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18.01 Single Variable Calculus, Fall 2007
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18.01 Single Variable Calculus, Fall 2007
Transcript – Lecture 30
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PROFESSOR: Now, to start out today we're going to finish up what we did last time.
Which has to do with partial fractions. I told you how to do partial fractions in several
special cases and everybody was trying to figure out what the general picture was.
But I'd like to lay that out. I'll still only do it for an example. But it will be somehow a
bigger example so that you can see what the general pattern is. Partial fractions,
remember, is a method for breaking up socalled rational functions. Which are ratios
of polynomials. And it shows you that you can always integrate them. That's really
the theme here. And this is what's reassuring is that it always works. That's really
the bottom line. And that's good because there are a lot of integrals that don't have
formulas and these do. It always works. But, maybe with lots of help. So maybe
slowly.
Now, there's a little bit of bad news, and I have to be totally honest and tell you
what all the bad news is. Along with the good news. The first step, which maybe I
should be calling Step 0, I had a Step 1, 2 and 3 last time, is long division. That's the
step where you take your polynomial divided by your other polynomial, and you find
the quotient plus some remainder. And you do that by long division. And the
quotient is easy to take the antiderivative of up because it's just a polynomial. And
the key extra property here is that the degree of the numerator now over here, this
remainder, is strictly less than the degree of the denominator. So that you can do
the next step. Now, the next step which I called Step 1 last time, that's great
imagination, it's right after Step 0, Step 1 was to factor the denominator. And I'm
going to illustrate by example what the setup is here. I don't know maybe, we'll do
this.
Some polynomial here, maybe cube this one. So here I've factored the denominator.
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