1 00:00:00,120 --> 00:00:02,330 The following content is provided under a Creative 2 00:00:02,330 --> 00:00:03,610 Commons license. 3 00:00:03,610 --> 00:00:06,110 Your support will help MIT OpenCourseWare 4 00:00:06,110 --> 00:00:09,950 continue to offer high quality educational resources for free. 5 00:00:09,950 --> 00:00:12,540 To make a donation, or to view additional materials 6 00:00:12,540 --> 00:00:16,160 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,160 --> 00:00:22,530 at ocw.mit.edu. 8 00:00:22,530 --> 00:00:24,970 PROFESSOR: Today we're moving on from theoretical things, 9 00:00:24,970 --> 00:00:30,570 from the mean value theorem, to the introduction to what's 10 00:00:30,570 --> 00:00:33,120 going to occupy us for the whole rest of the course, which 11 00:00:33,120 --> 00:00:34,850 is integration. 12 00:00:34,850 --> 00:00:38,850 So, in order to introduce that subject, 13 00:00:38,850 --> 00:00:42,660 I need to introduce for you a new notation, which 14 00:00:42,660 --> 00:00:52,640 is called differentials. 15 00:00:52,640 --> 00:00:55,640 I'm going to tell you what a differential is, 16 00:00:55,640 --> 00:01:02,040 and we'll get used to using it over time. 17 00:01:02,040 --> 00:01:10,290 If you have a function which is y = f(x), 18 00:01:10,290 --> 00:01:27,650 then the differential of y is going to be denoted dy, 19 00:01:27,650 --> 00:01:29,500 and it's by definition f'(x) dx. 20 00:01:34,160 --> 00:01:41,410 So here's the notation. 21 00:01:41,410 --> 00:01:44,800 And because y is really equal to f, sometimes 22 00:01:44,800 --> 00:01:50,140 we also call it the differential of f. 23 00:01:50,140 --> 00:01:58,700 It's also called the differential of f. 24 00:01:58,700 --> 00:02:06,750 That's the notation, and it's the same thing 25 00:02:06,750 --> 00:02:11,630 as what happens if you formally just take this dx, 26 00:02:11,630 --> 00:02:14,910 act like it's a number and divide it into dy. 27 00:02:14,910 --> 00:02:22,670 So it means the same thing as this statement here. 28 00:02:22,670 --> 00:02:29,160 And this is more or less the Leibniz interpretation 29 00:02:29,160 --> 00:02:38,360 of derivatives. 30 00:02:38,360 --> 00:02:50,990 Of a derivative as a ratio of these so called differentials. 31 00:02:50,990 --> 00:03:04,290 It's a ratio of what are known as infinitesimals. 32 00:03:04,290 --> 00:03:09,050 Now, this is kind of a vague notion, this little bit 33 00:03:09,050 --> 00:03:12,830 here being an infinitesimal. 34 00:03:12,830 --> 00:03:16,110 It's sort of like an infinitely small quantity. 35 00:03:16,110 --> 00:03:21,300 And Leibniz perfected the idea of dealing 36 00:03:21,300 --> 00:03:23,050 with these intuitively. 37 00:03:23,050 --> 00:03:26,730 And subsequently, mathematicians use them all the time. 38 00:03:26,730 --> 00:03:33,350 They're way more effective than the notation that Newton used. 39 00:03:33,350 --> 00:03:36,250 You might think that notations are a small matter, 40 00:03:36,250 --> 00:03:40,960 but they allow you to think much faster, sometimes. 41 00:03:40,960 --> 00:03:43,565 When you have the right names and the right symbols 42 00:03:43,565 --> 00:03:44,190 for everything. 43 00:03:44,190 --> 00:03:47,860 And in this case it made it very big difference. 44 00:03:47,860 --> 00:03:52,850 Leibniz's notation was adopted on the continent and Newton 45 00:03:52,850 --> 00:03:56,650 dominated in Britain and, as a result, 46 00:03:56,650 --> 00:03:58,800 the British fell behind by one or two 47 00:03:58,800 --> 00:04:01,850 hundred years in the development of calculus. 48 00:04:01,850 --> 00:04:03,540 It was really a serious matter. 49 00:04:03,540 --> 00:04:05,860 So it's really well worth your while to get 50 00:04:05,860 --> 00:04:08,940 used to this idea of ratios. 51 00:04:08,940 --> 00:04:12,140 And it comes up all over the place, both in this class 52 00:04:12,140 --> 00:04:14,320 and also in multivariable calculus. 53 00:04:14,320 --> 00:04:17,380 It's used in many contexts. 54 00:04:17,380 --> 00:04:20,030 So first of all, just to go a little bit easy. 55 00:04:20,030 --> 00:04:25,320 We'll illustrate it by its use in linear approximations, 56 00:04:25,320 --> 00:04:36,420 which we've already done. 57 00:04:36,420 --> 00:04:38,780 The picture here, which we've drawn a number of times, 58 00:04:38,780 --> 00:04:41,020 is that you have some function. 59 00:04:41,020 --> 00:04:44,300 And here's a value of the function. 60 00:04:44,300 --> 00:04:47,060 And it's coming up like that. 61 00:04:47,060 --> 00:04:48,330 So here's our function. 62 00:04:48,330 --> 00:04:51,090 And we go forward a little increment 63 00:04:51,090 --> 00:04:56,540 to a place which is dx further along. 64 00:04:56,540 --> 00:04:59,350 The idea of this notation is that dx 65 00:04:59,350 --> 00:05:05,260 is going to replace the symbol delta x, 66 00:05:05,260 --> 00:05:07,370 which is the change in x. 67 00:05:07,370 --> 00:05:10,270 And we won't think too hard about-- well, 68 00:05:10,270 --> 00:05:12,560 this is a small quantity, this is a small quantity, 69 00:05:12,560 --> 00:05:16,030 we're not going to think too hard about what that means. 70 00:05:16,030 --> 00:05:20,780 Now, similarly, if you see how much we've gone up - well, 71 00:05:20,780 --> 00:05:26,600 this is kind of low, so it's a small bit here. 72 00:05:26,600 --> 00:05:31,240 So this distance here is, previously 73 00:05:31,240 --> 00:05:36,040 we called it delta y. 74 00:05:36,040 --> 00:05:41,810 But now we're just going to call it dy. 75 00:05:41,810 --> 00:05:51,200 So dy replaces delta y. 76 00:05:51,200 --> 00:05:57,450 So this is the change in level of the function. 77 00:05:57,450 --> 00:05:59,760 And we'll represent it symbolically this way. 78 00:05:59,760 --> 00:06:04,050 Very frequently, this just saves a little bit of notation. 