WEBVTT 00:00:01.803 --> 00:00:11.900 Topic 17. Integration and the Indefinite Integral. I have a challenge for you. Let's try something. 00:00:11.902 --> 00:00:21.400 I'm going to make a little table here, and I'm going to label the first column f of x and the second 00:00:21.402 --> 00:00:33.704 column capital F of x, and so here's the challenge. I'm going to give you a derivative, and you're going to 00:00:33.706 --> 00:00:40.305 try to decide what it is the derivative of. I'm going to write the derivative here. For example, for the 00:00:40.307 --> 00:00:50.500 first one, I'm going to write 3 x squared. Can you guess, remember, or figure out what this is the derivative of? 00:00:50.502 --> 00:01:04.202 Well, if you said x cubed, you are mostly correct. So the relationship here is capital F prime of x, 00:01:04.204 --> 00:01:12.802 that's the derivative of this function, equals f of x. That's what we're trying to take a look at. 00:01:12.804 --> 00:01:21.801 So let's take another function. Oh, let's look at a constant function, 6. What is this the derivative of? 00:01:21.803 --> 00:01:35.000 Well, if you remember your derivative rules, the derivative of 6x is 6. So let's try 1. What is 1 the 00:01:35.002 --> 00:01:45.702 derivative of? Well, the derivative of x is 1, and so if you guessed that, you are correct. So let's try 0. 00:01:45.704 --> 00:02:00.601 What is 0 the derivative of? Hmmm. You probably have lots of answers. It turns out here that any constant, 00:02:00.603 --> 00:02:11.303 and we'll use the letter c, any constant has the derivative 0. So this probably makes you realize that 00:02:11.305 --> 00:02:18.001 there could have been a constant here on this x cubed. So we'll just put plus c because the derivative of 00:02:18.003 --> 00:02:24.703 this constant would have been 0, and when you take the derivative, the constant in effect disappears. 00:02:24.705 --> 00:02:32.599 So that means for all of these functions, we could have added a constant to this and actually had a 00:02:32.601 --> 00:02:40.305 more correct answer. Of course the constant could be 0, but this would take care of all the possibilities. 00:02:40.307 --> 00:02:51.402 Alright, now let's try another one. Suppose I give you negative x to the negative 3/2. Now there's a challenge. 00:02:51.404 --> 00:02:58.900 What do I take the derivative of to get this function? Well it's time for some rules. 00:02:58.902 --> 00:03:06.001 Before we get to the rules though, we need some terms and maybe some new notation. So let's just 00:03:06.003 --> 00:03:15.204 leave this for now, and let's take a look. First of all, we need some terminology. A function, capital 00:03:15.206 --> 00:03:23.702 F of x, is called an antiderivative. Now this is a great name because we're not taking the derivative. 00:03:23.704 --> 00:03:29.702 We are finding the antiderivative. So if you get the feeling you're working backwards, that is exactly 00:03:29.704 --> 00:03:38.401 what we're doing. It's called an antiderivative of the function f, if for every x in the domain of f, 00:03:38.403 --> 00:03:48.101 the derivative of this function is this one. So you now have to train yourself to think backwards. 00:03:48.103 --> 00:03:55.900 We develop some notation for this. We practice it in several topics, and then later on you will discover 00:03:55.902 --> 00:04:02.200 that this has some very interesting real-life applications. So what we're after now here is mostly 00:04:02.202 --> 00:04:12.002 skill in finding the what? The antiderivative. Here's the notation and some more terminology. 00:04:12.004 --> 00:04:18.302 The indefinite integral, now that's indefinite, and you don't want to get this mixed up with the definite 00:04:18.304 --> 00:04:24.301 integral which is going to come later. The indefinite integral is defined as, and this may be a new 00:04:24.303 --> 00:04:35.403 symbol for you. This is read the integral of f of x dx is capital F of x plus c, there's your plus c, 00:04:35.405 --> 00:04:43.303 if F prime of x, that's capital F prime, this one, the antiderivative, if the derivative of that one 00:04:43.305 --> 00:04:53.402 equals f of x. Ok, let's go through the indefinite integral, and let's look at each piece of it 00:04:53.404 --> 00:05:06.800 and get familiar with the terms. The integral of f of x dx. This is called an integral symbol. 