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Calculus: Antiderivatives of Powers of x


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About this Lesson

  • Type: Video Tutorial
  • Length: 17:56
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 195 MB
  • Posted: 11/18/2008

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus Review (48 lessons, $95.04)
Calculus: Final Exam Test Prep and Review (45 lessons, $64.35)
Calculus: Basics of Integration (14 lessons, $23.76)
Calculus: Antiderivatives (3 lessons, $5.94)

This lesson is a more advanced view of antidifferentiation and integration. In it, you will be introduced to the proper notation for denoting an integral or antiderivative and you will also be introduced to the power rule for integration and some properties of indefinite integrals. You will also learn several rules about antiderivatives or integrals. If a function has an antiderivative, it has an infinite number of antiderivatives, and the properties of antidifferentiation mirror those of differentiation. Professor Burger will go over some of these properties, including the constant multiple rule for integration and the sum rule for integration.

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. 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'Hôpital'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.

About this Author

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Recent Reviews

Good lesson
~ nachan

Professor Burger helps you find a formula to find the antiderivative. I found this lesson to be a little bit more confusing but I thin it might be because explaining the antiderivatives of powers of x are more difficult than most math problems. He always does a great job explaining the meaning of each term and gives great formulas to answer the problem.

Good lesson
~ nachan

Professor Burger helps you find a formula to find the antiderivative. I found this lesson to be a little bit more confusing but I thin it might be because explaining the antiderivatives of powers of x are more difficult than most math problems. He always does a great job explaining the meaning of each term and gives great formulas to answer the problem.