79 00:06:04,050 --> 00:06:05,930 For the purposes of this, we'll be 80 00:06:05,930 --> 00:06:09,620 doing the same things we did with delta x and delta y, 81 00:06:09,620 --> 00:06:12,540 but this is the way that Leibniz thought of it. 82 00:06:12,540 --> 00:06:14,690 And he would just have drawn it with this. 83 00:06:14,690 --> 00:06:24,670 So this distance here is dx and this distance here is dy. 84 00:06:24,670 --> 00:06:30,320 So for an example of linear approximation, 85 00:06:30,320 --> 00:06:35,970 we'll say what's 64.1, say, to the 1/3 power, 86 00:06:35,970 --> 00:06:39,500 approximately equal to? 87 00:06:39,500 --> 00:06:43,470 Now, I'm going to carry this out in this new notation here. 88 00:06:43,470 --> 00:06:47,810 The function involved is x^1/3. 89 00:06:47,810 --> 00:06:50,990 And then it's a differential, dy. 90 00:06:50,990 --> 00:06:53,670 Now, I want to use this rule to get used to it. 91 00:06:53,670 --> 00:06:56,870 Because this is what we're going to be doing all of today is, 92 00:06:56,870 --> 00:07:00,010 we're differentiating, or taking the differential of y. 93 00:07:00,010 --> 00:07:02,490 So that is going to be just the derivative. 94 00:07:02,490 --> 00:07:11,980 That's 1/3 x^(-2/3) dx. 95 00:07:11,980 --> 00:07:18,060 And now I'm just going to fill in exactly what this is. 96 00:07:18,060 --> 00:07:25,450 At x = 64, which is the natural place close by where it's easy 97 00:07:25,450 --> 00:07:36,110 to do the evaluations, we have y = 64^(1/3), which is just 4. 98 00:07:36,110 --> 00:07:39,740 And how about dy? 99 00:07:39,740 --> 00:07:42,360 Well, so this is a little bit more complicated. 100 00:07:42,360 --> 00:07:43,540 Put it over here. 101 00:07:43,540 --> 00:07:50,690 So dy = 1/3 64^(-2/3) dx. 102 00:07:55,800 --> 00:08:16,680 And that is 1/3 * 1/16 dx, which is 1/48 dx. 103 00:08:16,680 --> 00:08:19,810 And now I'm going to work out what 104 00:08:19,810 --> 00:08:26,320 64 to the, whatever it is here, this strange fraction. 105 00:08:26,320 --> 00:08:31,210 I just want to be very careful to explain to you one more 106 00:08:31,210 --> 00:08:33,170 thing. 107 00:08:33,170 --> 00:08:37,780 Which is that we're using x = 64, 108 00:08:37,780 --> 00:08:45,240 and so we're thinking of x + dx is going to be 64.1. 109 00:08:45,240 --> 00:08:53,700 So that means that dx is going to be 1/10. 110 00:08:53,700 --> 00:08:59,760 So that's the increment that we're interested in. 111 00:08:59,760 --> 00:09:03,910 And now I can carry out the approximation. 112 00:09:03,910 --> 00:09:11,230 The approximation says that 64.1^(1/3) is, well, 113 00:09:11,230 --> 00:09:14,710 it's approximately what I'm going to call y + dy. 114 00:09:14,710 --> 00:09:18,060 Because really, the dy that I'm determining here 115 00:09:18,060 --> 00:09:26,450 is determined by this linear relation. dy = 1/48 dx. 116 00:09:26,450 --> 00:09:29,820 And so this is only approximately true. 117 00:09:29,820 --> 00:09:33,430 Because what's really true is that this 118 00:09:33,430 --> 00:09:37,690 is equal to y + delta y. 119 00:09:37,690 --> 00:09:39,590 In our previous notation. 120 00:09:39,590 --> 00:09:41,600 So this is in disguise. 121 00:09:41,600 --> 00:09:42,960 What this is equal to. 122 00:09:42,960 --> 00:09:44,500 And that's the only approximately 123 00:09:44,500 --> 00:09:47,470 equal to what the linear approximation would give you. 124 00:09:47,470 --> 00:09:51,730 So, really, even though I wrote dy is this increment here, 125 00:09:51,730 --> 00:09:54,830 what it really is if dx is exactly that, 126 00:09:54,830 --> 00:09:57,410 is it's the amount it would go up 127 00:09:57,410 --> 00:10:00,270 if you went straight up the tangent line. 128 00:10:00,270 --> 00:10:02,300 So I'm not going to do that because that's not 129 00:10:02,300 --> 00:10:03,370 what people write. 130 00:10:03,370 --> 00:10:06,040 And that's not even what they think. 131 00:10:06,040 --> 00:10:08,110 They're really thinking of both dx and dy 132 00:10:08,110 --> 00:10:10,750 as being infinitesimally small. 133 00:10:10,750 --> 00:10:15,900 And here we're going to the finite level and doing it. 134 00:10:15,900 --> 00:10:20,350 So this is just something you have to live with, 135 00:10:20,350 --> 00:10:28,120 is a little ambiguity in this notation. 136 00:10:28,120 --> 00:10:29,490 This is the approximation. 137 00:10:29,490 --> 00:10:32,970 And now I can just calculate these numbers here. 138 00:10:32,970 --> 00:10:36,130 y at this value is 4. 139 00:10:36,130 --> 00:10:43,630 And dy, as I said, is 1/48 dx. 140 00:10:43,630 --> 00:10:50,440 And that turns out to be 4 + 1/480, because dx is 1/10. 141 00:10:50,440 --> 00:10:54,390 So that's approximately 4.002. 142 00:10:54,390 --> 00:11:04,550 And that's our approximation. 143 00:11:04,550 --> 00:11:20,760 Now, let's just compare it to our previous notation. 144 00:11:20,760 --> 00:11:22,970 This will serve as a review of, if you 145 00:11:22,970 --> 00:11:35,870 like, of linear approximation. 146 00:11:35,870 --> 00:11:38,990 But what I want to emphasize is that these things 147 00:11:38,990 --> 00:11:43,210 are supposed to be the same. 148 00:11:43,210 --> 00:11:45,520 Just that it's really the same thing. 149 00:11:45,520 --> 00:11:52,280 It's just a different notation for the same thing. 150 00:11:52,280 --> 00:11:56,360 I remind you the basic formula for linear approximation is 151 00:11:56,360 --> 00:12:00,980 that f(x) is approximately f(a) + f'(a) (x-a). 152 00:12:05,160 --> 00:12:11,620 And we're applying it in the situation that a = 64 and f(x) 153 00:12:11,620 --> 00:12:12,910 = x^(1/3). 154 00:12:17,980 --> 00:12:27,580 And so f(a), which is f(64), is of course 4. 155 00:12:27,580 --> 00:12:43,210 And f'(a), which is 1/3 a^(-2/3), is in our case 1/16. 