18-01F07-L30 - MIT OpenCourseWare http/ocw.mit.edu 18.01...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus, Fall 2007 Please use the following citation format: David Jerison, 18.01 Single Variable Calculus, Fall 2007 . (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-Noncommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms
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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus, Fall 2007 Transcript – Lecture 30 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. 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 so-called 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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18-01F07-L30 - MIT OpenCourseWare http/ocw.mit.edu 18.01...

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