The Basics of Integration
Antiderivatives of Powers of x Page [1 of 3]
So that last little game show episode we just saw actually inspired the topic that I want to take up next. And that is the notion of antidifferentiation. In particular, finding antiderivatives. Functions that have the property that, given a particular function, the function we find has the feature that its derivative is the original function. So it’s untangling the derivative. So actually I’d like to take a look at a general example and see what a formula would be for finding antiderivative.
Yeah, let me show you how we’re going to write this, first of all. So symbolically, if I want to write an antiderivative, what I’m going to use is the following very unusual symbol. Watch this. See what you think of that? I’m going to write a very pretty, long f. And then I’m going to write f(x). And then I’m going to write dx here. So I think this is very cryptic. It’s very pretty though, I think it’s very pretty, but very cryptic. Why would we ever use this symbol to represent the antiderivative? Well that’s a great challenge question and a great question that we’re going to put on the griddle and address when we’re ready. At this point, just lets look at this as a very cryptic mysterious symbol, and as we go on, maybe you want to actually take a guess and see why we might use this. But the point is I’m going to use this to denote the antiderivative. So in fact, what this means is take this function, nestled inside here and find a new function. Let me call the new function maybe capital F(x), which you should think about in a different way as little f. This is a different function, but it has the property that if you take the derivative of this, you get the original function back.
That’s what we were doing in the game show before. We took—we were given a function and we found a new function that has the property that its derivative was the original function. So here we could say is that if you take the derivative of capital F(x), we get back to little f(x). And so we call capital F(x) an antiderivative, so an antiderivative. And we saw some examples. In fact, all the examples we looked at in the game show were examples of antiderivatives. They have a feature that they’re a function, whose derivative is the original function we were handed.
Okay now what I’d like to do is actually see if we can give a formula, a general formula, for how to find antiderivatives with some common household functions. So let’s take a look and see what this may look like. Well let’s try to find the antiderivative of x raised to a particular number power. Now this symbol—I have to write both this symbol—which sometimes, by the way, is called an integral or integration—this integral symbol, and it always is accompanied by a dx. So the point is think about taking the function and nestling it cozily between these two symbols. The principles, I admit, are completely, completely not clear. Notice I have them written down there in a box as a thing for us to do. We’ve got to figure out why we use this symbol, but for now this is the symbol that produces the antiderivative.
Now n here is some fixed number. It’s not a variable. It could be like seven, or six, or three, or –2. Now, what would this equal? Well we actually hit a lot of examples and we saw that what we have to do in some sense is up the exponent. So I take the derivative. It’ll settle down at n. So what I should do here is look at x and raise it to the n+1 power, because the derivative of that will be—well, I bring the n+1 out in front, and then I have x n+1-1 and that gives me the xn. Of course, there’s a price to pay for that. And the price is that if I bring the n+1 down in front, what I see is an n+1 factor in front. Well, that sort of stinks, because I’m supposed to have no factor in front here. So I better divide through by that constant number and that way, when I divide, that n+1 comes out in front and it will actually cancel with the n+1 on the bottom. And then I just have xn, and that’s exactly what I have here. So that is a general formula for the antiderivative of xn.
Now you may be looking at this and saying, “Gee, he got really sloppy, more sloppy than usual, because he wrote that very, very high up. Why didn’t he write it sort of on the same line here?” Well, the answer is because I think there may be a little problem with what we have here. Look at this and determine if you can in your mind, is there a particular choice of the number n, which makes this thing nonsensical? Can you think of a value for n, where if I plug in that n into this formula, I’ll get something that is completely garbage? Well the answer is yes. If I make the bottom zero, that would be a bad thing. And notice that if n were to be the number –1, I’d have a zero on the bottom. Well, that’s not going to be a good thing, so in fact, this formula must only hold whenever that fixed number n is not equal to –1. So if n is not = to –1, then our analysis is perfect. You could take the derivative and check and see that you get this original function back.
But what about the interesting case when, in fact, n equals -1? Well then what happens there? Well, let’s think about that for a second. If n equals the number –1, what is the original function that we’re trying to find the antiderivative of? Well if this were a –1, the original function would be 1/x. Now we’ve got to think to ourselves can we think of a function, which has the property that it’s derivative, is 1/x? Well the answer is yes. But it’s nothing that looks like this. In fact, we have to think back to our discussion about transcendental functions and recall that, in fact, the natural logarithm function has this derivative 1/x. Remember the natural log function of the derivative is the reciprocal, so 1/x. So that a very special number when the number up here as the exponent is –1, we actually get this natural log as the answer, because the derivative of natural log is 1/x. But in fact, what I can lay here is the natural log(x). Now you remember that when we graphed these functions we saw that the natural log—we can’t put a negative number in there. I can’t take natural log of –3. There’s no power I can raise the special number e to in order to get –3. If I raised e to any power, it’s always going to be positive. So in fact, I have to make sure that that x is always positive. Otherwise, natural log of a negative number is just as scary as having a zero in the denominator. So to avoid that possibility, I’ll just put an absolute value around here. And we understand that means just lop off any negative sign that may be an x. That way it’s completely okay, and I can always talk about the natural log as a positive number. So now here is the formula the f(x)n is going to equal one of two things depending upon what the power is. If the power is not equal to negative one, so anything but negative one, then it equals this simple formula. In the very special case when n = the power –1, then I have 1/x and therefore, it’s the natural log and we put absolute values so that, in fact, our answer will be okay and not garbage. So this is actually the analog—the integration or the antidifferentiation analog of that formula we saw for taking the derivative of xn. Remember we saw that taking the derivative of xn we got n, xn-1. In particular, if you think of xn, what you do to take the derivative is what? You bring the n out in front and then you subtract 1. Well now we’re seeing the analogous formula for finding the antiderivative of xn. And we see what the answer is. The answer—you add 1 to the exponent n+1, and then divide by that number n+1. So when you take the derivative, the n+1 comes out in front, cancels away, I’m left with xn+1-1, which is xn. So in fact, this is the formula for finding antiderivatives. Notice we don’t have to worry about, you know, antiderivatives by definitions, limits, the ?x goes to zero, all that really serious stuff that we’re talking about in order to inspire and learn what a derivative is. We don’t have to worry about that, because all we’re doing is taking a process we already understand and running it backwards. So there, in fact, is the formula.