156 00:12:43,210 --> 00:12:49,600 No, 1/48. 157 00:12:49,600 --> 00:12:52,980 OK, that's the same calculation as before. 158 00:12:52,980 --> 00:12:59,490 And then our relationship becomes x^(1/3) is 159 00:12:59,490 --> 00:13:08,420 approximately equal to 4 plus 1/48 times x minus a, which is 160 00:13:08,420 --> 00:13:12,250 64. 161 00:13:12,250 --> 00:13:14,820 So look, every single number that I've written over here 162 00:13:14,820 --> 00:13:20,030 has a corresponding number for this other method. 163 00:13:20,030 --> 00:13:23,250 And now if I plug in the value we happen to want, 164 00:13:23,250 --> 00:13:32,500 which is the 64.1, this would be 4 + 1/48 1/10, 165 00:13:32,500 --> 00:13:38,580 which is just the same thing we had before. 166 00:13:38,580 --> 00:13:45,270 So again, same answer. 167 00:13:45,270 --> 00:13:55,300 Same method, new notation. 168 00:13:55,300 --> 00:14:02,300 Well, now I get to use this notation in a novel way. 169 00:14:02,300 --> 00:14:04,640 So again, here's the notation. 170 00:14:04,640 --> 00:14:16,410 This notation of differential. 171 00:14:16,410 --> 00:14:20,520 The way I'm going to use it is in discussing something 172 00:14:20,520 --> 00:14:32,560 called antiderivative Again, this is a new notation now. 173 00:14:32,560 --> 00:14:33,680 But it's also a new idea. 174 00:14:33,680 --> 00:14:37,850 It's one that we haven't discussed yet. 175 00:14:37,850 --> 00:14:42,500 Namely, the notation that I want to describe here 176 00:14:42,500 --> 00:14:48,090 is what's called the integral of g(x) dx. 177 00:14:48,090 --> 00:14:51,560 And I'll denote that by a function capital G of x. 178 00:14:51,560 --> 00:14:53,820 So it's, you start with a function g(x) 179 00:14:53,820 --> 00:14:55,380 and you produce a function capital 180 00:14:55,380 --> 00:15:12,190 G(x), which is called the antiderivative of g. 181 00:15:12,190 --> 00:15:15,330 Notice there's a differential sitting in here. 182 00:15:15,330 --> 00:15:31,380 This symbol, this guy here, is called an integral sign. 183 00:15:31,380 --> 00:15:34,600 Or an integral, or this whole thing is called an integral. 184 00:15:34,600 --> 00:15:37,830 And another name for the antiderivative of g 185 00:15:37,830 --> 00:15:50,700 is the indefinite integral of g. 186 00:15:50,700 --> 00:15:58,050 And I'll explain to you why it's indefinite in just-- 187 00:15:58,050 --> 00:16:04,330 very shortly here. 188 00:16:04,330 --> 00:16:13,330 Well, so let's carry out some examples. 189 00:16:13,330 --> 00:16:16,840 Basically what I'd like to do is as many examples 190 00:16:16,840 --> 00:16:18,510 along the lines of all the derivatives 191 00:16:18,510 --> 00:16:21,329 that we derived at the beginning of the course. 192 00:16:21,329 --> 00:16:22,870 In other words, in principle you want 193 00:16:22,870 --> 00:16:26,550 to be able to integrate as many things as possible. 194 00:16:26,550 --> 00:16:34,680 We're going to start out with the integral of sin x dx. 195 00:16:34,680 --> 00:16:40,670 That's a function whose derivative is sin x. 196 00:16:40,670 --> 00:16:44,990 So what function would that be? 197 00:16:44,990 --> 00:16:48,400 Cosine x, minus, right. 198 00:16:48,400 --> 00:16:49,590 It's -cos x. 199 00:16:52,700 --> 00:16:56,240 So -cos x differentiated gives you sin x. 200 00:16:56,240 --> 00:17:00,540 So that is an antiderivative of sine. 201 00:17:00,540 --> 00:17:02,210 And it satisfies this property. 202 00:17:02,210 --> 00:17:09,520 So this function, G(x) = - cos x, 203 00:17:09,520 --> 00:17:15,170 has the property that its derivative is sin x. 204 00:17:15,170 --> 00:17:20,700 On the other hand, if you differentiate a constant, 205 00:17:20,700 --> 00:17:22,230 you get 0. 206 00:17:22,230 --> 00:17:25,090 So this answer is what's called indefinite. 207 00:17:25,090 --> 00:17:28,910 Because you can also add any constant here. 208 00:17:28,910 --> 00:17:33,610 And the same thing will be true. 209 00:17:33,610 --> 00:17:38,060 So, c is constant. 210 00:17:38,060 --> 00:17:41,660 And as I said, the integral is called indefinite. 211 00:17:41,660 --> 00:17:45,410 So that's an explanation for this modifier 212 00:17:45,410 --> 00:17:46,810 in front of the "integral". 213 00:17:46,810 --> 00:17:48,960 It's indefinite because we actually didn't 214 00:17:48,960 --> 00:17:50,540 specify a single function. 215 00:17:50,540 --> 00:17:52,170 We don't get a single answer. 216 00:17:52,170 --> 00:17:54,400 Whenever you take the antiderivative of something 217 00:17:54,400 --> 00:18:08,110 it's ambiguous up to a constant. 218 00:18:08,110 --> 00:18:12,340 Next, let's do some other standard functions 219 00:18:12,340 --> 00:18:13,890 from our repertoire. 220 00:18:13,890 --> 00:18:17,820 We have the integral of x^a dx. 221 00:18:17,820 --> 00:18:20,730 Some power, the integral of a power. 222 00:18:20,730 --> 00:18:24,680 And if you think about it, what you should be differentiating 223 00:18:24,680 --> 00:18:27,590 is one power larger than that. 224 00:18:27,590 --> 00:18:33,250 But then you have to divide by 1/(a+1), 225 00:18:33,250 --> 00:18:36,780 in order that the differentiation be correct. 226 00:18:36,780 --> 00:18:44,360 So this just is the fact that d/dx of x^(a+1), 227 00:18:44,360 --> 00:18:46,600 or maybe I should even say it this way. 228 00:18:46,600 --> 00:18:49,620 Maybe I'll do it in differential notation. 229 00:18:49,620 --> 00:18:54,230 d(x^(a+1)) = (a+1) x^a dx. 230 00:18:57,370 --> 00:19:02,950 So if I divide that through by a+1, 231 00:19:02,950 --> 00:19:06,280 then I get the relation above. 232 00:19:06,280 --> 00:19:11,800 And because this is ambiguous up to a constant, 233 00:19:11,800 --> 00:19:15,820 it could be any additional constant 234 00:19:15,820 --> 00:19:20,830 added to that function. 