1 00:00:00 --> 00:00:00 2 00:00:00 --> 00:00:02 The following content is provided under a Creative 3 00:00:02 --> 00:00:03 Commons license. 4 00:00:03 --> 00:00:06 Your support will help MIT OpenCourseWare continue to 5 00:00:06 --> 00:00:09 offer high quality educational resources for free. 6 00:00:09 --> 00:00:12 To make a donation or to view additional materials from 7 00:00:12 --> 00:00:16 hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:16 --> 00:00:23 at ocw.mit.edu. 9 00:00:23 --> 00:00:28 PROFESSOR: Today we're going to continue our discussion 10 00:00:28 --> 00:00:33 of methods of integration. 11 00:00:33 --> 00:00:36 The method that I'm going to describe today handles a 12 00:00:36 --> 00:00:40 whole class of functions of the following form. 13 00:00:40 --> 00:00:45 You take P (x) / Q (x) and this is known as 14 00:00:45 --> 00:00:53 a rational function. 15 00:00:53 --> 00:01:05 And all that means is that it's a ratio off two polynomials, 16 00:01:05 --> 00:01:13 which are these functions P ( x) and Q ( x). 17 00:01:13 --> 00:01:17 We'll handle all such functions by a method which is known 18 00:01:17 --> 00:01:26 as partial fractions. 19 00:01:26 --> 00:01:36 And what this does is, it splits P / Q into what you 20 00:01:36 --> 00:01:45 could call easier pieces. 21 00:01:45 --> 00:01:48 So that's going to be some kind of algebra. 22 00:01:48 --> 00:01:49 And that's what we're going to spend most of our 23 00:01:49 --> 00:01:52 time doing today. 24 00:01:52 --> 00:01:55 I'll start with an example. 25 00:01:55 --> 00:01:58 And all of my examples will be illustrating 26 00:01:58 --> 00:02:01 more general methods. 27 00:02:01 --> 00:02:07 The example is to integrate the function 1 / x - 1 28 00:02:07 --> 00:02:15 +, say, 3 / x + 2 dx. 29 00:02:15 --> 00:02:17 That's easy to do. 30 00:02:17 --> 00:02:19 It's just, we already know the answer. 31 00:02:19 --> 00:02:26 It's ln x - 1 + 3 ln x + 3. 32 00:02:26 --> 00:02:28 Plus a constant. 33 00:02:28 --> 00:02:37 So that's done. 34 00:02:37 --> 00:02:41 So now, here's the difficulty that is going to arise. 35 00:02:41 --> 00:02:45 The difficulty is that I can start with this function, which 36 00:02:45 --> 00:02:49 is perfectly manageable. 37 00:02:49 --> 00:02:52 And than I can add these two functions together. 38 00:02:52 --> 00:02:55 The way I add fractions. 39 00:02:55 --> 00:03:00 So that's getting a common denominator. 40 00:03:00 --> 00:03:06 And so that gives me x + 2 here + 3 ( x - 1). 41 00:03:06 --> 00:03:11 And now if I combine together all of these terms, then 42 00:03:11 --> 00:03:16 altogether I have 4x + 2 - 3, that's - 1. 43 00:03:16 --> 00:03:20 And if I multiply out the denominator that's x ^2 44 00:03:20 --> 00:03:28 + that 2 turned into a 3, that's interesting. 45 00:03:28 --> 00:03:31 Hope there aren't too many more of those transformations. 46 00:03:31 --> 00:03:32 Is there another one here? 47 00:03:32 --> 00:03:36 STUDENT: [INAUDIBLE] 48 00:03:36 --> 00:03:39 PROFESSOR: Oh, it happened earlier on. 49 00:03:39 --> 00:03:42 Wow that's an interesting vibration there. 50 00:03:42 --> 00:03:45 OK. 51 00:03:45 --> 00:03:50 Thank you. 52 00:03:50 --> 00:03:56 So, I guess my 3's were speaking to my 2's. 53 00:03:56 --> 00:03:58 Somewhere in my past. 54 00:03:58 --> 00:04:07 OK, anyway, I think this is now correct. 55 00:04:07 --> 00:04:10 So the problem is the following. 56 00:04:10 --> 00:04:11 This is the problem with this. 57 00:04:11 --> 00:04:14 This integral was easy. 58 00:04:14 --> 00:04:18 I'm calling it easy, we already know how to do it. 59 00:04:18 --> 00:04:19 Over here. 60 00:04:19 --> 00:04:28 But now over here, it's disguised. 61 00:04:28 --> 00:04:32 It's the same function, but it's no longer clear 62 00:04:32 --> 00:04:32 how to integrate it. 63 00:04:32 --> 00:04:34 If you're faced with this one, you say what am 64 00:04:34 --> 00:04:38 I supposed to do. 65 00:04:38 --> 00:04:41 And we have to get around that difficulty. 66 00:04:41 --> 00:04:44 And so what we're going to do is we're going to 67 00:04:44 --> 00:04:46 unwind this disguise. 68 00:04:46 --> 00:04:55 So we have the algebra problem that we have. 69 00:04:55 --> 00:04:57 Oh, wow. 70 00:04:57 --> 00:04:59 There must be something in the water. 71 00:04:59 --> 00:05:06 Impressive. 72 00:05:06 --> 00:05:06 Wow. 73 00:05:06 --> 00:05:08 OK, let's see. 74 00:05:08 --> 00:05:14 Is 2/3 = 3/2? 75 00:05:14 --> 00:05:17 Holy cow. 76 00:05:17 --> 00:05:18 Well that's good. 77 00:05:18 --> 00:05:21 Well, I'll keep you awake today with several other 78 00:05:21 --> 00:05:23 transpositions here. 79 00:05:23 --> 00:05:29 So our algebra problem is to detect the easy 80 00:05:29 --> 00:05:36 pieces which are inside. 81 00:05:36 --> 00:05:39 And the method that we're going to use, the one that we'll 82 00:05:39 --> 00:05:43 emphasize anyway, is one algebraic trick which is a 83 00:05:43 --> 00:05:49 shortcut, which is called the cover-up method. 84 00:05:49 --> 00:05:51 But we're going to talk about even more general 85 00:05:51 --> 00:05:52 things than that. 86 00:05:52 --> 00:05:54 But anyway, this is where we're headed. 87 00:05:54 --> 00:05:57 Is something called the cover-up method. 88 00:05:57 --> 00:05:59 Alright. 89 00:05:59 --> 00:06:02 So that's our intro. 90 00:06:02 --> 00:06:05 And I'll just have to remember that 2 is not 3. 91 00:06:05 --> 00:06:08 I'll keep on repeating that. 92 00:06:08 --> 00:06:10 So now here I'm going to describe to you how we 93 00:06:10 --> 00:06:13 unwind this disguise. 94 00:06:13 --> 00:06:17 The first step is, we write down the function we 95 00:06:17 --> 00:06:18 want to integrate. 96 00:06:18 --> 00:06:21 Which was this. 97 00:06:21 --> 00:06:26 And now we have to undo the first damage that we did. 98 00:06:26 --> 00:06:31 So the first step is to factor the denominator. 99 00:06:31 --> 00:06:34 And that factors, we happen to know the factors, so I'm not 100 00:06:34 --> 00:06:36 going to carry this out. 101 00:06:36 --> 00:06:38 But this can be a rather difficult step. 102 00:06:38 --> 00:06:41 But we're going to assume that it's done. 103 00:06:41 --> 00:06:44 For the purposes of illustration here. 104 00:06:44 --> 00:06:46 So I factor the denominator. 105 00:06:46 --> 00:06:51 And now, the second thing that I'm going to do is what I'm 106 00:06:51 --> 00:06:54 going to call the setup here. 107 00:06:54 --> 00:06:56 How I'm going to set things up. 108 00:06:56 --> 00:06:59 And I'll tell you what these things are more 109 00:06:59 --> 00:07:01 systematically in a second. 