00:05:06.802 --> 00:05:26.004 The entire expression here is called an indefinite integral. The dx right here is called the differential, 00:05:26.006 --> 00:05:33.201 and it later turns out to be very important. It tells you that the derivative, when it was taken of capital 00:05:33.203 --> 00:05:40.200 F of x, was taken with respect to the variable x, and it always has to be there. There has to be d-something, 00:05:40.202 --> 00:05:45.004 and that something tells you what the variable was when the derivative was taken. 00:05:45.006 --> 00:06:01.305 Now this entire thing, this nice little expression, is called an indefinite integral. 00:06:01.307 --> 00:06:10.903 The process of finding the antiderivative is called integration. That comes from the word integral, 00:06:10.905 --> 00:06:22.502 and when we find this, the capital F of x is called, can you remember? It's the antiderivative. 00:06:22.504 --> 00:06:30.003 Of course you have to always remember to put plus c because there could have been a constant here. Ok, 00:06:30.005 --> 00:06:34.403 Indefinite integral 00:06:34.405 --> 00:06:36.502 Antiderivative 00:06:36.504 --> 00:06:39.503 Differential 00:06:39.505 --> 00:06:47.408 Now before we go any further we need to review the three simplest rules of derivatives before we 00:06:47.410 --> 00:06:58.101 practice going backwards. The first one is the derivative of a function of the form x to the n. 00:06:58.103 --> 00:07:05.499 Now I'm not using u to the n here. This is just the simple power rule. The derivative, and by the way 00:07:05.501 --> 00:07:16.003 there's your dx. The derivative of x to the n is n x to the n minus 1. Ok, so what did we do? 00:07:16.005 --> 00:07:24.202 We multiplied by the power, and then we subtracted 1 from the power. 00:07:24.204 --> 00:07:32.002 The second rule comes from a class of functions called exponential functions, and in particular, the one 00:07:32.004 --> 00:07:42.203 we're interested in is the natural exponential function. The derivative of this function is e to the x, 00:07:42.205 --> 00:07:49.605 and the third one that we're interested in is the derivative with respect to x of the natural log of x. 00:07:49.607 --> 00:07:59.600 What is that derivative? 1 over x. So these are the three basic functions or classes of functions. 00:07:59.602 --> 00:08:05.904 This is the power function, exponential, natural exponential, and natural log. Now what we want is a 00:08:05.906 --> 00:08:16.204 set of rules that takes us from that direction back to the original function. So let's take a look. 00:08:16.206 --> 00:08:27.400 Notice how these three functions, or these three rules, are the reverse of the ones I have on the board. The 00:08:27.402 --> 00:08:36.103 integral of x to the n is going to be undoing what we did here. So here we multiplied by n and subtracted 1, 00:08:36.105 --> 00:08:42.103 so here we're going to add 1 and then divide by the power to undo the multiplication that we did. 00:08:42.105 --> 00:08:52.200 Notice here that n cannot be negative 1. This would give us a 0 denominator. So this is the 00:08:52.202 --> 00:09:01.004 integration rule for power functions. Again, what is the process? Add 1, divide by the power. 00:09:01.006 --> 00:09:07.402 The rule for the integral of e to the x dx is e to the x. Why? Because the derivative is e to the x. 00:09:07.404 --> 00:09:13.501 So the antiderivative is also e to the x, except remember you have to put plus c, and then 00:09:13.503 --> 00:09:21.004 the third rule handles what happens when n is negative 1. When you have a function x to the negative 1 dx, 00:09:21.006 --> 00:09:30.402 the antiderivative is the natural log. Now why is that? That's because 1 over x is x to the negative 1. 