235 00:19:20,830 --> 00:19:26,290 Now, the identity that I wrote down below is correct. 236 00:19:26,290 --> 00:19:30,700 But this one is not always correct. 237 00:19:30,700 --> 00:19:35,200 What's the exception? 238 00:19:35,200 --> 00:19:38,211 Yeah. a equals-- 239 00:19:38,211 --> 00:19:38,710 STUDENT: 0. 240 00:19:38,710 --> 00:19:42,060 PROFESSOR: Negative 1. 241 00:19:42,060 --> 00:19:47,440 So this one is OK for all a. 242 00:19:47,440 --> 00:19:52,550 But this one fails because we've divided by 0 when a = -1. 243 00:19:52,550 --> 00:20:04,770 So this is only true when a is not equal to -1. 244 00:20:04,770 --> 00:20:07,790 And in fact, of course, what's happening when a = 0, 245 00:20:07,790 --> 00:20:11,670 you're getting 0 when you differentiate the constant. 246 00:20:11,670 --> 00:20:15,510 So there's a third case that we have to carry out. 247 00:20:15,510 --> 00:20:25,230 Which is the exceptional case, namely the integral of dx/x. 248 00:20:25,230 --> 00:20:31,265 And this time, if we just think back 249 00:20:31,265 --> 00:20:32,640 to what our-- So what we're doing 250 00:20:32,640 --> 00:20:35,080 is thinking backwards here, which a very important thing 251 00:20:35,080 --> 00:20:38,690 to do in math at all stages. 252 00:20:38,690 --> 00:20:41,800 We got all of our formulas, now we're reading them backwards. 253 00:20:41,800 --> 00:20:49,890 And so this one, you may remember, is ln x. 254 00:20:49,890 --> 00:20:53,370 The reason why I want to do this carefully and slowly now, 255 00:20:53,370 --> 00:20:57,299 is right now I also want to write the more standard form 256 00:20:57,299 --> 00:20:58,090 which is presented. 257 00:20:58,090 --> 00:21:01,540 So first of all, first we have to add a constant. 258 00:21:01,540 --> 00:21:04,160 And please don't put the parentheses here. 259 00:21:04,160 --> 00:21:10,360 The parentheses go there. 260 00:21:10,360 --> 00:21:14,700 But there's another formula hiding in the woodwork 261 00:21:14,700 --> 00:21:16,500 here behind this one. 262 00:21:16,500 --> 00:21:19,260 Which is that you can also get the correct formula 263 00:21:19,260 --> 00:21:20,870 when x is negative. 264 00:21:20,870 --> 00:21:27,130 And that turns out to be this one here. 265 00:21:27,130 --> 00:21:32,670 So I'm treating the case, x positive, as being something 266 00:21:32,670 --> 00:21:34,460 that you know. 267 00:21:34,460 --> 00:21:43,530 But let's check the case, x negative. 268 00:21:43,530 --> 00:21:45,810 In order to check the case x negative, 269 00:21:45,810 --> 00:21:51,750 I have to differentiate the logarithm of the absolute value 270 00:21:51,750 --> 00:21:55,680 of x in that case. 271 00:21:55,680 --> 00:21:57,612 And that's the same thing, again, 272 00:21:57,612 --> 00:22:02,170 for x negative as the derivative of the logarithm of negative x. 273 00:22:02,170 --> 00:22:08,410 That's the formula, when x is negative. 274 00:22:08,410 --> 00:22:10,980 And if you carry that out, what you 275 00:22:10,980 --> 00:22:18,400 get, maybe I'll put this over here, is, 276 00:22:18,400 --> 00:22:20,800 well, it's the chain rule. 277 00:22:20,800 --> 00:22:27,160 It's 1/(-x) times the derivative with respect to x of -x. 278 00:22:27,160 --> 00:22:30,730 So see that there are two minus signs. 279 00:22:30,730 --> 00:22:32,750 There's a -x in the denominator and then 280 00:22:32,750 --> 00:22:35,610 there's the derivative of -x in the numerator. 281 00:22:35,610 --> 00:22:38,070 That's just -1. 282 00:22:38,070 --> 00:22:39,160 This part is -1. 283 00:22:39,160 --> 00:22:43,100 So this -1 over -x, which is 1/x. 284 00:22:43,100 --> 00:22:53,200 So the negative signs cancel. 285 00:22:53,200 --> 00:23:00,090 If you just keep track of this in terms of ln(-x) and its 286 00:23:00,090 --> 00:23:05,480 graph, that's a function that looks like this. 287 00:23:05,480 --> 00:23:08,370 For x negative. 288 00:23:08,370 --> 00:23:14,170 And its derivative is 1/x, I claim. 289 00:23:14,170 --> 00:23:17,120 And if you just look at it a little bit carefully, 290 00:23:17,120 --> 00:23:23,431 you see that the slope is always negative. 291 00:23:23,431 --> 00:23:23,930 Right? 292 00:23:23,930 --> 00:23:26,950 So here the slope is negative. 293 00:23:26,950 --> 00:23:30,480 So it's going to be below the axis. 294 00:23:30,480 --> 00:23:32,890 And, in fact, it's getting steeper and steeper negative 295 00:23:32,890 --> 00:23:34,980 as we go down. 296 00:23:34,980 --> 00:23:37,980 And it's getting less and less negative as we go horizontally. 297 00:23:37,980 --> 00:23:40,790 So it's going like this, which is indeed 298 00:23:40,790 --> 00:23:43,770 the graph of this function, for x negative. 299 00:23:43,770 --> 00:23:53,320 Again, x negative. 300 00:23:53,320 --> 00:23:56,850 So that's one other standard formula. 301 00:23:56,850 --> 00:24:00,501 And very quickly, very often, we won't put the absolute value 302 00:24:00,501 --> 00:24:01,000 signs. 303 00:24:01,000 --> 00:24:03,180 We'll only consider the case x positive here. 304 00:24:03,180 --> 00:24:06,280 But I just want you to have the tools to do it 305 00:24:06,280 --> 00:24:08,710 in case we want to use, we want to handle, 306 00:24:08,710 --> 00:24:14,040 both positive and negative x. 307 00:24:14,040 --> 00:24:28,620 Now, let's do two more examples. 308 00:24:28,620 --> 00:24:35,870 The integral of sec^2 x dx. 309 00:24:35,870 --> 00:24:38,270 These are supposed to get you to remember 310 00:24:38,270 --> 00:24:41,130 all of your differentiation formulas, the standard ones. 311 00:24:41,130 --> 00:24:48,200 So this one, integral of sec^2 dx is what? tan x. 312 00:24:48,200 --> 00:24:50,690 And here we have + c, all right? 313 00:24:50,690 --> 00:24:55,520 And then the last one of, a couple of, this type would be, 314 00:24:55,520 --> 00:24:56,800 let's see. 315 00:24:56,800 --> 00:25:04,139 I should do at least this one here, square root of 1 - x^2. 