110 00:07:01 --> 00:07:03 And the setup is that I want to somehow detect 111 00:07:03 --> 00:07:05 what I did before. 112 00:07:05 --> 00:07:11 And I'm going to write some unknowns here. 113 00:07:11 --> 00:07:15 What I expect is that this will break up into two pieces. 114 00:07:15 --> 00:07:17 One with the denominator x - 1, and the other with 115 00:07:17 --> 00:07:23 the denominator x + 2. 116 00:07:23 --> 00:07:32 So now, my third step is going to be to solve for A and B. 117 00:07:32 --> 00:07:34 And then I'm done, if I do that. 118 00:07:34 --> 00:07:41 That's the complete unwinding of this disguise. 119 00:07:41 --> 00:07:43 And this is where the cover-up method comes in handy. 120 00:07:43 --> 00:07:46 This is this method that I'm about to describe. 121 00:07:46 --> 00:07:49 Now, you can do the algebra in a clumsy way, or you 122 00:07:49 --> 00:07:50 can do it in a quick way. 123 00:07:50 --> 00:07:54 And we'd like to get efficient about the algebra involved. 124 00:07:54 --> 00:07:58 And so let me show you what the first step in the trick is. 125 00:07:58 --> 00:08:12 We're going to solve for A by multiplying by (x - 1). 126 00:08:12 --> 00:08:15 Now, notice if you multiply by (x - 1) in that equation 127 00:08:15 --> 00:08:16 2, what you get is this. 128 00:08:16 --> 00:08:21 You got 4x - 2 / the x - 1's cancel. 129 00:08:21 --> 00:08:23 You get this on the left-hand side. 130 00:08:23 --> 00:08:26 And on the right-hand side you get A. 131 00:08:26 --> 00:08:29 The x - 1's cancel again. 132 00:08:29 --> 00:08:31 And then we get this extra term. 133 00:08:31 --> 00:08:35 Which is B/ ( x + 2)( x - 1). 134 00:08:35 --> 00:08:38 Now, the trick here, and we're going to get even better 135 00:08:38 --> 00:08:40 trick in just a second. 136 00:08:40 --> 00:08:42 The trick here is that I didn't try to clear the 137 00:08:42 --> 00:08:44 denominators completely. 138 00:08:44 --> 00:08:46 I was very efficient about the way I did it. 139 00:08:46 --> 00:08:51 It just cleared one factor. 140 00:08:51 --> 00:08:54 And the result here is very useful. 141 00:08:54 --> 00:09:03 Namely, if I plug in now x = 1, this term drops out too. 142 00:09:03 --> 00:09:10 So what I'm going to do now is I'm going to plug in x = 1. 143 00:09:10 --> 00:09:15 And what I get on the left-hand side here is 4 - 1 and 1 + 2, 144 00:09:15 --> 00:09:17 and on the left-hand side I get A. 145 00:09:17 --> 00:09:19 That's the end. 146 00:09:19 --> 00:09:21 This is my formula for A. 147 00:09:21 --> 00:09:27 A happens to be equal to 1. 148 00:09:27 --> 00:09:29 And that's, of course, what I expect. 149 00:09:29 --> 00:09:32 A had better be 1, because the thing broke up into 150 00:09:32 --> 00:09:37 1 / x - 1 + 3 / x + 2. 151 00:09:37 --> 00:09:39 So this is the correct answer. 152 00:09:39 --> 00:09:41 There was a question out here, which I missed. 153 00:09:41 --> 00:09:48 STUDENT: Aren't polynomials defined as functions with 154 00:09:48 --> 00:09:56 whole powers, or could they be square roots? 155 00:09:56 --> 00:09:58 PROFESSOR: Are polynomials defined as functions with 156 00:09:58 --> 00:10:00 whole powers, or can they be square roots? 157 00:10:00 --> 00:10:01 That's the question. 158 00:10:01 --> 00:10:04 The answer is, they only have whole powers. 159 00:10:04 --> 00:10:06 So for instance here I only have the power 1 and 0. 160 00:10:06 --> 00:10:10 Here I have the powers 2, 1 and 0 in the denominator. 161 00:10:10 --> 00:10:16 Square roots are no good for this method. 162 00:10:16 --> 00:10:17 Another question. 163 00:10:17 --> 00:10:18 STUDENT: [INAUDIBLE] 164 00:10:18 --> 00:10:22 PROFESSOR: Why did I say x = 1? 165 00:10:22 --> 00:10:26 The reason why I said x = 1 was that it works really fast. 166 00:10:26 --> 00:10:30 You can't know this in advance, that's part of the method. 167 00:10:30 --> 00:10:32 It just turns out to be the best thing to do. 168 00:10:32 --> 00:10:35 The fastest way of getting at the coefficient A. 169 00:10:35 --> 00:10:38 Now the curious thing, let me just pause for a 170 00:10:38 --> 00:10:39 second before I do it. 171 00:10:39 --> 00:10:44 If I had plugged x = 1 into the original equation, I would 172 00:10:44 --> 00:10:45 have gotten nonsense. 173 00:10:45 --> 00:10:48 Because I would've gotten 0 in the denominator. 174 00:10:48 --> 00:10:50 And that seems like the most horrible thing to do. 175 00:10:50 --> 00:10:54 The worst possible thing to do, is to set x = 1. 176 00:10:54 --> 00:10:56 On the other hand, what we did is a trick. 177 00:10:56 --> 00:10:58 We multiplied by x - 1. 178 00:10:58 --> 00:11:01 And that turned the equation into this. 179 00:11:01 --> 00:11:05 So now, in disguise, I multiplied by 0. 180 00:11:05 --> 00:11:07 But that turns out to be legitimate. 181 00:11:07 --> 00:11:11 Because really this equation is true for all x except 1. 182 00:11:11 --> 00:11:13 And then instead of taking x = 1, I can really 183 00:11:13 --> 00:11:15 take x tends to 1. 184 00:11:15 --> 00:11:17 That's really what I need. 185 00:11:17 --> 00:11:18 The limit is x goes to one. 186 00:11:18 --> 00:11:20 The equation is still valid then. 187 00:11:20 --> 00:11:23 So I'm using the worst case, the case that looks like 188 00:11:23 --> 00:11:24 it's dividing by 0. 189 00:11:24 --> 00:11:26 And it's helping me because it's cancelling out all the 190 00:11:26 --> 00:11:29 information in terms of B. 191 00:11:29 --> 00:11:33 So the advantage here is this cancellation that 192 00:11:33 --> 00:11:36 occurs in this part. 193 00:11:36 --> 00:11:37 So that's the method. 194 00:11:37 --> 00:11:39 We're going to shorten it much, much more in a second. 195 00:11:39 --> 00:11:44 But let me carry it out for the other coefficient as well. 196 00:11:44 --> 00:11:51 So the other coefficient I'm going to solve for B, I'm 197 00:11:51 --> 00:11:57 going to multiply by x + 2. 198 00:11:57 --> 00:12:02 And when I do that, I get 4x - 1 / x - 1, that's the left-hand 199 00:12:02 --> 00:12:05 side, the very top expression there. 200 00:12:05 --> 00:12:10 And then down below I get A/ ( x - 1)( x + 2). 201 00:12:10 --> 00:12:14 And then again the x + 2's cancel. 202 00:12:14 --> 00:12:15 So I get B sitting alone. 203 00:12:15 --> 00:12:18 And now I'm going to do the same trick. 