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Calculus: Effective Function Decompositions

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About this Lesson

  • Type: Video Tutorial
  • Length: 11:42
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 127 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus: Basics of Integration (14 lessons, $23.76)
Calculus: Integration by Substitution Illustrated (4 lessons, $7.92)

Taught by Professor Edward Burger, this lesson comes from a comprehensive Calculus course. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/calculus. The full course covers limits, derivatives, implicit differentiation, integration or antidifferentiation, L'Hopital's Rule, functions and their inverses, improper integrals, integral calculus, differential calculus, sequences, series, differential equations, parametric equations, polar coordinates, vector calculus and a variety of other AP Calculus, College Calculus and Calculus II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, "Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas" and of the textbook "The Heart of Mathematics: An Invitation to Effective Thinking". He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The "Journal of Number Theory" and "American Mathematical Monthly". His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

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The Basics of Integration
Illustrating u-Substitution
Choosing Effective Function Decompositions Page [1 of 1]
Well I thought we'd close this section like we began the introduction to antidifferentiation, with a game. We looked at Math Jeopardy before, where you had to actually produce the right question for the answer. Here I'd like to be more traditional. I want you to actually give me the answer to my question. And the question's always going to be, what is a good choice of u in each of the integrals I'm going to show you. And the reason for this, that I want you to think about this is a few-fold. First of all, it's good practice to get in the habit of seeing if you can guess what the right choice of u is. And secondly, to know what to look for when you're looking for those choices of u. And thirdly, to point out the basic fact that in reality, when you're trying these problems, whether it's in the library or in your dorm room, or where ever, inevitably you might make some mistakes in your choice of u. You might make a guess for u, do some work, and realize that wasn't a good choice of u. You have to realize it's okay to go back and try again with a different choice of u. And in fact, it's only after you practice and practice as I have that you can look at these things and actually make a pretty good guess at what u might be. So here's an opportunity for you to practice that. We're not going to actually evaluate the integrals. All we're going to do is make a guess as to what u should be. You want to go out and try these integrals on your own, great. The goal is to figure out what u should be. And here's a chance for you to actually contribute and interact.
For the first question, is dealing with this long integral, (4x^3+x^2-x+1)(12x^2+2x-1). What is a good choice for u? Here are your options. Should it be this first thing right here in this first product? Should it be the second term here, all this stuff? Or should it be the product of all these things. What is a good choice for u? Make a guess right now. Well, you may have noticed that if we take a look at this term and differentiate it, notice what we get. We get 12x^2+2x-1. And that's exactly what's here, 12x^2+2x-1. And so, if you let u be this, the derivative is sitting right over here. So this is a good choice for u. You let u be that, you will succeed in actually evaluating this derivative--this antiderivative.
Okay let's try another one. The integral of x^3 x sin(4x^4). So here are your choices right now. Should u be x^3? Should u be sin(4x^4)? Or should u be 4x^4? Or should u be the whole thing, x^3 x sin(4x^4)? Think about it. Make your guess now. Well if I look at just the inside term right here, you'll notice the derivative is 16x^3. Even though that's not exactly what this term is, it's just off by a constant multiple. So in fact, this choice of u, 4x^4, would be one that would actually lead us to an easier but equivalent integral to evaluate. We will have take care of that 16 by dividing by 16, but we could certainly do that. And therefore, this is the right choice for u. Why? Because it's derivative is basically sitting right here.
Okay let's try another one. Here's a long fraction. (6x^2-8x+6)/(x^3-2x^2+3x+1). What is a good choice of u here? Is it the top, (6x^2-8x+6)? Is it the bottom, (x^3-2x^2+3x+1)? Or should u be the whole thing, this quotient? Enter your answer now. Well, you may have noticed that if you look at the bottom here, and take the derivative, I see 3x^2-4x+3. And that is exactly the top if I were to multiply the answer by 2. So the derivative of this bottom is actually equal to half of the top. So since they differ only by a constant multiple, it appears that this is a good choice for u. And by the way, if you were to actually perform this integration, you would see that you would have something over u, a natural log of u. You try that. Anyway, this question though, is just to find out the right choice of u, and I believe it's the bottom here.