316 00:25:04,139 --> 00:25:05,680 This is another notation, by the way, 317 00:25:05,680 --> 00:25:07,240 which is perfectly acceptable. 318 00:25:07,240 --> 00:25:10,280 Notice I've put the dx in the numerator 319 00:25:10,280 --> 00:25:13,400 and the function in the denominator here. 320 00:25:13,400 --> 00:25:18,880 So this one turns out to be sin^(-1) x. 321 00:25:18,880 --> 00:25:23,710 And, finally, let's see. 322 00:25:23,710 --> 00:25:28,470 About the integral of dx / (1 + x^2). 323 00:25:28,470 --> 00:25:41,320 That one is tan^(-1) x. 324 00:25:41,320 --> 00:25:43,652 For a little while, because you're reading these things 325 00:25:43,652 --> 00:25:45,110 backwards and forwards, you'll find 326 00:25:45,110 --> 00:25:46,970 this happens to you on exams. 327 00:25:46,970 --> 00:25:49,540 It gets slightly worse for a little while. 328 00:25:49,540 --> 00:25:53,120 You will antidifferentiate when you meant to differentiate. 329 00:25:53,120 --> 00:25:54,620 And you'll differentiate when you're 330 00:25:54,620 --> 00:25:57,230 meant to antidifferentiate. 331 00:25:57,230 --> 00:26:00,190 Don't get too frustrated by that. 332 00:26:00,190 --> 00:26:05,270 But eventually, you'll get them squared away. 333 00:26:05,270 --> 00:26:08,130 And it actually helps to do a lot of practice 334 00:26:08,130 --> 00:26:15,410 with antidifferentiations, or integrations, 335 00:26:15,410 --> 00:26:17,100 as they're sometimes called. 336 00:26:17,100 --> 00:26:19,790 Because that will solidify your remembering 337 00:26:19,790 --> 00:26:25,820 all of the differentiation formulas. 338 00:26:25,820 --> 00:26:30,080 So, last bit of information that I 339 00:26:30,080 --> 00:26:32,730 want to emphasize before we go on some more 340 00:26:32,730 --> 00:26:44,130 complicated examples is this. 341 00:26:44,130 --> 00:26:49,250 It's obvious because the derivative of a constant is 0. 342 00:26:49,250 --> 00:26:55,330 That the antiderivative is ambiguous up to a constant. 343 00:26:55,330 --> 00:26:56,920 But it's very important to realize 344 00:26:56,920 --> 00:27:01,410 that this is the only ambiguity that there is. 345 00:27:01,410 --> 00:27:04,390 So the last thing that I want to tell you about 346 00:27:04,390 --> 00:27:24,070 is uniqueness of antiderivatives up to a constant. 347 00:27:24,070 --> 00:27:30,290 The theorem is the following. 348 00:27:30,290 --> 00:27:41,230 The theorem is if F' = G', then F equals G-- 349 00:27:41,230 --> 00:27:43,620 so F(x) equals G(x) plus a constant. 350 00:27:48,420 --> 00:27:54,050 But that means, not only that these are antiderivatives, 351 00:27:54,050 --> 00:27:56,640 all these things with these plus c's are antiderivatives. 352 00:27:56,640 --> 00:28:02,080 But these are the only ones. 353 00:28:02,080 --> 00:28:03,300 Which is very reassuring. 354 00:28:03,300 --> 00:28:06,020 And that's a kind of uniqueness, although its uniqueness up 355 00:28:06,020 --> 00:28:09,660 to a constant, it's acceptable to us. 356 00:28:09,660 --> 00:28:12,850 Now, the proof of this is very quick. 357 00:28:12,850 --> 00:28:18,216 But this is a fundamental fact. 358 00:28:18,216 --> 00:28:19,340 The proof is the following. 359 00:28:19,340 --> 00:28:29,340 If F' = G', then if you take the difference between the two 360 00:28:29,340 --> 00:28:33,520 functions, its derivative, which of course is F' - 361 00:28:33,520 --> 00:28:40,110 G', is equal to 0. 362 00:28:40,110 --> 00:28:55,300 Hence, F(x) - G(x) is a constant. 363 00:28:55,300 --> 00:28:58,940 Now, this is a key fact. 364 00:28:58,940 --> 00:28:59,870 Very important fact. 365 00:28:59,870 --> 00:29:03,840 We deduced it last time from the mean value theorem. 366 00:29:03,840 --> 00:29:05,370 It's not a small matter. 367 00:29:05,370 --> 00:29:06,850 It's a very, very important thing. 368 00:29:06,850 --> 00:29:08,690 It's the basis for calculus. 369 00:29:08,690 --> 00:29:11,550 It's the reason why calculus make sense. 370 00:29:11,550 --> 00:29:14,120 If we didn't have the fact that the derivative is 371 00:29:14,120 --> 00:29:18,890 0 implied that the function was constant, we would be done. 372 00:29:18,890 --> 00:29:23,012 We would have-- Calculus would be just useless for us. 373 00:29:23,012 --> 00:29:24,470 The point is, the rate of change is 374 00:29:24,470 --> 00:29:27,330 supposed to determine the function up 375 00:29:27,330 --> 00:29:29,650 to this starting value. 376 00:29:29,650 --> 00:29:32,330 So this conclusion is very important. 377 00:29:32,330 --> 00:29:35,010 And we already checked it last time, this conclusion. 378 00:29:35,010 --> 00:29:37,710 And now just by algebra, I can rearrange 379 00:29:37,710 --> 00:30:03,340 this to say that F(x) is equal to G(x) plus a constant. 380 00:30:03,340 --> 00:30:07,530 Now, maybe I should leave differentials up here. 381 00:30:07,530 --> 00:30:21,390 Because I want to illustrate-- So let's 382 00:30:21,390 --> 00:30:29,050 go on to some trickier, slightly trickier, integrals. 383 00:30:29,050 --> 00:30:35,740 Here's an example. 384 00:30:35,740 --> 00:30:42,010 The integral of, say, x^3 (x^4 + 2)^5 dx. 385 00:30:51,210 --> 00:30:53,480 This is a function which you actually 386 00:30:53,480 --> 00:30:56,500 do know how to integrate, because we already 387 00:30:56,500 --> 00:30:59,840 have a formula for all powers. 388 00:30:59,840 --> 00:31:03,280 Namely, the integral of x^a is equal to this. 389 00:31:03,280 --> 00:31:06,520 And even if it were a negative power, we could do it. 390 00:31:06,520 --> 00:31:08,630 So it's OK. 391 00:31:08,630 --> 00:31:14,290 On the other hand, to expand the 5th power here is quite a mess. 392 00:31:14,290 --> 00:31:18,480 And this is just a very, very bad idea. 393 00:31:18,480 --> 00:31:21,320 There's another trick for doing this that evaluates this 394 00:31:21,320 --> 00:31:23,060 much more efficiently. 