204 00:12:18 --> 00:12:21 I'm going to set x = - 2. 205 00:12:21 --> 00:12:28 That's the value which is going to knock out this A term here. 206 00:12:28 --> 00:12:30 So that cancels this term completely. 207 00:12:30 --> 00:12:37 And what we get here all told is - 8 - 1 / - 2 - 1 = B. 208 00:12:37 --> 00:12:43 In other words, B = 3, which was also what it 209 00:12:43 --> 00:12:44 was supposed to be. 210 00:12:44 --> 00:12:48 B was this number 3, right here. 211 00:12:48 --> 00:12:50 Which I'm now not going to change to 2. 212 00:12:50 --> 00:12:52 Because I know that it's not 2. 213 00:12:52 --> 00:12:53 There was a question. 214 00:12:53 --> 00:12:59 STUDENT: [INAUDIBLE] 215 00:12:59 --> 00:13:01 PROFESSOR: All right. 216 00:13:01 --> 00:13:05 Now, this is the method which is called cover-up. 217 00:13:05 --> 00:13:09 But it's really carried out much, much faster than this. 218 00:13:09 --> 00:13:11 So I'm going to review the method and I'm going to show 219 00:13:11 --> 00:13:14 you what it is in general. 220 00:13:14 --> 00:13:22 So the first step is to factor the denominator, Q. 221 00:13:22 --> 00:13:24 That's what I labeled 1 over there. 222 00:13:24 --> 00:13:29 That was the factorization of the denominator up top. 223 00:13:29 --> 00:13:36 The second step is what I'm going to call the setup. 224 00:13:36 --> 00:13:37 That's step 2. 225 00:13:37 --> 00:13:41 And that's where I knew what I was aiming for in advance. 226 00:13:41 --> 00:13:42 And I'm going to have to explain to you in every 227 00:13:42 --> 00:13:45 instance exactly what this setup should be. 228 00:13:45 --> 00:13:49 That is, what the unknowns should be and what target, 229 00:13:49 --> 00:13:53 simplified expression, we're aiming for. 230 00:13:53 --> 00:13:54 So that's the setup. 231 00:13:54 --> 00:14:01 And then the third step is what I'll now call cover-up. 232 00:14:01 --> 00:14:04 Which is just a very fast way of doing what I did on this 233 00:14:04 --> 00:14:09 last board, which is solving for the unknown coefficients. 234 00:14:09 --> 00:14:12 So now, let me perform it for you again. 235 00:14:12 --> 00:14:14 Over here. 236 00:14:14 --> 00:14:18 So it's 4x - 1 divided by, so this is to eliminate 237 00:14:18 --> 00:14:19 writing here. 238 00:14:19 --> 00:14:24 Handwriting it makes it much faster. 239 00:14:24 --> 00:14:28 So this part just factoring the denominator, that 240 00:14:28 --> 00:14:31 was 1, that was step 1. 241 00:14:31 --> 00:14:34 And then step 2, again, is the setup, which is 242 00:14:34 --> 00:14:39 setting it up like this. 243 00:14:39 --> 00:14:42 Alright, that's the setup. 244 00:14:42 --> 00:14:46 And now I claim that without writing very much, I can 245 00:14:46 --> 00:14:49 figure out what A and B are. 246 00:14:49 --> 00:14:51 Just by staring at this. 247 00:14:51 --> 00:14:54 So now what I'm going to do is I'm just going to think 248 00:14:54 --> 00:14:55 what I did over there. 249 00:14:55 --> 00:14:57 And I'm just going to do it directly. 250 00:14:57 --> 00:15:02 So let me show you what the method consists of visually. 251 00:15:02 --> 00:15:09 I'm going to cover up, that is, knock out this factor, and 252 00:15:09 --> 00:15:13 focus on this number here. 253 00:15:13 --> 00:15:15 And I'm going to plug in the thing that makes 254 00:15:15 --> 00:15:17 the 0, which is x = 1. 255 00:15:17 --> 00:15:20 So I'm plugging in x = 1. 256 00:15:20 --> 00:15:21 To this left-hand side. 257 00:15:21 --> 00:15:28 And what I get is 4 - 1 / 1 + 2 = A. 258 00:15:28 --> 00:15:29 Now, that's the same thing I did over there. 259 00:15:29 --> 00:15:33 I just did it by skipping the intermediate algebra step, 260 00:15:33 --> 00:15:35 which is a lot of writing. 261 00:15:35 --> 00:15:38 So the cover-up method really amounts to the following thing. 262 00:15:38 --> 00:15:40 I'm thinking of multiplying this over here. 263 00:15:40 --> 00:15:43 It cancels this and it gets rid of everything else. 264 00:15:43 --> 00:15:45 And it just leaves me with A on the right-hand side. 265 00:15:45 --> 00:15:47 And I have to get rid of it on this side. 266 00:15:47 --> 00:15:51 So in other words, by eliminating this, I'm isolating 267 00:15:51 --> 00:15:53 a on the right-hand side. 268 00:15:53 --> 00:15:55 So the cover-up is that I'm covering this and 269 00:15:55 --> 00:15:57 getting A out of it. 270 00:15:57 --> 00:16:01 Similarly, I can do the same thing with B. 271 00:16:01 --> 00:16:04 It's focused on the value x = - 2. 272 00:16:04 --> 00:16:07 And b is what I'm getting on the right-hand side. 273 00:16:07 --> 00:16:10 And then I have to cover up this. 274 00:16:10 --> 00:16:14 So if I cover up that, then what's left over with x = - 2 275 00:16:14 --> 00:16:22 is again, - 8 - 1 / - 2 - 1. 276 00:16:22 --> 00:16:26 So this is the way the method gets carried out in practice. 277 00:16:26 --> 00:16:32 Writing, essentially, the least you can. 278 00:16:32 --> 00:16:40 Now, when you get to several variables, it becomes just way 279 00:16:40 --> 00:16:42 more convenient to do this. 280 00:16:42 --> 00:16:45 So now, let me just review when cover-up works. 281 00:16:45 --> 00:17:00 So this cover-up method works if Q ( x) has distinct 282 00:17:00 --> 00:17:06 linear factors. 283 00:17:06 --> 00:17:11 And, so you need two things here. 284 00:17:11 --> 00:17:14 It has to factor completely, the denominator has 285 00:17:14 --> 00:17:15 to factor completely. 286 00:17:15 --> 00:17:21 And the degree of the numerator has to be strictly less than 287 00:17:21 --> 00:17:27 the degree of the denominator. 288 00:17:27 --> 00:17:30 I'm going to give you an example here. 289 00:17:30 --> 00:17:35 So, for instance, and this tells you the general 290 00:17:35 --> 00:17:38 pattern of the setup also. 291 00:17:38 --> 00:17:44 Say you had x ^2 + 3x + 8, let's say. 292 00:17:44 --> 00:17:50 Over (x - 1) ( x - 2)( x + 5). 293 00:17:50 --> 00:17:52 So here I'm going to tell you the setup. 294 00:17:52 --> 00:18:02 The setup is going to be A / (x - 1) + B / (x - 2) + c / x + 5. 295 00:18:02 --> 00:18:04 And it will always break up into something. 296 00:18:04 --> 00:18:07 So however many factors you have, you'll have to put in 297 00:18:07 --> 00:18:09 a term for each of those. 298 00:18:09 --> 00:18:13 And then you can find each number here by 299 00:18:13 --> 00:18:26 this cover-up method. 