Okay great. I didn't make it out very well. I hope you're faring okay. I'm getting these, which is sort of surprising. Okay, here's a green one, and let's make sure that you understand what this says here. This says x multiplied by (e^x)^2, so that x^2 is in the power, divided by (e^x)^2+5. What is a good choice for u? One answer is x. The next answer is x(e^x)^2. The next answer is just (e^x)^2. The next answer is just x^2. And the last answer is (e^x)^2-5, a lot of choices there. Take a thought about it. Think about it for a second. And now enter your answer when you're ready. Well, there are a lot of possibilities there. Which one is the one that might actually pan out? Well, if we let u be this bottom here, what's the derivative of that? Well the derivative of the 5, +5 just drops out. And what's the derivative of this? Well that actually requires a little chain rule. I see e to the blop. And the derivative of e to the blop is e to the blop--and then I have to multiply by the derivative of the blop, which is 2x. So I see that the derivative of this entire bottom is almost equal to this. It's just off by a factor of 2 in the front. And so really the choice for u might be (e^x)^2+5. If you let u be that, the derivative is basically all across here, except you have to modify it with the 2. This guess might have been a good guess by the way, if you would have let u be that. And that actually would simplify things dramatically, but then you would have a u+5, and you'd have to do another u substitution again. So if you said that, that's a good runner up. Feel good about that. But I think this choice of u would actually simplify things even more.
Okay, let's try just a couple more here. This is a penultimate one. We're almost done. This is the orange one. It's the integral of sin of the square root of x all divided by the square root of x, dx. What are the choices for u here? Well should u be sin of the square root of x? Should u be just the square root of x? Or should u be 1/the square root of x? Think about this and make your guess when you're ready. Well let's think about the following possibility. What if we let u equal just that square root of x right here? What's the derivative of the square root of x? Well the derivative of the square root of x is 1 divided by 2 square root of x. Let me write that down for you. Remember that the derivative of the square root of x is equal to 1 over 2 square root of x. And so that 1 over the square root of x actually is up here, 1 over the square root of x. So in fact, this will simplify quite nicely with a u choice of square root of x. Then you just have sin(u), and I guess you have to take care of that factor so you'll have a 2sin(u). And integrate 2sin(u), you know that'd be -2 cos(u). So in fact, the right choice here will be the square root of x. That was a hard one.
And I'm going to close with a really hard one now. So if you don't get this, don't be discouraged. Look at that sec^3x x tangent x. We want to find the antiderivative. So we're looking now for a good choice of u here. This is, I think, really tricky. Let me give you some choices. Should u be tangent x? Should u be sec x? Should u be sec^3 x? Should u be sin x? Should u be cos x? A lot of choices there, this one, I think is hard. Think about it and, if you can, if you've been watching these videos sort of in succession, try to think about trig functions the way I think about trig functions. That may be a hint. If you don't know what I'm talking about, I'll explain that in a second. But first try your darnedest to make a guess at which of these things you think might work. Don't feel bad if you don't get this one right, but give it a try. All right. There may be a lot of ways of looking at this, but let me tell you the sort of naïve, silly way that I look at these problems. When it's not completely apparent what to do, I usually take a trig problem and convert it back into sin's and cos's and hope that things go well. And if I try that, if I convert this back into sin's and cos's, what I would see is--I'd remember that sec is actually 1/cos. So I would see 1/cos^3x. And tangent I would revert to sin/cos. And if I now combine this, I see sin x divided by well cos^3 x cos^4 x. So this is identical to that, and now it's a little bit clearer. I'm making it slightly more clearer for you. I write this as sin x/(cos x)^4. This is the same thing, but I'm trying to write it so you can really see the inside there. Notice that if I call that quantity u, the inside stuff right there, cos, the derivative of it is sort of sitting on the top right there, because the derivative of cos is -sin. So as long as I prefer things with a -sin in front, I'm okay. So really this is a tricky problem. I first converted back to sin's and cos's, simplified and realized that there's an inside and the derivative is over here. So the choice here would be the cos x, very sneaky, very tricky, but at least I wanted to see what one of these looks like, and now try these on your own if you want, it would be a good idea to actually work through. This one and the other ones, in fact, too, I invite you to try that. Anyway, well congratulations on conquering this idea of substitution and the notion of finding integrals and antiderivatives. And up next and lastly, for this course, we're going to take a look at that second fundamental question of calculus. I'll remind you what that question is, and I'll remind you where we are in this course, and we're finally going to begin to see the big picture and head toward the finish line. Congratulations, and I'll see you in just a bit.

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