395 00:31:23,060 --> 00:31:26,950 And it's the only device that we're going 396 00:31:26,950 --> 00:31:31,550 to learn now for integrating. 397 00:31:31,550 --> 00:31:36,690 Integration actually is much harder than differentiation, 398 00:31:36,690 --> 00:31:37,520 symbolically. 399 00:31:37,520 --> 00:31:42,610 It's quite difficult. And occasionally impossible. 400 00:31:42,610 --> 00:31:45,550 And so we have to go about it gently. 401 00:31:45,550 --> 00:31:47,450 But for the purposes of this unit, 402 00:31:47,450 --> 00:31:50,069 we're only going to use one method. 403 00:31:50,069 --> 00:31:50,860 Which is very good. 404 00:31:50,860 --> 00:31:52,526 That means whenever you see an integral, 405 00:31:52,526 --> 00:31:56,180 either you'll be able to divine immediately what the answer is, 406 00:31:56,180 --> 00:31:57,830 or you'll use this method. 407 00:31:57,830 --> 00:31:59,090 So this is it. 408 00:31:59,090 --> 00:32:09,470 The trick is called the method of substitution. 409 00:32:09,470 --> 00:32:17,860 And it is tailor-made for notion of differentials. 410 00:32:17,860 --> 00:32:36,510 So tailor-made for differential notation. 411 00:32:36,510 --> 00:32:37,840 The idea is the following. 412 00:32:37,840 --> 00:32:40,260 I'm going to to define a new function. 413 00:32:40,260 --> 00:32:43,200 And it's the messiest function that I see here. 414 00:32:43,200 --> 00:32:50,290 It's u = x^4 + 2. 415 00:32:50,290 --> 00:32:56,300 And then, I'm going to take its differential and what 416 00:32:56,300 --> 00:32:58,840 I discover, if I look at its formula, 417 00:32:58,840 --> 00:33:02,570 and the rule for differentials, which is right here. 418 00:33:02,570 --> 00:33:06,070 Its formula is what? 419 00:33:06,070 --> 00:33:10,180 4x^3 dx. 420 00:33:10,180 --> 00:33:14,000 Now, lo and behold with these two quantities, 421 00:33:14,000 --> 00:33:17,940 I can substitute, I can plug in to this integral. 422 00:33:17,940 --> 00:33:21,760 And I will simplify it considerably. 423 00:33:21,760 --> 00:33:23,350 So how does that happen? 424 00:33:23,350 --> 00:33:34,740 Well, this integral is the same thing as, well, really 425 00:33:34,740 --> 00:33:36,350 I should combine it the other way. 426 00:33:36,350 --> 00:33:41,420 So let me move this over. 427 00:33:41,420 --> 00:33:43,340 So there are two pieces here. 428 00:33:43,340 --> 00:33:46,440 And this one is u^5. 429 00:33:46,440 --> 00:33:54,990 And this one is 1/4 du. 430 00:33:54,990 --> 00:34:01,840 Now, that makes it the integral of u^5 du / 4. 431 00:34:01,840 --> 00:34:04,040 And that's relatively easy to integrate. 432 00:34:04,040 --> 00:34:05,460 That is just a power. 433 00:34:05,460 --> 00:34:06,410 So let's see. 434 00:34:06,410 --> 00:34:11,250 It's just 1/20 u to the-- whoops, not 1/20. 435 00:34:11,250 --> 00:34:15,480 The antiderivative of u^5 is u^6. 436 00:34:15,480 --> 00:34:25,480 With the 1/6, so it's 1/24 u^6 + c. 437 00:34:25,480 --> 00:34:29,050 Now, that's not the answer to the question. 438 00:34:29,050 --> 00:34:32,260 It's almost the answer to the question. 439 00:34:32,260 --> 00:34:33,287 Why isn't it the answer? 440 00:34:33,287 --> 00:34:35,120 It isn't the answer because now the answer's 441 00:34:35,120 --> 00:34:37,480 expressed in terms of u. 442 00:34:37,480 --> 00:34:41,750 Whereas the problem was posed in terms of this variable x. 443 00:34:41,750 --> 00:34:45,960 So we must change back to our variable here. 444 00:34:45,960 --> 00:34:47,990 And we do that just by writing it in. 445 00:34:47,990 --> 00:34:56,190 So it's 1/24 (x^4 + 2)^6 + c. 446 00:34:56,190 --> 00:35:02,120 And this is the end of the problem. 447 00:35:02,120 --> 00:35:02,990 Yeah, question. 448 00:35:02,990 --> 00:35:16,350 STUDENT: [INAUDIBLE] 449 00:35:16,350 --> 00:35:19,330 PROFESSOR: The question is, can you see it directly? 450 00:35:19,330 --> 00:35:20,160 Yeah. 451 00:35:20,160 --> 00:35:23,820 And we're going to talk about that in just one second. 452 00:35:23,820 --> 00:35:30,290 OK. 453 00:35:30,290 --> 00:35:35,500 Now, I'm going to do one more example 454 00:35:35,500 --> 00:35:44,310 and illustrate this method. 455 00:35:44,310 --> 00:35:45,405 Here's another example. 456 00:35:45,405 --> 00:35:51,430 The integral of x dx over the square root of 1 + x^2. 457 00:35:51,430 --> 00:35:56,610 Now, here's another example. 458 00:35:56,610 --> 00:36:03,475 Now, the method of substitution leads us to the idea u = 1 + 459 00:36:03,475 --> 00:36:05,190 x^2. 460 00:36:05,190 --> 00:36:11,540 du = 2x dx, etc. 461 00:36:11,540 --> 00:36:14,450 It takes about as long as this other problem did. 462 00:36:14,450 --> 00:36:15,720 To figure out what's going on. 463 00:36:15,720 --> 00:36:17,440 It's a very similar sort of thing. 464 00:36:17,440 --> 00:36:20,870 You end up integrating u^(-1/2). 465 00:36:20,870 --> 00:36:23,480 It leads to the integral of u^(-1/2) du. 466 00:36:28,350 --> 00:36:31,630 Is everybody seeing where this...? 467 00:36:31,630 --> 00:36:37,540 However, there is a slightly better method. 468 00:36:37,540 --> 00:36:46,070 So recommended method. 469 00:36:46,070 --> 00:36:59,250 And I call this method advanced guessing. 470 00:36:59,250 --> 00:37:01,077 What advanced guessing means is that you've 471 00:37:01,077 --> 00:37:02,660 done enough of these problems that you 472 00:37:02,660 --> 00:37:04,750 can see two steps ahead. 473 00:37:04,750 --> 00:37:08,030 And you know what's going to happen. 474 00:37:08,030 --> 00:37:10,440 So the advanced guessing leads you 475 00:37:10,440 --> 00:37:12,817 to believe that here you had a power -1/2, 476 00:37:12,817 --> 00:37:14,650 here you have the differential of the thing. 477 00:37:14,650 --> 00:37:16,790 So it's going to work out somehow. 