300 00:18:26 --> 00:18:29 Now we're done with that. 301 00:18:29 --> 00:18:33 And now we have to go on to the algebraic complications. 302 00:18:33 --> 00:18:38 So would the first typical algebraic complication be. 303 00:18:38 --> 00:18:50 It would be repeated roots or repeated factors. 304 00:18:50 --> 00:18:54 Let me get one that doesn't come out to be 305 00:18:54 --> 00:18:56 extremely ugly here. 306 00:18:56 --> 00:19:02 So this is what we'll call Example 2. 307 00:19:02 --> 00:19:06 And this is going to work when the degree, you always need 308 00:19:06 --> 00:19:08 that the degree of the numerator is less than the 309 00:19:08 --> 00:19:12 degree of the denominator. 310 00:19:12 --> 00:19:22 And Q has now repeated linear factors. 311 00:19:22 --> 00:19:26 So let's see which example I wanted to show you. 312 00:19:26 --> 00:19:28 So let's just give this here. 313 00:19:28 --> 00:19:34 I'll just repeat the denominator. 314 00:19:34 --> 00:19:39 With an extra factor like this. 315 00:19:39 --> 00:19:41 Now, the main thing you need to know, since I've 316 00:19:41 --> 00:19:44 already performed the factorization for you. 317 00:19:44 --> 00:19:46 Already performed Step 1. 318 00:19:46 --> 00:19:49 This is Step 1 here. 319 00:19:49 --> 00:19:51 You have to factor things all the way, and that's 320 00:19:51 --> 00:19:53 already been done for you. 321 00:19:53 --> 00:19:56 And here's what this setup is. 322 00:19:56 --> 00:20:00 The setup is that it's of the form A / (x - 323 00:20:00 --> 00:20:05 1) + B / (x - 1) ^2. 324 00:20:05 --> 00:20:15 We need another term for the square here. + C / (x + 2). 325 00:20:15 --> 00:20:17 In general, if you have more powers you just need to keep 326 00:20:17 --> 00:20:19 on putting in those powers. 327 00:20:19 --> 00:20:24 You need one for each of the powers. 328 00:20:24 --> 00:20:26 Why does it have to be squared? 329 00:20:26 --> 00:20:27 OK. 330 00:20:27 --> 00:20:28 Good question. 331 00:20:28 --> 00:20:31 So why in the world am I doing this? 332 00:20:31 --> 00:20:37 Let me just give you one hint as to why I'm doing this. 333 00:20:37 --> 00:20:40 It's very, very much like the decimal expansion of a 334 00:20:40 --> 00:20:44 number or, say, the base 2 expansion of a number. 335 00:20:44 --> 00:20:57 So, for, example the number 7/16 is 0 / 2 + 1 / 2 ^2 + 336 00:20:57 --> 00:21:02 1/2 ^3 +, is that right? 337 00:21:02 --> 00:21:07 So it's 4/16 + 1/2 ^4. 338 00:21:07 --> 00:21:09 It's this sort of thing. 339 00:21:09 --> 00:21:11 And I'm getting this power and this power. 340 00:21:11 --> 00:21:13 If I have higher powers, I'm going to have to 341 00:21:13 --> 00:21:14 have more and more. 342 00:21:14 --> 00:21:17 So this is what happens when I have a 2 ^ 4. 343 00:21:17 --> 00:21:20 I have to represent things like this. 344 00:21:20 --> 00:21:23 That's what's coming out of this piece with 345 00:21:23 --> 00:21:24 the repetitious here. 346 00:21:24 --> 00:21:26 Of the powers. 347 00:21:26 --> 00:21:31 This is just an analogy. 348 00:21:31 --> 00:21:31 Of what we're doing. 349 00:21:31 --> 00:21:33 Yeah, another question over here. 350 00:21:33 --> 00:21:34 STUDENT: [INAUDIBLE] 351 00:21:34 --> 00:21:35 PROFESSOR: Yes. 352 00:21:35 --> 00:21:37 So this is an example, but it's meant to represent the general 353 00:21:37 --> 00:21:41 case and I will also give you a general picture. 354 00:21:41 --> 00:21:43 For sure, once you have the second power here, you'll need 355 00:21:43 --> 00:21:46 both the first and the second power mentioned over here. 356 00:21:46 --> 00:21:48 And since there's only a first power over here I only have 357 00:21:48 --> 00:21:52 to mention a first power over there. 358 00:21:52 --> 00:21:55 If this were a 3 here, there would be one more term which 359 00:21:55 --> 00:22:00 would be the one for x - 1 ^2 in the denominator. 360 00:22:00 --> 00:22:03 That's what you just said. 361 00:22:03 --> 00:22:09 OK, now, what's different about this setup is that the cover-up 362 00:22:09 --> 00:22:13 method, although it works, it doesn't work so well. 363 00:22:13 --> 00:22:14 It doesn't work quite as well. 364 00:22:14 --> 00:22:33 The cover-up works for the coefficients B and C, not A. 365 00:22:33 --> 00:22:38 We'll have a quick method for the numbers B and C. 366 00:22:38 --> 00:22:40 To figure out what they are. 367 00:22:40 --> 00:22:43 But it will be a little slower to get to A, 368 00:22:43 --> 00:22:47 which we will do last. 369 00:22:47 --> 00:22:56 Let me show you how it works. 370 00:22:56 --> 00:23:01 First of all, I'm going to do the ordinary cover-up with C. 371 00:23:01 --> 00:23:05 So for C, I just want to do the same old thing 372 00:23:05 --> 00:23:06 that I did before. 373 00:23:06 --> 00:23:10 I cover up this, and that's going to get rid of all the 374 00:23:10 --> 00:23:12 junk except for the C term. 375 00:23:12 --> 00:23:16 So I have to plug x = - 2. 376 00:23:16 --> 00:23:21 And I get x -- sorry, I get (-2 ) ^2 + 2 in the numerator. 377 00:23:21 --> 00:23:25 And I get (- 2 - 1)^2 in the denominator. 378 00:23:25 --> 00:23:28 Remember I'm covering this up. 379 00:23:28 --> 00:23:30 So that's all there is on the left-hand side. 380 00:23:30 --> 00:23:37 And on the right-hand side all there is C. 381 00:23:37 --> 00:23:39 Everything else got killed off, because it was 382 00:23:39 --> 00:23:40 x - 2 times that. 383 00:23:40 --> 00:23:42 That's 0 times all that other stuff. 384 00:23:42 --> 00:23:45 And the x - 2 over here canceled. 385 00:23:45 --> 00:23:47 This is the shortcut that I just described, and this is 386 00:23:47 --> 00:23:50 much faster than doing all that arithmetic. 387 00:23:50 --> 00:23:52 And algebra. 388 00:23:52 --> 00:23:57 So all together this is a 6/9, right? 389 00:23:57 --> 00:24:09 So it's C = 6/9, which is 2/3. 390 00:24:09 --> 00:24:13 Now, the other one which is easy to do, I'm going to do 391 00:24:13 --> 00:24:14 by the slow method first. 392 00:24:14 --> 00:24:17 But you omit a term. 393 00:24:17 --> 00:24:23 The idea is to cover up the other bad factor. 394 00:24:23 --> 00:24:27 Cover ups, I'll do it both the way and the slow way. 395 00:24:27 --> 00:24:29 I'll do it the fast way first, and then I'll 396 00:24:29 --> 00:24:30 show you the slow way. 397 00:24:30 --> 00:24:32 The fast way is to cover this up. 398 00:24:32 --> 00:24:34 And then I have to cover up everything else. 399 00:24:34 --> 00:24:35 That gets eliminated. 