I
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volunteer
feel like I applied it wrong.
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volunteer
Anyway.
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volunteer
TFX is a polynomial in X and
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volunteer
R(x) over Qx is a proper rational fraction.
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volunteer
as we know how to integrate polynomials, the integration of any rational function is reduced to the integration of a proper rational function.
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volunteer
So basically, if you get a
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volunteer
um
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volunteer
rational function
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volunteer
and you find that the degree of the enumerator is greater than the degree of the denominator
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volunteer
Then you perform long division.
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volunteer
to get a form.
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volunteer
um to get a uh
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volunteer
the function into a form of a proper function.
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volunteer
right? So that PE is always equal to or less than q. The the degree, I'm sorry.
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volunteer
I think specifically it has to be
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volunteer
it doesn't say what happens when the degrees are equal to each other.
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volunteer
We're just assuming from the table
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volunteer
that
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student
wait, can i get 5 mins pls
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volunteer
the degree of the numerator
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volunteer
yeah
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volunteer
take your time
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volunteer
Mhm
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student
i'm back
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volunteer
Welcome back.
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student
thank you!
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volunteer
Hm
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student
let's continue
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volunteer
Yeah, um.
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volunteer
so
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volunteer
what um
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volunteer
so I said, I, I went right where we left off. So thus, if P over q is improper, i.e.
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volunteer
that
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volunteer
um
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volunteer
if
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volunteer
oh, OK, if, if the degree of P
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volunteer
is greater than or equal to
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volunteer
the degree of Q
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volunteer
then their function is called proper.
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volunteer
and if it's improper
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volunteer
then we need to um
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volunteer
do long division to make sure that P over q is improper, which means that
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volunteer
even if P has the same degree as Q.
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volunteer
you have to perform long you have to perform long division to get
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volunteer
this form
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volunteer
to be able to make it um
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volunteer
integradable, eat more or, or be able to redefine it such that it's easier to integrate.
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student
convert into mixed fraction?
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volunteer
kind of
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volunteer
Yeah, you want to convert it into mixed fraction, basically.
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student
okay
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volunteer
Yeah
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volunteer
Um
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volunteer
yeah, it's, and then, uh, let's see. And R(x) is a polynomial inex.
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volunteer
P Q y over Q is a proper rational function, as we know how to integrate polynomials, integration of inter rational function is reduced to the integration of a proper rational function.
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volunteer
So they're saying that integration of any rational function.
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volunteer
can be reduced to the innovation of a proper rational function or saying more than likely that they're saying that you pretty much have to do it.
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volunteer
The rational fractions, which
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volunteer
we shall consider here for integration purposes will be those whose denominators can be factored into linear and quadratic factors.
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volunteer
So it's saying right here that the ones that they will consider, means that there are other forms with higher degree denominators, but they're just only considering linear and 1st and 2nd order.
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volunteer
denominators
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student
ohh okay
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volunteer
And then assuming we want to evaluate where P overq is a proper rational function it's always possible to write the innergra.
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volunteer
as the sum of simpler rational functions by using partial fraction decomposition, which is basically what you're saying.
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volunteer
Um, kind of
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volunteer
you know, if it's greater, then you have to kind of mixed fractions, um, but
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volunteer
just trying to write it as something that's more understandable.
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volunteer
Then after this, the integration could be carried out easily using the already known methods, and that's the
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volunteer
big thing here, the past few chapters, it's just they've been giving you known methods.
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volunteer
right? All these different tools
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volunteer
so if you can figure out how
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volunteer
the, the square peg can fit this can fit the whole.
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volunteer
then, you know
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volunteer
it certainly helps to
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