478 00:37:16,790 --> 00:37:19,670 And the advanced guessing allows you to guess that the answer 479 00:37:19,670 --> 00:37:23,520 should be something like this. (1 + x^2)^(1/2). 480 00:37:26,050 --> 00:37:27,800 So this is your advanced guess. 481 00:37:27,800 --> 00:37:31,670 And now you just differentiate it, and see whether it works. 482 00:37:31,670 --> 00:37:32,550 Well, here it is. 483 00:37:32,550 --> 00:37:39,330 It's 1/2 (1 + x^2)^(-1/2) 2x, that's the chain rule here. 484 00:37:39,330 --> 00:37:44,770 Which, sure enough, gives you x over square root of 1 + x^2. 485 00:37:44,770 --> 00:37:45,480 So we're done. 486 00:37:45,480 --> 00:37:56,960 And so the answer is square root of (1 + x^2) + c. 487 00:37:56,960 --> 00:38:02,160 Let me illustrate this further with another example. 488 00:38:02,160 --> 00:38:06,010 I strongly recommend that you do this, 489 00:38:06,010 --> 00:38:09,270 but you have to get used to it. 490 00:38:09,270 --> 00:38:11,490 So here's another example. 491 00:38:11,490 --> 00:38:18,900 e^(6x) dx. 492 00:38:18,900 --> 00:38:26,360 My advanced guess is e^(6x). 493 00:38:26,360 --> 00:38:30,020 And if I check, when I differentiate it, 494 00:38:30,020 --> 00:38:33,180 I get 6e^(6x). 495 00:38:33,180 --> 00:38:35,020 That's the derivative. 496 00:38:35,020 --> 00:38:38,030 And so I know that the answer, so now I 497 00:38:38,030 --> 00:38:39,030 know what the answer is. 498 00:38:39,030 --> 00:38:46,300 It's 1/6 e^(6x) + c. 499 00:38:46,300 --> 00:38:54,510 Now, OK, you could, it's also OK, but slow, 500 00:38:54,510 --> 00:39:02,550 to use a substitution, to use u = 6x. 501 00:39:02,550 --> 00:39:07,920 Then you're going to get du = 6dx, dot, dot, dot. 502 00:39:07,920 --> 00:39:23,220 It's going to work, it's just a waste of time. 503 00:39:23,220 --> 00:39:26,760 Well, I'm going to give you a couple more examples. 504 00:39:26,760 --> 00:39:27,910 So how about this one. 505 00:39:33,560 --> 00:39:35,120 x e^(-x^2) dx. 506 00:39:41,340 --> 00:39:45,600 What's the guess? 507 00:39:45,600 --> 00:39:51,270 Anybody have a guess? 508 00:39:51,270 --> 00:39:52,480 Well, you could also correct. 509 00:39:52,480 --> 00:39:54,438 So I don't want you to bother - yeah, go ahead. 510 00:39:54,438 --> 00:39:56,787 STUDENT: [INAUDIBLE] 511 00:39:56,787 --> 00:39:59,120 PROFESSOR: Yeah, so you're already one step ahead of me. 512 00:39:59,120 --> 00:40:02,050 Because this is too easy. 513 00:40:02,050 --> 00:40:04,110 When they get more complicated, you just 514 00:40:04,110 --> 00:40:05,690 want to make this guess here. 515 00:40:05,690 --> 00:40:08,884 So various people have said 1/2, and they understand 516 00:40:08,884 --> 00:40:10,050 that there's 1/2 going here. 517 00:40:10,050 --> 00:40:13,890 But let me just show you what happens, OK? 518 00:40:13,890 --> 00:40:19,140 If you make this guess and you differentiate it, 519 00:40:19,140 --> 00:40:23,940 what you get here is e^(-x^2) times the derivative 520 00:40:23,940 --> 00:40:28,130 of negative 2x, so that's -2x. 521 00:40:28,130 --> 00:40:30,400 x^2, so it's -2x. 522 00:40:30,400 --> 00:40:37,970 So now you see that you're off by a factor of not 2, but -2. 523 00:40:37,970 --> 00:40:39,820 So a number of you were saying that. 524 00:40:39,820 --> 00:40:43,120 So the answer is -1/2 e^(-x^2) + c. 525 00:40:46,510 --> 00:40:49,090 And I can guarantee you, having watched 526 00:40:49,090 --> 00:40:55,290 this on various problems, that people who don't write this out 527 00:40:55,290 --> 00:40:57,360 make arithmetic mistakes. 528 00:40:57,360 --> 00:41:00,140 In other words, there is a limit to how much 529 00:41:00,140 --> 00:41:02,760 people can think ahead and guess correctly. 530 00:41:02,760 --> 00:41:04,760 Another way of doing it, by the way, 531 00:41:04,760 --> 00:41:06,680 is simply to write this thing in and then 532 00:41:06,680 --> 00:41:10,160 fix the coefficient by doing the differentiation here. 533 00:41:10,160 --> 00:41:14,850 That's perfectly OK as well. 534 00:41:14,850 --> 00:41:18,920 All right, one more example. 535 00:41:18,920 --> 00:41:30,840 We're going to integrate sin x cos x dx. 536 00:41:30,840 --> 00:41:33,750 So what's a good guess for this one? 537 00:41:33,750 --> 00:41:36,520 STUDENT: [INAUDIBLE] 538 00:41:36,520 --> 00:41:38,850 PROFESSOR: Someone suggesting sin^2 x. 539 00:41:38,850 --> 00:41:41,490 So let's try that. 540 00:41:41,490 --> 00:41:45,020 Over 2 - well, we'll get the coefficient in just a second. 541 00:41:45,020 --> 00:41:50,970 So sin^2 x, if I differentiate, I get 2 sin x cos x. 542 00:41:50,970 --> 00:41:53,380 So that's off by a factor of 2. 543 00:41:53,380 --> 00:42:04,540 So the answer is 1/2 sin^2 x. 544 00:42:04,540 --> 00:42:12,320 But now I want to point out to you 545 00:42:12,320 --> 00:42:17,120 that there's another way of doing this problem. 546 00:42:17,120 --> 00:42:31,240 It's also true that if you differentiate cos^2 x, 547 00:42:31,240 --> 00:42:34,600 you get 2 cos x (-sin x). 548 00:42:38,130 --> 00:42:51,030 So another answer is that the integral of sin x cos x dx is 549 00:42:51,030 --> 00:43:01,840 equal to -1/2 cos^2 x + c. 550 00:43:01,840 --> 00:43:03,740 So what is going on here? 551 00:43:03,740 --> 00:43:06,890 What's the problem with this? 552 00:43:06,890 --> 00:43:10,785 STUDENT: [INAUDIBLE] 553 00:43:10,785 --> 00:43:11,660 PROFESSOR: Pardon me? 554 00:43:11,660 --> 00:43:15,060 STUDENT: [INAUDIBLE] 555 00:43:15,060 --> 00:43:18,130 PROFESSOR: Integrals aren't unique. 556 00:43:18,130 --> 00:43:21,390 That's part of the-- but somehow these two answers still 557 00:43:21,390 --> 00:43:22,320 have to be the same. 558 00:43:22,320 --> 00:43:32,910 STUDENT: [INAUDIBLE] 559 00:43:32,910 --> 00:43:35,910 PROFESSOR: OK. 560 00:43:35,910 --> 00:43:36,660 What do you think? 