400 00:24:35 --> 00:24:38 And that includes everything but B. 401 00:24:38 --> 00:24:40 So I get B on this side. 402 00:24:40 --> 00:24:42 And I get 1 on that side. 403 00:24:42 --> 00:24:48 So that's 1 ^2 + 2 / 1 + 2. 404 00:24:48 --> 00:24:56 So in other words, B = 1. 405 00:24:56 --> 00:24:58 That was pretty fast, so let me show you what arithmetic 406 00:24:58 --> 00:25:00 was hiding behind that. 407 00:25:00 --> 00:25:01 What algebra was hiding behind it. 408 00:25:01 --> 00:25:06 What I was really doing is this. 409 00:25:06 --> 00:25:16 In multiplying through by x - 1 ^2, so I got this. 410 00:25:16 --> 00:25:19 So this canceled here, so this C just stands alone. 411 00:25:19 --> 00:25:26 And then I have here C /x + 2 (x - 1) ^2. 412 00:25:26 --> 00:25:30 Notice again, I cleared out that 1, this term from the 413 00:25:30 --> 00:25:33 denominator and sent it over to the other side. 414 00:25:33 --> 00:25:38 Now, what's happening is that when I set x = 1 415 00:25:38 --> 00:25:43 here, this term is dying. 416 00:25:43 --> 00:25:45 This term is going away, because there's more powers 417 00:25:45 --> 00:25:47 in the numerator than in the denominator. 418 00:25:47 --> 00:25:50 This is still 0. 419 00:25:50 --> 00:25:52 And this one is gone also. 420 00:25:52 --> 00:25:56 So all that's left is B. 421 00:25:56 --> 00:25:59 Now, I cannot pull that off with a single power of x - 1. 422 00:25:59 --> 00:26:02 I can't expose the A term. 423 00:26:02 --> 00:26:04 It's the B term that I can expose. 424 00:26:04 --> 00:26:06 Because I can multiply through by this thing squared. 425 00:26:06 --> 00:26:10 If I multiply through by just x - 1, what'll happen here is I 426 00:26:10 --> 00:26:12 won't have canceled this (x - 1 )^2. 427 00:26:12 --> 00:26:13 It's useless. 428 00:26:13 --> 00:26:15 I still have a 0 in the denominator. 429 00:26:15 --> 00:26:17 I'll have B / 0 when I plug in x = 1. 430 00:26:17 --> 00:26:22 Which I can't use. 431 00:26:22 --> 00:26:32 Again, the cover-up method is giving us B and C, not A. 432 00:26:32 --> 00:26:38 Now, for the last term, for A, I'm going to just have to be 433 00:26:38 --> 00:26:40 straightforward about it. 434 00:26:40 --> 00:26:51 And so I'll just suggest for A, plug in your favorite number. 435 00:26:51 --> 00:26:59 So plug in my favorite number. 436 00:26:59 --> 00:27:01 Which is x = 0. 437 00:27:01 --> 00:27:04 And you won't be able to plug in x = 0 if you've 438 00:27:04 --> 00:27:05 already used it. 439 00:27:05 --> 00:27:08 Here the two numbers we've already used are 440 00:27:08 --> 00:27:13 x = 1 and x = - 2. 441 00:27:13 --> 00:27:18 But we haven't used x = 0 yet, so that's good. 442 00:27:18 --> 00:27:21 I'm going to plug in now x = 0 into the equation. 443 00:27:21 --> 00:27:22 What do I get? 444 00:27:22 --> 00:27:35 I get 0 ^2 + 2 / (- 1) ^2 * 2 is equal to, let's see. 445 00:27:35 --> 00:27:36 A is the thing that I don't know. 446 00:27:36 --> 00:27:44 So it's A / - 1 +, B / x - 1 ^2 so B = 1, so 447 00:27:44 --> 00:27:48 that's 1 / (- 1) ^2. 448 00:27:48 --> 00:27:51 And then C was 2/3. 449 00:27:51 --> 00:27:55 2/3 / x + 2. 450 00:27:55 --> 00:28:00 So that's 0 + 2. 451 00:28:00 --> 00:28:02 Don't give up at this point. 452 00:28:02 --> 00:28:03 This is a lot of algebra. 453 00:28:03 --> 00:28:06 You really have to plug in all these numbers. 454 00:28:06 --> 00:28:08 You make one arithmetic mistake and you're always going 455 00:28:08 --> 00:28:09 to get the wrong answer. 456 00:28:09 --> 00:28:16 This is very arithmetically intensive. 457 00:28:16 --> 00:28:18 However, it does simplify at this point. 458 00:28:18 --> 00:28:22 We have 2 / 2, that's 1. 459 00:28:22 --> 00:28:27 Is equal to - A + 1 + 1/3. 460 00:28:27 --> 00:28:29 So let's see. 461 00:28:29 --> 00:28:34 A on the other side, this becomes A = 1/3. 462 00:28:34 --> 00:28:35 And that's it. 463 00:28:35 --> 00:28:36 This is the end. 464 00:28:36 --> 00:28:40 We've we've simplified our function. 465 00:28:40 --> 00:28:48 And now it's easy to integrate. 466 00:28:48 --> 00:28:48 Question. 467 00:28:48 --> 00:28:49 Another question. 468 00:28:49 --> 00:29:02 STUDENT: [INAUDIBLE] 469 00:29:02 --> 00:29:04 PROFESSOR: So the question is, if x = 0 has already 470 00:29:04 --> 00:29:05 been used, what do I do? 471 00:29:05 --> 00:29:10 And the answer is, pick something else. 472 00:29:10 --> 00:29:11 And you said pick a random number. 473 00:29:11 --> 00:29:13 And that's right, except that if you really picked a random 474 00:29:13 --> 00:29:19 number it would be 4.12567843, which would be difficult. 475 00:29:19 --> 00:29:21 What you want to pick is the easiest possible number 476 00:29:21 --> 00:29:24 you can think of. 477 00:29:24 --> 00:29:24 Yeah. 478 00:29:24 --> 00:29:32 STUDENT: [INAUDIBLE] 479 00:29:32 --> 00:29:37 PROFESSOR: If you had, as in this sort of situation here. 480 00:29:37 --> 00:29:39 More powers. 481 00:29:39 --> 00:29:41 Wouldn't you have more variables. 482 00:29:41 --> 00:29:42 Very good question. 483 00:29:42 --> 00:29:44 That's absolutely right. 484 00:29:44 --> 00:29:48 This was a 3 by 3 system in disguise, for these three 485 00:29:48 --> 00:29:50 unknowns, A, B and C. 486 00:29:50 --> 00:29:52 What we started with in the previous problem 487 00:29:52 --> 00:29:53 was two variables. 488 00:29:53 --> 00:29:56 It's over here, the variables A and B. 489 00:29:56 --> 00:29:59 And as the degree of the denominator goes up, the 490 00:29:59 --> 00:30:02 number of variables goes up. 491 00:30:02 --> 00:30:05 It gets more and more and more complicated. 492 00:30:05 --> 00:30:06 More and more arithmetically intensive. 493 00:30:06 --> 00:30:09 STUDENT: [INAUDIBLE] 494 00:30:09 --> 00:30:09 PROFESSOR: Well, so. 495 00:30:09 --> 00:30:11 The question is, how would you solve it if you 496 00:30:11 --> 00:30:12 have two unknowns. 497 00:30:12 --> 00:30:16 That's exactly the point here. 498 00:30:16 --> 00:30:19 This is a system of simultaneous equations 499 00:30:19 --> 00:30:19 for unknowns. 500 00:30:19 --> 00:30:23 And we have little tricks for isolating single variables. 501 00:30:23 --> 00:30:25 Otherwise we're stuck with solving the whole system. 502 00:30:25 --> 00:30:28 And you'd have to solve the whole system by elimination, 503 00:30:28 --> 00:30:35 various other tricks. 504 00:30:35 --> 00:30:38 I'll say a little more about that later. 