00:09:30.404 --> 00:09:36.704 So in this rule I've written it as x to the negative 1. Of course it's the same thing as 1 over x. 00:09:36.706 --> 00:09:45.305 Alright now before we go any further you may ask yourself, why the absolute value bars? Because in this process 00:09:45.307 --> 00:09:53.301 you've created a natural log function. If you remember, the restriction on the domain of the log function 00:09:53.303 --> 00:10:00.301 says that what sits in this position must be greater than 0. When you create it, it's possible that this 00:10:00.303 --> 00:10:09.302 x could be negative, and that would be out of the domain. So you always put absolute value bars in this position. 00:10:09.304 --> 00:10:19.401 When you integrate you have format rules similar to those that you have in taking the derivative, but 00:10:19.403 --> 00:10:29.402 there are some shortcomings. You have the constant rule. When you take the integral of a constant, 00:10:29.404 --> 00:10:38.801 this would be for example something similar to what I had earlier. If 6 is the derivative, then 6x plus 00:10:38.803 --> 00:10:44.502 possibly c was the original function. So let's go back and look. What this means is when you have 00:10:44.504 --> 00:10:51.504 a constant in here, and you can figure this out from the power rule, but it's much easier just to remember 00:10:51.506 --> 00:10:58.005 that all you have to do is put in an x here. While we're on the subject, check all of these. 00:10:58.007 --> 00:11:06.005 What's the derivative of a constant times x plus c? It is the constant. So this is the antiderivative, 00:11:06.007 --> 00:11:16.502 and it does check. The derivative of kx would be k. The derivative of c is 0. Alright, you have the 00:11:16.504 --> 00:11:24.205 constant times a function rule. This one is a big rule. When you have a constant times a function, 00:11:24.207 --> 00:11:33.402 that constant can float in and out of the integral. You see here it's inside. Here it's outside of the integral. 00:11:33.404 --> 00:11:39.105 You can't do this with a variable of course, but a constant can because remember when you take the derivative 00:11:39.107 --> 00:11:44.700 of a constant times a function you take the derivative of the function and you multiply by the constant. 00:11:44.702 --> 00:11:49.403 Well when you undo it the same thing happens. You just move the constant out of the way; you find the 00:11:49.405 --> 00:11:57.001 antiderivative; and you put it back together using the power rule, the exponential rule, or the log rule. 00:11:57.003 --> 00:12:05.602 The sum and difference rule. If you have the sum or difference of two functions inside an integral, the 00:12:05.604 --> 00:12:13.200 correct antiderivative will be found by integrating each one separately using one of the three rules 00:12:13.202 --> 00:12:20.102 and then putting the correct plus or minus between them. Now, what do you not see up here? 00:12:20.104 --> 00:12:26.600 You do not see a product rule, and that's because there really is no product rule for integration. 00:12:26.602 --> 00:12:33.302 Once you have a product in here it is somewhat challenging to figure out what the original problem was. 00:12:33.304 --> 00:12:39.302 Sometimes you can look at it and tell. Other times you have to make adjustments, and other times it's 00:12:39.304 --> 00:12:45.904 almost impossible. What else do you not see? You don't see a quotient rule, and the same thing applies as 00:12:45.906 --> 00:12:54.304 applied to the product rule. There is no quotient rule, and that's where integration becomes a challenge. 