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Calculus: Integration by Parts & the Natural Log


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About this Lesson

  • Type: Video Tutorial
  • Length: 8:12
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 88 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus: Techniques of Integration (28 lessons, $40.59)
Calculus: Integration by Parts (5 lessons, $8.91)

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 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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Techniques of Integration
Integration by Parts
Applying Integration by Parts to the Natural Log Function Page [1 of 2]
Okay, so I really want to try to figure out what the integral of the natural log is. And so I inspired this with a technique called integration by parts. Integration by parts is just a technique, which allows us to look at something and see it as a product of two things, one of which can be integrated very easily, one of which can be differentiated very easily. And the integration by parts, which is just the product rule in reverse, tells us that I can then swap the roles, integrate the easily integratable function, differentiate the easily differentiable functions, and then produce a new integral which will be more simple and easier to do.
Now, that's a mouthful and conceptually it's a brainful, so what I want to try to do is to really drive home and make this idea real by looking at a collection of inspirational examples. So I want to now try to inspire this method and really give you a sense of how to actually use this method in practice. So much for the philosophy, let's get down to the nuts and bolts.
So, in fact, I want to begin with this example. So let's try to integrate the natural log. Now, to use this method, what I want to do is try to think of this integral, and this is how we want to proceed in all these questions, as the product of two people, one person which is easily integrated and one person which is easily differentiated. Now you look at this and you say, "Well, there's only one person sort of here. It's just sort of this term right here." And if I knew how to integrate it, then I'd be done. So obviously what's the second term? Well, there's actually a second term there that's sort of invisible, maybe you don't see it. It's actually an invisible 1. In fact, there's an invisible 1 shadow that follows every math thing all the time, because 1 times anything is anything. So, in this case, let's notice that if I think about the 1, I can easily integrate 1. The integral of 1 is just x. And notice I can easily differentiate the natural log. Taking the derivative of it is real easy. We know it's just . So maybe I can use this technique. And I have to pick now my choices of u and v carefully. So think of this now if you want a collection of characters, and I've got to now give you the cast of characters here. Here's how I do it: I actually write down u, and then I'll write down v, and then I'll write down their derivatives, du, dv. So this is how I always set these integration by parts questions up. I always write down a little schematic like this. I think of it like a playbill and this is going to tell me who the characters are. Now, what am I going to do? I have to now take a look at this integral and I have to realize that one of the characters is going to represent the u and the other character is going to represent the dv. So I'm going to fill in these slots first. So these are the first slots to be filled up. And then once I fill those slots in, I'll figure out the other ones. So what do I do here? Well, I want to put the character here that I can differentiate very easily. And I want to put the character here that I can integrate very easily. This is saying the derivative of v, and to get back to v, I have to integrate. So what are the choices that I have? Well, I could have 1 be the u and the natural log be the dv. In that case, taking the derivative of 1 is really easy. The derivative of 1 = 0. However, taking the integral of is the entire question to begin with. So, that I don't know how to do. So that's a bad choice. A better choice would be perhaps to let this thing be the u, and then take what's left over, and what's left over is that 1 - everything has to be taken into account, so here's the , that's here, and I take everything that's left over, the 1 dx, and I write it as the dv person. So that's going to be my substitution in trying to use this integration by parts. Notice that I'm always going to put someone in for u and someone in for dv. Now, let's see if we can fill out the rest of our table here. Well, to go from here to here, this is saying take the derivative of u. So the derivative of the natural log, that's actually okay. That's dx. So that's pretty easy. And what about going from here to here? Here I'm given the derivative is 1. What's the function? The function is just x. So I integrated to get to this, I differentiated to get to here. And now, everything is filled up. I have my playbill in good shape, I'm ready now to actually see the performance. And the performance is basically this, that if you have this integral, which is here, it's equal to this integral. And what is that integral? Well, let's see. I'll put this right here and so we'll see that the integral of 1 dx equals - well, it equals u v, so I take the product of these two things. So that's x, and then I subtract, so we have to subtract now, a new integral. Now what's the new integral? Well, look the original integral was u dv, and so our new integral is the opposite direction. Our new integral will now be this. So this was the original integral and now I could swap that by v du. So this is the first integral, it gets converted into the this integral. So it's an integral of v du. So that's going to be x times dx. So the integration by parts technique allows us to convert this integral into this fact. Now let's see if that's any easier to look at or not.
Well, let's see. First of all, this is no integral at all, that's a function, minus - and , that's just 1. And look, that's actually a pretty easy function to integrate. Even I can do that. That just equals x - x + c. So that should be the answer. Now, let's, in fact, check and see if we take the derivative of this, if we really do get the natural log. So let's just do a quick check. So if we take the derivative of this, let's see what happens. Well, notice this piece here requires the product rule, and that is not a coincidence. Since integration by parts is sort of doing the product rule in reverse, if you want to check your answer, you definitely have to use the product rule in taking the derivative. Otherwise, something is really messed up. So I'm going to use the product rule right here, which says the first times the derivative of the second, which is , plus the second, which is the natural log, times the derivative of the first, which is just 1, and then I take the derivative of -x, which is -1. And = 1, so I'm left with 1 +- 1. And look, the 1's cancel. I'm left with . So this checks. So, in fact, the original question, which was what is that thing, it turns out the answer is it's x- x + c. And we see that this technique of integration by parts, where I write a u, a v a du and a dv and place them in the appropriate way so that I get a simpler integral, actually is a powerful technique. And we'll see this technique in a variety of examples coming up next. And I'll see you there.

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