505 00:30:38 --> 00:30:47 Now, I have to get one step more complicated with 506 00:30:47 --> 00:30:53 my next example. 507 00:30:53 --> 00:31:02 My next example is going to have a quadratic factor. 508 00:31:02 --> 00:31:05 So still I'm sticking to the degree of the polynomial and 509 00:31:05 --> 00:31:08 the numerator is less than the degree of the polynomial 510 00:31:08 --> 00:31:09 in the denominator. 511 00:31:09 --> 00:31:21 And I'm going to take the case where Q has a quadratic factor. 512 00:31:21 --> 00:31:26 Let me just again illustrate this by example. 513 00:31:26 --> 00:31:30 I have here (x - 1) (x^2 + 1). 514 00:31:30 --> 00:31:34 I'll make it about as easy as they come. 515 00:31:34 --> 00:31:38 Now, the setup will be slightly different here. 516 00:31:38 --> 00:31:40 Here's the setup. 517 00:31:40 --> 00:31:42 It's already factored. 518 00:31:42 --> 00:31:44 I've already done as much as I can do. 519 00:31:44 --> 00:31:48 I can't factor this x^2 + 1 into linear factors unless you 520 00:31:48 --> 00:31:49 know about complex numbers. 521 00:31:49 --> 00:31:52 If you know about complex numbers this method 522 00:31:52 --> 00:31:53 becomes much easier. 523 00:31:53 --> 00:31:55 And it comes back to the cover-up method. 524 00:31:55 --> 00:31:58 Which is the way that the cover-up method was originally 525 00:31:58 --> 00:32:01 conceived by heavy side. 526 00:32:01 --> 00:32:04 But you won't get to that until 18.03. 527 00:32:04 --> 00:32:05 So we'll wait. 528 00:32:05 --> 00:32:08 This, by the way, is a method which is used for integration. 529 00:32:08 --> 00:32:11 But it was invented to do something with Laplace 530 00:32:11 --> 00:32:14 transforms and inversion of certain kinds of 531 00:32:14 --> 00:32:15 differential equations. 532 00:32:15 --> 00:32:17 By heavy side. 533 00:32:17 --> 00:32:21 And so it came much later than integration. 534 00:32:21 --> 00:32:26 But anyway, it's a very convenient method. 535 00:32:26 --> 00:32:30 So here's the set up with this one. 536 00:32:30 --> 00:32:34 Again, we want a term for this (x - 1) factor. 537 00:32:34 --> 00:32:36 And now we're going to also have a term with the 538 00:32:36 --> 00:32:39 denominator x squared plus 1. 539 00:32:39 --> 00:32:40 But this is the difference. 540 00:32:40 --> 00:32:44 It's now going to be a first degree polynomial. 541 00:32:44 --> 00:32:53 One degree down from the quadratic here. 542 00:32:53 --> 00:32:55 So this is what I keep on calling the setup, 543 00:32:55 --> 00:32:57 this is number 2. 544 00:32:57 --> 00:32:59 You have to know that in advance based on the pattern 545 00:32:59 --> 00:33:03 that you see on the left-hand side. 546 00:33:03 --> 00:33:03 Yes. 547 00:33:03 --> 00:33:13 STUDENT: [INAUDIBLE] 548 00:33:13 --> 00:33:15 PROFESSOR: The question is, if the degree of the numerator. 549 00:33:15 --> 00:33:19 So in this case, if this were cubed, and this is matching 550 00:33:19 --> 00:33:23 with the denominator, which is total of degree 3. 551 00:33:23 --> 00:33:26 The answer is that this does not work. 552 00:33:26 --> 00:33:29 STUDENT: [INAUDIBLE] 553 00:33:29 --> 00:33:31 PROFESSOR: It definitely doesn't work. 554 00:33:31 --> 00:33:32 And we're going to have to do something totally 555 00:33:32 --> 00:33:34 different to handle it. 556 00:33:34 --> 00:33:37 Which turns out, fortunately, to be much easier than this. 557 00:33:37 --> 00:33:41 But we'll deal with that at the end. 558 00:33:41 --> 00:33:43 Keep this in mind. 559 00:33:43 --> 00:33:45 This is an easy way to make a mistake if you start with 560 00:33:45 --> 00:33:47 a higher degree numerator. 561 00:33:47 --> 00:33:51 You'll never get the right answer. 562 00:33:51 --> 00:33:54 So now, so I have my setup now. 563 00:33:54 --> 00:33:56 And now what can I do? 564 00:33:56 --> 00:34:03 Well, I claim that I can still do cover-up for A. 565 00:34:03 --> 00:34:05 It's the same idea. 566 00:34:05 --> 00:34:07 I cover this guy up. 567 00:34:07 --> 00:34:09 And if I really multiply by it it would knock 568 00:34:09 --> 00:34:11 everything out but A. 569 00:34:11 --> 00:34:14 So I cover this up and I plug in x = 1. 570 00:34:14 --> 00:34:20 So I get here 1 ^2 / 1 ^2 + 1 = A. 571 00:34:20 --> 00:34:25 In other words, A = 1/2. 572 00:34:25 --> 00:34:28 Again cover-up is pretty fast, as you can see. 573 00:34:28 --> 00:34:32 It's not too bad. 574 00:34:32 --> 00:34:41 Now, at this next stage, I want to find B and C. 575 00:34:41 --> 00:34:45 And the best idea is the slow way. 576 00:34:45 --> 00:34:48 Here, it's not too terrible. 577 00:34:48 --> 00:34:50 But it's just what we're going to do. 578 00:34:50 --> 00:34:54 Which is to clear the denominators completely. 579 00:34:54 --> 00:35:05 So for B and C, just clear the denominator. 580 00:35:05 --> 00:35:08 That means multiply through by that whole business. 581 00:35:08 --> 00:35:09 Now, when you do that on the left-hand side you're 582 00:35:09 --> 00:35:10 going to get x ^2. 583 00:35:10 --> 00:35:12 Because I got rid of the whole denominator. 584 00:35:12 --> 00:35:16 On the right-hand side when I bring this up, the x - 585 00:35:16 --> 00:35:18 1 will cancel with this. 586 00:35:18 --> 00:35:23 So the a term will be A ( x ^2 + 1). 587 00:35:23 --> 00:35:29 And the Bx + C term will have a remaining factor of x - 1. 588 00:35:29 --> 00:35:33 Because the x ^2 + 1 will cancel. 589 00:35:33 --> 00:35:38 Again, the arithmetic here is not too terrible. 590 00:35:38 --> 00:35:41 Now I'm going to do the following. 591 00:35:41 --> 00:35:46 I'm going to look at the x ^2 term. 592 00:35:46 --> 00:35:49 On the left-hand side and the right-hand side. 593 00:35:49 --> 00:35:51 And that will give me one equation for B and C. 594 00:35:51 --> 00:35:54 And then I'm going to do the same thing with another term. 595 00:35:54 --> 00:35:57 The x^2 term on the left-hand side, the coefficient is 1. 596 00:35:57 --> 00:35:59 It's 1 ( x ^2). 597 00:35:59 --> 00:36:06 On the other side, it's A. remember I actually have A. 598 00:36:06 --> 00:36:08 So I'm going to put it in, it's 1/2. 599 00:36:08 --> 00:36:11 So this is the A term. 600 00:36:11 --> 00:36:14 And so I get 1/2 ( x ^2). 601 00:36:14 --> 00:36:18 And then the only other x^2 is when this Bx multiplies this x. 602 00:36:18 --> 00:36:23 So Bx * x is Bx ^2, so this is the other coefficient 603 00:36:23 --> 00:36:25 on x ^2 is B. 604 00:36:25 --> 00:36:36 And that forces B to be 1/2. 