561 00:43:36,660 --> 00:43:38,743 STUDENT: If you add them together, you just get c. 562 00:43:38,743 --> 00:43:40,900 PROFESSOR: If you add them together you get c. 563 00:43:40,900 --> 00:43:44,185 Well, actually, that's almost right. 564 00:43:44,185 --> 00:43:45,810 That's not what you want to do, though. 565 00:43:45,810 --> 00:43:47,620 You don't want to add them. 566 00:43:47,620 --> 00:43:50,930 You want to subtract them. 567 00:43:50,930 --> 00:43:53,750 So let's see what happens when you subtract them. 568 00:43:53,750 --> 00:43:56,840 I'm going to ignore the c, for the time being. 569 00:43:56,840 --> 00:44:05,520 I get sin^2 x, 1/2 sin^2 x - (-1/2 cos^2 x). 570 00:44:05,520 --> 00:44:08,880 So the difference between them, we hope to be 0. 571 00:44:08,880 --> 00:44:10,760 But actually of course it's not 0. 572 00:44:10,760 --> 00:44:17,680 What it is, is it's 1/2 sine squared plus cosine squared, 573 00:44:17,680 --> 00:44:18,850 which is 1/2. 574 00:44:18,850 --> 00:44:24,200 It's not 0, it's a constant. 575 00:44:24,200 --> 00:44:27,310 So what's really going on here is that these two formulas 576 00:44:27,310 --> 00:44:29,290 are the same. 577 00:44:29,290 --> 00:44:31,740 But you have to understand how to interpret them. 578 00:44:31,740 --> 00:44:34,450 The two constants, here's a constant up here. 579 00:44:34,450 --> 00:44:37,900 There's a constant, c_1 associated to this one. 580 00:44:37,900 --> 00:44:43,250 There's a different constant, c_2 associated to this one. 581 00:44:43,250 --> 00:44:45,960 And this family of functions for all possible c_1's 582 00:44:45,960 --> 00:44:49,860 and all possible c_2's, is the same family of functions. 583 00:44:49,860 --> 00:44:52,940 Now, what's the relationship between c_1 and c_2? 584 00:44:52,940 --> 00:44:57,210 Well, if you do the subtraction, c_1 - c_2 585 00:44:57,210 --> 00:44:59,240 has to be equal to 1/2. 586 00:44:59,240 --> 00:45:06,610 They're both constants, but they differ by 1/2. 587 00:45:06,610 --> 00:45:08,404 So this explains, when you're dealing 588 00:45:08,404 --> 00:45:10,820 with families of things, they don't have to look the same. 589 00:45:10,820 --> 00:45:12,560 And there are lots of trig functions 590 00:45:12,560 --> 00:45:16,280 which look a little different. 591 00:45:16,280 --> 00:45:19,050 So there can be several formulas that actually are the same. 592 00:45:19,050 --> 00:45:21,960 And it's hard to check that they're actually the same. 593 00:45:21,960 --> 00:45:28,950 You need some trig identities to do it. 594 00:45:28,950 --> 00:45:55,210 Let's do one more example here. 595 00:45:55,210 --> 00:46:06,250 Here's another one. 596 00:46:06,250 --> 00:46:13,510 Now, you may be thinking, and a lot of people 597 00:46:13,510 --> 00:46:22,250 are, thinking ugh, it's got a ln in it. 598 00:46:22,250 --> 00:46:24,400 If you're experienced, you actually 599 00:46:24,400 --> 00:46:26,130 can read off the answer just the way 600 00:46:26,130 --> 00:46:28,088 there were several people who were shouting out 601 00:46:28,088 --> 00:46:31,000 the answers when we were doing the rest of these problems. 602 00:46:31,000 --> 00:46:32,970 But, you do need to relax. 603 00:46:32,970 --> 00:46:35,790 Because in this case, now this is definitely not 604 00:46:35,790 --> 00:46:37,490 true in general when we do integrals. 605 00:46:37,490 --> 00:46:39,080 But, for now, when we do integrals, 606 00:46:39,080 --> 00:46:40,570 they'll all be manageable. 607 00:46:40,570 --> 00:46:42,670 And there's only one method. 608 00:46:42,670 --> 00:46:47,390 Which is substitution. 609 00:46:47,390 --> 00:46:49,680 And in the substitution method, you 610 00:46:49,680 --> 00:46:52,200 want to go for the trickiest part. 611 00:46:52,200 --> 00:46:55,220 And substitute for that. 612 00:46:55,220 --> 00:46:57,630 So the substitution that I proposed 613 00:46:57,630 --> 00:47:02,200 to you is that this should be, u should be ln x. 614 00:47:02,200 --> 00:47:06,270 And the advantage that that has is that its differential 615 00:47:06,270 --> 00:47:08,720 is simpler than itself. 616 00:47:08,720 --> 00:47:15,570 So du = dx / x. 617 00:47:15,570 --> 00:47:18,500 Remember, we use that in logarithmic differentiation, 618 00:47:18,500 --> 00:47:21,550 too. 619 00:47:21,550 --> 00:47:28,810 So now we can express this using this substitution. 620 00:47:28,810 --> 00:47:32,290 And what we get is, the integral of, 621 00:47:32,290 --> 00:47:33,990 so I'll divide the two parts here. 622 00:47:33,990 --> 00:47:36,515 It's 1 / ln x, and then it's dx / x. 623 00:47:36,515 --> 00:47:43,370 And this part is 1 / u, and this part is du. 624 00:47:43,370 --> 00:47:49,260 So it's the integral of du / u. 625 00:47:49,260 --> 00:47:58,490 And that is ln u + c. 626 00:47:58,490 --> 00:48:11,030 Which altogether, if I put back in what u is, is ln (ln x) + c. 627 00:48:11,030 --> 00:48:14,290 And now we see some uglier things. 628 00:48:14,290 --> 00:48:15,900 In fact, technically speaking, we 629 00:48:15,900 --> 00:48:18,730 could take the absolute value here. 630 00:48:18,730 --> 00:48:28,130 And then this would be absolute values there. 631 00:48:28,130 --> 00:48:33,090 So this is the type of example where I really 632 00:48:33,090 --> 00:48:35,740 would recommend that you actually use the substitution, 633 00:48:35,740 --> 00:48:39,030 at least for now. 634 00:48:39,030 --> 00:48:41,820 All right, tomorrow we're going to be 635 00:48:41,820 --> 00:48:43,080 doing differential equations. 636 00:48:43,080 --> 00:48:45,130 And we're going to review for the test. 637 00:48:45,130 --> 00:48:47,649 I'm going to give you a handout telling you just exactly 638 00:48:47,649 --> 00:48:48,940 what's going to be on the test. 639 00:48:48,940 --> 00:48:52,298 So, see you tomorrow.