00:12:54.306 --> 00:13:01.402 Ok, it's time to practice. Since we know these format rules and they follow very closely from what we knew 00:13:01.404 --> 00:13:07.204 about derivatives, we're going to go back to the screen here that has the new integration rules, and we're 00:13:07.206 --> 00:13:17.302 going to do lots of practice here. Let's start off slow. Find the antiderivative. That's the plan. 00:13:17.304 --> 00:13:29.700 Number one. The integral of x dx. What's this called? An indefinite integral, and we are looking for the 00:13:29.702 --> 00:13:39.501 antiderivative. So what is the rule? This is x to the 1, so we're going to add 1 which makes 2. So we 00:13:39.503 --> 00:13:49.304 have x squared. That's add 1 and then divide by 2 and put plus c, and like every other rule, it becomes 00:13:49.306 --> 00:14:00.300 almost automatic after you practice this a while. Number two. Let's take the integral of x squared dx. 00:14:00.302 --> 00:14:12.200 The antiderivative would be, this is still x to the n, the same as the first rule, so it's x cubed over 3 plus c. 00:14:12.202 --> 00:14:24.899 Let's try the integral of x to the 3/4 dx. Again, power rule. Now when we add 1, we're going to 00:14:24.901 --> 00:14:36.700 add 1 in the form of 4/4, so that makes this 7/4. Now we have to divide by 7/4 and put plus c. 00:14:36.702 --> 00:14:43.200 I would recommend to you that you very quickly get in the habit, when this power is a fraction, 00:14:43.202 --> 00:14:51.800 instead of setting it up this way with 7/4, just divide by 7/4 which means multiply by 4/7. So just get 00:14:51.802 --> 00:14:57.800 in the habit. Take this number. Take the reciprocal. Multiply by it, and instead of what you see here, 00:14:57.802 --> 00:15:09.700 you would write x to the 7/4 times 4/7, and just sort of squeeze it in right there in the front, plus c. 00:15:09.702 --> 00:15:28.601 Number four. The integral of x to the negative 3 dx. Power rule again. Add 1, divide by the power, plus c. 00:15:28.603 --> 00:15:37.500 We're not interested in simplifying right now, and so that's just fine. Example five. Let's try something 00:15:37.502 --> 00:15:48.501 a little different. Suppose we have the integral of 5 dx. Well, that was the first format rule that we had 00:15:48.503 --> 00:16:02.400 and the one I suggest you just get used to. There's the 5, and so the antiderivative would be 5x plus c. 00:16:02.402 --> 00:16:07.800 You can justify this as a spinoff of the power rule, but it's probably much more efficient to think of it 00:16:07.802 --> 00:16:21.500 in this way. Number six. This one looks strange. The integral of dx. I expect you're saying to yourself whoa, 00:16:21.502 --> 00:16:30.501 there's no function here. Ah but there is. This is 1 dx, and 1 is a perfectly good constant, and so we use this rule, 00:16:30.503 --> 00:16:40.400 and we get 1x plus c. You quickly get used to this. This comes up periodically, and you write x plus c. 00:16:40.402 --> 00:16:54.400 Example seven. So what if we have a constant times a function raised to a power? Don't forget the dx. 00:16:54.402 --> 00:17:08.701 What do we do with this constant? The rule says take the constant and pull it out. And then do what? 00:17:08.703 --> 00:17:24.300 Find the antiderivative using this rule. And what is it? It's add 1, over 5, multiply by the negative 6, plus c. 00:17:24.302 --> 00:17:40.600 Let's try something like this. Number eight. What if we have the integral of x to the 2/3 minus 1 dx? 00:17:40.602 --> 00:17:48.600 So now we're going to use the sum and difference rule. We can find this antiderivative by integrating 00:17:48.602 --> 00:18:06.273 x to the 2/3 dx minus the integral of 1 dx. This would be x to the, add 3/3. That makes this 5/3. 00:18:06.275 --> 00:18:20.074 Now it's divided by 5/3 which is times 3/5. Then minus and the antiderivative 1 is x. What goes next? 00:18:20.076 --> 00:18:28.670 Plus c. So that's the sum and difference rule. Now what happens if we have a product because we know 00:18:28.672 --> 00:18:45.773 there is no product rule. Example nine. The integral of x plus 3 times x minus 5 dx. Well, you cannot 00:18:45.775 --> 00:18:51.874 just look at this and say the antiderivative is ... add 1, divide by. It just doesn't work that way. 00:18:51.876 --> 00:19:01.972 There is no rule, so we try to do other things. The first technique we try here is to multiply, and in 00:19:01.974 --> 00:19:12.074 this particular example, that's going to work just fine. We can rewrite this as x squared minus 2x minus 15. 