605 00:36:36 --> 00:36:41 And last of all, I'm going to do the x^ 0 power term. 606 00:36:41 --> 00:36:44 Or, otherwise known as the constant term. 607 00:36:44 --> 00:36:48 And on the left-hand side, the constant term is 0. 608 00:36:48 --> 00:36:51 There is no constant term. 609 00:36:51 --> 00:36:57 On the right-hand side there's a constant term, 1/2 * 1. 610 00:36:57 --> 00:36:58 That's 1/2 here. 611 00:36:58 --> 00:37:01 And then there's another constant term, which is this 612 00:37:01 --> 00:37:03 constant times this - 1. 613 00:37:03 --> 00:37:09 Which is - C. 614 00:37:09 --> 00:37:18 And so the conclusion here is that C = 1/2. 615 00:37:18 --> 00:37:18 Another question. 616 00:37:18 --> 00:37:19 Yeah. 617 00:37:19 --> 00:37:33 STUDENT: [INAUDIBLE] 618 00:37:33 --> 00:37:37 PROFESSOR: There's also an x^ 0 power hidden in here. 619 00:37:37 --> 00:37:40 Sorry, an x ^ 1 , that's what you were asking about, sorry. 620 00:37:40 --> 00:37:42 There's also an x ^ 1 . 621 00:37:42 --> 00:37:45 The only reason why I didn't go to the x^ 1 is that it turns 622 00:37:45 --> 00:37:49 out with these two I didn't need it. 623 00:37:49 --> 00:37:51 The other thing is that by experience, I know that the 624 00:37:51 --> 00:37:53 extreme ends of the multiplication are 625 00:37:53 --> 00:37:54 the easiest ends. 626 00:37:54 --> 00:37:57 And the middle terms have tons of cross terms. 627 00:37:57 --> 00:37:59 And so I don't like the middle term as much because it always 628 00:37:59 --> 00:38:01 involves more arithmetic. 629 00:38:01 --> 00:38:05 So I stick to the lowest and the highest terms if I can. 630 00:38:05 --> 00:38:07 So that was really a sneaky thing. 631 00:38:07 --> 00:38:10 I did that without saying anything. 632 00:38:10 --> 00:38:10 Yes. 633 00:38:10 --> 00:38:14 STUDENT: [INAUDIBLE] 634 00:38:14 --> 00:38:15 PROFESSOR: Another good question. 635 00:38:15 --> 00:38:17 Could I just set equals 0? 636 00:38:17 --> 00:38:17 Absolutely. 637 00:38:17 --> 00:38:22 In fact, that's equivalent to picking out the x^ 0 term. 638 00:38:22 --> 00:38:24 And you could plug in numbers. 639 00:38:24 --> 00:38:25 If you wanted. 640 00:38:25 --> 00:38:27 That's another way of doing this besides doing that. 641 00:38:27 --> 00:38:30 So you can also plug in numbers. 642 00:38:30 --> 00:38:38 Can plug in numbers. x = 0. 643 00:38:38 --> 00:38:44 Actually, not x = 1, right? - 1, 2, etc. 644 00:38:44 --> 00:38:46 Not 1 just because we've already used it. 645 00:38:46 --> 00:38:48 We won't get interesting information out. 646 00:38:48 --> 00:38:48 Yes. 647 00:38:48 --> 00:38:56 STUDENT: [INAUDIBLE] 648 00:38:56 --> 00:38:58 PROFESSOR: So the question is, could I have done 649 00:38:58 --> 00:38:59 it this other way. 650 00:38:59 --> 00:39:02 With the polynomial, with this other one. 651 00:39:02 --> 00:39:04 Yes, absolutely. 652 00:39:04 --> 00:39:07 So in other words what I've taught you now is two choices 653 00:39:07 --> 00:39:09 which are equally reasonable. 654 00:39:09 --> 00:39:12 The one that I picked was the one that was the fastest for 655 00:39:12 --> 00:39:15 this guy and the one that was fastest for this one, but I 656 00:39:15 --> 00:39:19 could've done the other way around. 657 00:39:19 --> 00:39:22 There are a lot of ways of solving simultaneous equations. 658 00:39:22 --> 00:39:23 Yeah, another question. 659 00:39:23 --> 00:39:24 STUDENT: [INAUDIBLE] 660 00:39:24 --> 00:39:25 PROFESSOR: The question is the following. 661 00:39:25 --> 00:39:28 So now everybody can understand the question. 662 00:39:28 --> 00:39:33 If this, instead of being x ^2 + 1, this were x^3 + 1. 663 00:39:33 --> 00:39:36 So that's an important case to understand. 664 00:39:36 --> 00:39:39 That's a case in which this denominator is 665 00:39:39 --> 00:39:41 not fully factored. 666 00:39:41 --> 00:39:46 So it's an x^3 + 1, you would have to factor out an x + 1. 667 00:39:46 --> 00:39:49 So that would be a situation like this, you have an x^3 + 1, 668 00:39:49 --> 00:39:57 but that's (x + 1)( x^2 + x + 1), this kind of thing. 669 00:39:57 --> 00:40:01 If that's the right, there must be a minus sign in here maybe. 670 00:40:01 --> 00:40:03 OK, something like this. 671 00:40:03 --> 00:40:07 Right? 672 00:40:07 --> 00:40:08 Isn't that what it is? 673 00:40:08 --> 00:40:12 STUDENT: [INAUDIBLE] 674 00:40:12 --> 00:40:12 PROFESSOR: I think it's right. 675 00:40:12 --> 00:40:15 But anyway, the point is that you have to factor it. 676 00:40:15 --> 00:40:17 And then you have a linear and a quadratic. 677 00:40:17 --> 00:40:21 So you're always going to be faced eventually with linear 678 00:40:21 --> 00:40:23 factors and quadratic factors. 679 00:40:23 --> 00:40:25 If you have a cubic, that means you haven't 680 00:40:25 --> 00:40:28 factored sufficiently. 681 00:40:28 --> 00:40:32 So you're still back in Step 1. 682 00:40:32 --> 00:40:32 STUDENT: [INAUDIBLE] 683 00:40:32 --> 00:40:34 PROFESSOR: In the x^3 + 1 case? 684 00:40:34 --> 00:40:36 STUDENT: [INAUDIBLE] 685 00:40:36 --> 00:40:39 PROFESSOR: In the x^3 + 1 case, we are out of luck until we've 686 00:40:39 --> 00:40:41 completed the factorization. 687 00:40:41 --> 00:40:44 Once we've completed the factorization, we're going to 688 00:40:44 --> 00:40:48 have to deal with these two factors as denominators. 689 00:40:48 --> 00:40:54 So it'll be this plus something over x + 1 + a Bx + C type of 690 00:40:54 --> 00:40:58 thing over this thing here. 691 00:40:58 --> 00:41:01 That's what's eventually going to happen. 692 00:41:01 --> 00:41:03 But hold on to that idea. 693 00:41:03 --> 00:41:14 Let me carry out one more example here. 694 00:41:14 --> 00:41:17 So I've figured out what all the values are. 695 00:41:17 --> 00:41:21 But I think it's also worth it to remember now that we also 696 00:41:21 --> 00:41:29 have to carry out the integration. 697 00:41:29 --> 00:41:36 What I've just shown you is that the integral of x ^2 dx 698 00:41:36 --> 00:41:42 over (x - 1)( x ^2 + 1) is equal to, and I've split 699 00:41:42 --> 00:41:43 up into these pieces. 700 00:41:43 --> 00:41:45 So what are the pieces? 701 00:41:45 --> 00:42:01 The pieces are, 1/2, x - 1 + 1/2 x / x ^2 + 1. 702 00:42:01 --> 00:42:02 This is the A term. 703 00:42:02 --> 00:42:04 This is the B term. 704 00:42:04 --> 00:42:12 And then there's the C term. 705 00:42:12 --> 00:42:16 So we'd better remember that we know how to antidifferentiate 706 00:42:16 --> 00:42:18 these things. 