Okay well now, let me point out something that I think is sort of interesting in finding antiderivatives, and let’s try to inspire it with an example. So let’s look at the following example. Let’s find the antiderivative or the f(2x)dx. Now remember that dx is something that I just write in there. That’s notational. It goes along with that beautiful, fancy f sign. These 2 things all go together. But what it means is we want to find the antiderivative of 2x. We want to find a function whose derivative is 2x. Okay well, we’ve already seen this example on our game. And we saw that this actually could be x2, because if you check it, you’ll see that the derivative of this is 2x. So this is actually okay.
But now I want to actually take a look at a picture of this answer. See what it looks like. Well, it’s a parabolic course. We all know that. And it looks just like this. And that is a function, you’ll notice, which has the wonderful property that it’s derivative is 2x. And what does that mean? Well, it means that if you wanted to look at the slopes of the tangent lines, you know how to find it. You would just take the value you’re at and plug it into the 2x formula, and that will tell you what the slope is. Notice for example here the slope is zero because 2 X 0 = 0. At 1, the slope would be 2, because I plug in 1 into here and so on. You know the whole story. So we’re given the slope function and we’re trying to find the function function.
Well now here’s something interesting. What happens if I take this curve and just shift if up by 1? Do the slopes change? Do the slopes of the tangent lines change? The answer is no. They don’t change at all. The slope is still the same. I just moved the whole picture up, but the curvature hasn’t changed. The function hasn’t changed. I just lifted it up by 1 unit. So the point is that, in fact, when I’m looking for an antiderivative of this, there’s actually more than 1 answer. This is an answer, but if I actually added 1 to that, I’d get another answer, which also works. In fact, let’s take a look at that to make sure that that really is okay. If I write down this function, this is going to be x2+1. What’s the derivative of that? Well we know that the derivative is 2x, but that’s a constant. So it’s derivative is zero. So in fact, we get 2x again. So in fact, I can add any constant I want and I’m still going to get the same derivative. In particular, if I take a function and add a constant to it, it’s derivative won’t change. All I’m doing is shifting up or shifting down, if I subtract a number. Like if I subtract 3. It just moves down. It doesn’t change the curvature or the relative position or anything. It just moves it up or down, up or down. So in fact, there are many answers to finding the f(2x). There are infinitely many answers. x2 Is just one out of many. x2+1 Is another one. Can you think of another? How about x2+Pi? The derivative of pi is zero. It’s a constant. How about x2-17.38? That just shifts it down by 17.38, but the derivative remains the same, because the derivative of a constant is zero. So in fact, there are infinitely many different answers to this question. Namely, can you find an antiderivative? The answer is there are infinitely many. So how do we capture that? Well, what we do is we write a + C. We add a constant, because in fact, we can put in any constant there that we want, and the derivative of that will still be zero. And so the derivative of this whole thing will be 2x. So when we talk about antiderivatives or finding integrals, we always have to remember that the answer is only unique up to a constant. So you have to always add a constant, the constant of integration, because the derivative of that will be zero, and we’ll get an answer. The most general answer requires me to stick on a constant.
Okay so on with that little fact. I thought we would take a look at some basic properties of antiderivatives. Not surprisingly, they all mirror exactly what we’ve already discovered about derivatives, because the derivative process is just the antiderivative process, but in reverse. So the first thing I want to tell you guys is the following. Suppose that I want to take the antiderivative, I hope you’re getting used to this symbol, of some number, so some number like maybe 16 times a function. So this symbol just means find an antiderivative. Well when we find derivatives and we have a constant in front times a function, what do we know? You put the constant in the holding pattern, and we just take the derivative as the function, and then take that answer and multiply it by the constant. So it turns out that that exact property for the exact reason holds here. In particular, this is equal to just that number multiplied by the antiderivative of f(x) alone. So if you have a constant times a function and you want to find an antiderivative, it equals pulling out that constant and just finding the antiderivative of this piece and multiplying it through by the constant. This is only true for a constant. You can’t pull out another function. You can’t pull out x’s. Just like you can’t pull out x’s when you’re taking derivatives. You’ve got to use a [inaudible] or something. Well here if it’s a constant, you can always do this—for example, the f(5x3)dx. Well what I can do for thinking purposes is take that 5, which is a constant, and actually pull it outside of the antiderivative symbol or the integral symbol, and I could write this. And now I’m home free, back to my original formula. I know how to do this. I keep the 5 out in front. And then here what I do is I add 1 to the exponent, so I get a 4, and divide by the 4. Whoops. Divide by the 4. So that’s my answer, except don’t forget I have to add on a constant, because any constant can be added. So in fact, I see the final answer would be 5/4x4+C. That’s the answer I get. And let me tell you, I think the most wonderful thing about finding antiderivatives, the most wonderful thing, is that the answer is in front of you always. That is to say, you will always know for certain if you have the right answer or not, because the answer was given to us. So if you want to see if you were right, there’s only one thing you have to do. Take the derivative of this, and see if you get that. Isn’t it great? You actually have the answer as the question. It’s that reverse question process. You might want to check to make sure. Let’s check. Let’s take the derivative of this, see if we get the original function back. Taking the derivative, I bring the 4 out in front. So that 4 and this 4 cancel. And I’m left with 5x4-1—5x3. The derivative of a constant is zero. So I see the derivative is 5x3, and that’s exactly what we started with. So we know with certainty this is correct. You see that? So in fact, that is a very empowering feeling if you know that we can always check our answer. There’s no ambiguity.
The other really important formula I want to tell you is the analog of the fact that if I take two functions and add them up, and want to take the derivative, all I have to do is take the derivative of the first and add it to the derivative of the second. Well, it turns out for that exact same reason, antiderivatives work the same way. So if you want to take the antiderivative of the sum of two functions, let’s say f(x)+g(x), then in fact, all you have to do is first take the antiderivative of the first function and then add that to the antiderivative of the second function. And similarly, if you subtract two functions, you can just subtract their individual integrals or their individual antiderivatives. So these two properties I think are identical to the kind of things that we’ve seen before. The new twist that we have to be careful of is that when we take one of these integrals, we have to remember to tack on the +C. This is really important, because in fact, if we don’t tack it on, we don’t have all the answers. We just have one particular answer, and we might want all the answers to see exactly what we’re up against. Okay.
Up next what I’d like to tell you about are some formulas for integrals that actually will involve more things that just x’s to powers. Let’s see what we can do in the next section. I’ll meet you there.

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