00:19:12.076 --> 00:19:23.473 We need this in parentheses, dx. Now let's check our multiplication. x squared minus 2x minus 15. 00:19:23.475 --> 00:19:35.973 Now, it's the sum and difference rule. It's the integral of x squared dx minus the integral of 2x dx 00:19:35.975 --> 00:19:45.473 minus the integral of 15 dx. Now as soon as you get highly skilled at this, like in about five minutes, 00:19:45.475 --> 00:19:51.771 you won't need to write this step. You will recognize that this is a sum and difference rule. We can go 00:19:51.773 --> 00:19:59.773 straight to the antiderivative. It would be x cubed over 3, and let's see what this is. This is going 00:19:59.775 --> 00:20:15.275 to be minus 2 x squared over 2, which is actually going to be just x squared, and then minus 15x plus c. 00:20:15.277 --> 00:20:21.374 So we were able to avoid the problem here because even though there was a product in here we could 00:20:21.376 --> 00:20:30.073 actually just multiply. Had any of these numbers, these binomials, been raised to anything but a whole number 00:20:30.075 --> 00:20:36.073 power, 1 or any other number... Say this was squared you could square this first. But if this was raised 00:20:36.075 --> 00:20:41.272 to, say the negative 2 power you would not be able to multiply. If it was raised to a fractional power 00:20:41.274 --> 00:20:46.074 or a negative fractional power, you couldn't do that either. This gives you a hint that maybe there are 00:20:46.076 --> 00:20:51.973 some other things we have to learn how to do in order to do much integration at all. Example ten. 00:20:51.975 --> 00:21:03.470 Let's take a look at the integral of oh, let's try something like x squared minus x over x dx. 00:21:03.472 --> 00:21:13.175 Well, it is not so terribly apparent what this is the derivative of, and there is no quotient rule. So, 00:21:13.177 --> 00:21:19.573 why don't we try to divide? That's what the quotient means. It's divide. Well let's see what this would be. 00:21:19.575 --> 00:21:32.773 This would be x squared over x minus x over x dx. Further simplifying, notice I'm still carrying the 00:21:32.775 --> 00:21:40.773 integral symbol because I have not found the antiderivative yet, so this would be x squared over x, 00:21:40.775 --> 00:21:58.273 which is x, minus 1 dx, and now it's a difference. So the antiderivative is x squared over 2 minus x plus c. 00:21:58.275 --> 00:22:03.973 Could we check this? Of course. This is the antiderivative of this function, or it should be. 00:22:03.975 --> 00:22:12.473 So you can go backwards. The derivative of this would be 2x over 2 which is x. The derivative of negative x 00:22:12.475 --> 00:22:21.174 is negative 1, so that term is correct. Then if I multiply by x over x and distribute it, I can get 00:22:21.176 --> 00:22:25.774 all the way back to here. Usually when you get back to this point, you know you're on the right track. 00:22:25.776 --> 00:22:42.373 Let's try one more. Example eleven. So what if I have the integral of x cubed over x squared minus 1 dx. 00:22:42.375 --> 00:22:49.074 This is a quotient. There is no quotient rule. Can I separate this, reduce it, and get it to be something 00:22:49.076 --> 00:22:58.872 that's a sum or a difference? No. And so at this point, with the rules we have available, we cannot 00:22:58.874 --> 00:23:05.272 complete this problem and find the antiderivative. But we have learned a lot here, and we can find the 00:23:05.274 --> 00:23:16.775 antiderivative of these simple functions, power function, natural exponential function, and the 1 over x 00:23:16.777 --> 00:23:24.773 which takes care of what happens when n is negative 1 here. So, maybe we have an indication that we have 00:23:24.775 --> 00:23:32.673 more to learn. Before we go on to anymore, you need to practice this. This is a skill, and a lot of 00:23:32.675 --> 00:23:40.700 understanding comes from practice. Go practice this, and when you're ready come on to the next topic.