707 00:42:18 --> 00:42:20 In other words, I want to finish the problem. 708 00:42:20 --> 00:42:22 The others were pretty easy, so I didn't bother to finish my 709 00:42:22 --> 00:42:26 sentence, but here I want to be careful and have you realize 710 00:42:26 --> 00:42:28 that there's something a little more to do. 711 00:42:28 --> 00:42:31 First of all we have the, the first one is no problem. 712 00:42:31 --> 00:42:35 That's this. 713 00:42:35 --> 00:42:39 The second one actually is not too bad either. 714 00:42:39 --> 00:42:45 This is, by the advanced guessing method, my favorite 715 00:42:45 --> 00:42:48 method, something like the logarithm, because that's 716 00:42:48 --> 00:42:50 what's going to appear in the denominator. 717 00:42:50 --> 00:42:51 And then, if you differentiate this, you're going 718 00:42:51 --> 00:42:53 to get 2x over this. 719 00:42:53 --> 00:42:55 But here we have 1/2. 720 00:42:55 --> 00:42:59 So altogether it's 1/4 of this. 721 00:42:59 --> 00:43:02 So I fixed the coefficient here. 722 00:43:02 --> 00:43:06 And then the last one, you have to think back to some level of 723 00:43:06 --> 00:43:10 memorization here and remember that this is 1/2 724 00:43:10 --> 00:43:15 the arc tangent. 725 00:43:15 --> 00:43:20 STUDENT: [INAUDIBLE] 726 00:43:20 --> 00:43:22 PROFESSOR: Why did I go to 1/4? 727 00:43:22 --> 00:43:26 Because in disguise, for this guy, I was thinking d / dx of 728 00:43:26 --> 00:43:35 ln (x ^2 + 1) = 2x / x^2 + 1. 729 00:43:35 --> 00:43:39 Because it's the derivative of this divided by itself. 730 00:43:39 --> 00:43:46 This is the derivative of ln u is u ' / u. 731 00:43:46 --> 00:43:50 Ln u' = u' / u. 732 00:43:50 --> 00:43:54 That was what I applied. 733 00:43:54 --> 00:43:58 And what I had was 1/2, so I need a total of 1/4 to cancel. 734 00:43:58 --> 00:44:06 So 2/4 is 1/2. 735 00:44:06 --> 00:44:09 Now I've got to get you out of one more deep hole. 736 00:44:09 --> 00:44:12 And I'm going to save the general pattern for next time. 737 00:44:12 --> 00:44:27 But I do want to clarify one thing. 738 00:44:27 --> 00:44:30 So let's handle this thing. 739 00:44:30 --> 00:44:34 What if the degree of P is bigger than or equal 740 00:44:34 --> 00:44:39 to the degree of Q. 741 00:44:39 --> 00:44:42 That's the thing that I claimed was easier. 742 00:44:42 --> 00:44:45 And I'm going to describe to you how it's done. 743 00:44:45 --> 00:44:49 Now, in analogy, with numbers you might call 744 00:44:49 --> 00:44:55 this an improper fraction. 745 00:44:55 --> 00:44:59 That's the thing that should echo in your mind when 746 00:44:59 --> 00:45:01 you're thinking about this. 747 00:45:01 --> 00:45:04 And I'm just going to do it by example here. 748 00:45:04 --> 00:45:08 Let's see., I cooked up an example so that I don't 749 00:45:08 --> 00:45:11 make an arithmetistic mistake along the way. 750 00:45:11 --> 00:45:16 So there are two or three steps that I need to explain. 751 00:45:16 --> 00:45:18 So here's an example. 752 00:45:18 --> 00:45:21 The denominator's degree 2, the numerator is degree 3. 753 00:45:21 --> 00:45:24 It well exceeds, so there's a problem here. 754 00:45:24 --> 00:45:27 Our method is not going to work. 755 00:45:27 --> 00:45:32 And the first step that I want to carry out 756 00:45:32 --> 00:45:36 is to reverse Step 1. 757 00:45:36 --> 00:45:38 That is, I don't want the factorization for what 758 00:45:38 --> 00:45:39 I'm going to do next. 759 00:45:39 --> 00:45:42 I want it multiplied out. 760 00:45:42 --> 00:45:48 That means I have to multiply through, so I get x ^2 + x - 2. 761 00:45:48 --> 00:45:52 I'm back to the starting place here. 762 00:45:52 --> 00:45:57 And now, the next thing that I'm going to do is, I'm 763 00:45:57 --> 00:46:01 going to use long division. 764 00:46:01 --> 00:46:04 That's how you convert an improper fraction 765 00:46:04 --> 00:46:07 to a proper fraction. 766 00:46:07 --> 00:46:12 This is something you were supposed to learn in, I don't 767 00:46:12 --> 00:46:16 know, Grade 4, I know. 768 00:46:16 --> 00:46:23 Grade 3, Grade 4, Grade 5, Grade 6, etc. 769 00:46:23 --> 00:46:27 So here's how it works in the case of polynomials. 770 00:46:27 --> 00:46:31 It's kind of amusing. 771 00:46:31 --> 00:46:37 So we're dividing this polynomial into that one. 772 00:46:37 --> 00:46:41 And so the quotient to first order here is x. 773 00:46:41 --> 00:46:44 That is, that's going to match the top terms. 774 00:46:44 --> 00:46:48 So I get x^3 + x ^2 - 2x. 775 00:46:48 --> 00:46:50 That's the product. 776 00:46:50 --> 00:46:51 And now I subtract. 777 00:46:51 --> 00:46:53 And it cancels. 778 00:46:53 --> 00:46:58 So we get here - x ^2 + 2x. 779 00:46:58 --> 00:47:01 That's the difference. 780 00:47:01 --> 00:47:04 And now I need to divide this next term in. 781 00:47:04 --> 00:47:08 And I need a - 1. 782 00:47:08 --> 00:47:14 So - 1 times this is - x ^2 - x + 2. 783 00:47:14 --> 00:47:16 And I subtract. 784 00:47:16 --> 00:47:17 And the x^2's cancel. 785 00:47:17 --> 00:47:24 And here I get + 3x - 2. 786 00:47:24 --> 00:47:27 Now, this thing has a name. 787 00:47:27 --> 00:47:30 This is called the quotient. 788 00:47:30 --> 00:47:33 And this thing also has a name. 789 00:47:33 --> 00:47:39 This is called the remainder. 790 00:47:39 --> 00:47:43 And now I'll show you how it works by sticking it 791 00:47:43 --> 00:47:44 into the answer here. 792 00:47:44 --> 00:47:47 The quotient is x - 1. 793 00:47:47 --> 00:47:51 And the remainder is, let's get down there. 794 00:47:51 --> 00:47:58 3x - 2 / x ^2 + x - 2. 795 00:47:58 --> 00:48:03 So the punchline here is that this thing is 796 00:48:03 --> 00:48:05 easy to integrate. 797 00:48:05 --> 00:48:08 This is easy. 798 00:48:08 --> 00:48:17 And this one, you can use, now you can use cover-up, The 799 00:48:17 --> 00:48:18 method that we had before. 800 00:48:18 --> 00:48:21 Because the degree of the numerator is now below the 801 00:48:21 --> 00:48:22 degree of the denominator. 802 00:48:22 --> 00:48:25 It's now first degree and this is second degree. 803 00:48:25 --> 00:48:27 What you can't do is use cover-up to 804 00:48:27 --> 00:48:29 start out with here. 805 00:48:29 --> 00:48:32 That will give you the wrong answer. 806 00:48:32 --> 00:48:37 So that's the end for today, and see you next time. 807 00:48:37 --> 00:48:37