1 00:00:00,000 --> 00:00:02,330 The following content is provided under a Creative 2 00:00:02,330 --> 00:00:03,950 Commons license. 3 00:00:03,950 --> 00:00:05,990 Your support will help MIT OpenCourseWare 4 00:00:05,990 --> 00:00:09,410 continue to offer high quality educational resources for free. 5 00:00:09,410 --> 00:00:12,600 To make a donation, or to view additional materials 6 00:00:12,600 --> 00:00:16,150 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,150 --> 00:00:23,080 at ocw.mit.edu. 8 00:00:23,080 --> 00:00:25,240 PROFESSOR: Now, to start out today 9 00:00:25,240 --> 00:00:27,850 we're going to finish up what we did last time. 10 00:00:27,850 --> 00:00:30,950 Which has to do with partial fractions. 11 00:00:30,950 --> 00:00:33,160 I told you how to do partial fractions 12 00:00:33,160 --> 00:00:35,080 in several special cases and everybody 13 00:00:35,080 --> 00:00:37,920 was trying to figure out what the general picture was. 14 00:00:37,920 --> 00:00:39,180 But I'd like to lay that out. 15 00:00:39,180 --> 00:00:41,730 I'll still only do it for an example. 16 00:00:41,730 --> 00:00:43,810 But it will be somehow a bigger example 17 00:00:43,810 --> 00:00:53,310 so that you can see what the general pattern is. 18 00:00:53,310 --> 00:01:04,280 Partial fractions, remember, is a method for breaking up 19 00:01:04,280 --> 00:01:06,730 so-called rational functions. 20 00:01:06,730 --> 00:01:09,510 Which are ratios of polynomials. 21 00:01:09,510 --> 00:01:13,110 And it shows you that you can always integrate them. 22 00:01:13,110 --> 00:01:14,920 That's really the theme here. 23 00:01:14,920 --> 00:01:21,190 And this is what's reassuring is that it always works. 24 00:01:21,190 --> 00:01:23,670 That's really the bottom line. 25 00:01:23,670 --> 00:01:26,550 And that's good because there are 26 00:01:26,550 --> 00:01:34,100 a lot of integrals that don't have formulas and these do. 27 00:01:34,100 --> 00:01:35,560 It always works. 28 00:01:35,560 --> 00:01:43,340 But, maybe with lots of help. 29 00:01:43,340 --> 00:01:46,257 So maybe slowly. 30 00:01:46,257 --> 00:01:47,840 Now, there's a little bit of bad news, 31 00:01:47,840 --> 00:01:50,820 and I have to be totally honest and tell you 32 00:01:50,820 --> 00:01:52,230 what all the bad news is. 33 00:01:52,230 --> 00:01:54,930 Along with the good news. 34 00:01:54,930 --> 00:02:00,070 The first step, which maybe I should be calling Step 0, 35 00:02:00,070 --> 00:02:08,710 I had a Step 1, 2 and 3 last time, is long division. 36 00:02:08,710 --> 00:02:11,790 That's the step where you take your polynomial divided 37 00:02:11,790 --> 00:02:17,440 by your other polynomial, and you find the quotient 38 00:02:17,440 --> 00:02:22,470 plus some remainder. 39 00:02:22,470 --> 00:02:24,370 And you do that by long division. 40 00:02:24,370 --> 00:02:27,520 And the quotient is easy to take the antiderivative 41 00:02:27,520 --> 00:02:30,010 of because it's just a polynomial. 42 00:02:30,010 --> 00:02:32,380 And the key extra property here is 43 00:02:32,380 --> 00:02:35,680 that the degree of the numerator now over here, this remainder, 44 00:02:35,680 --> 00:02:40,510 is strictly less than the degree of the denominator. 45 00:02:40,510 --> 00:02:44,140 So that you can do the next step. 46 00:02:44,140 --> 00:02:48,900 Now, the next step which I called Step 1 last time, that's 47 00:02:48,900 --> 00:02:52,290 great imagination, it's right after Step 0, Step 1 48 00:02:52,290 --> 00:02:54,930 was to factor the denominator. 49 00:02:54,930 --> 00:03:00,630 And I'm going to illustrate by example what the setup is here. 50 00:03:00,630 --> 00:03:09,120 I don't know maybe, we'll do this. 51 00:03:09,120 --> 00:03:12,920 Some polynomial here, maybe cube this one. 52 00:03:12,920 --> 00:03:21,440 So here I've factored the denominator. 53 00:03:21,440 --> 00:03:24,840 That's what I called Step 1 last time. 54 00:03:24,840 --> 00:03:27,650 Now, here's the first piece of bad news. 55 00:03:27,650 --> 00:03:31,620 In reality, if somebody gave you a multiplied 56 00:03:31,620 --> 00:03:35,530 out degree-whatever polynomial here, 57 00:03:35,530 --> 00:03:40,710 you would be very hard pressed to factor it. 58 00:03:40,710 --> 00:03:44,132 A lot of them are extremely difficult to factor. 59 00:03:44,132 --> 00:03:46,590 And so that's something you would have to give to a machine 60 00:03:46,590 --> 00:03:47,940 to do. 61 00:03:47,940 --> 00:03:50,880 And it's just basically a hard problem. 62 00:03:50,880 --> 00:03:54,050 So obviously, we're only going to give you ones 63 00:03:54,050 --> 00:03:55,480 that you can do by hand. 64 00:03:55,480 --> 00:03:58,060 So very low degree examples. 65 00:03:58,060 --> 00:03:59,320 And that's just the way it is. 66 00:03:59,320 --> 00:04:03,600 So this is really a hard step in disguise, in real life. 67 00:04:03,600 --> 00:04:06,000 Anyway, we're just going to take it as given. 68 00:04:06,000 --> 00:04:07,820 And we have this numerator, which 69 00:04:07,820 --> 00:04:10,720 is of degree less than the denominator. 70 00:04:10,720 --> 00:04:14,830 So let's count up what its degree has to be. 71 00:04:14,830 --> 00:04:18,640 This is 4 + 2 + 6. 72 00:04:18,640 --> 00:04:22,440 So this is degree 4 + 2 + 6. 73 00:04:22,440 --> 00:04:24,190 I added that up because this is degree 4, 74 00:04:24,190 --> 00:04:28,600 this is degree 2 and (x^2)^3 is the 6th power. 75 00:04:28,600 --> 00:04:32,330 So all together it's this, which is 12. 76 00:04:32,330 --> 00:04:39,822 And so this thing is of degree <= 11. 77 00:04:39,822 --> 00:04:41,280 All the way up to degree 11, that's 78 00:04:41,280 --> 00:04:44,240 the possibilities for the numerator here. 79 00:04:44,240 --> 00:04:49,810 Now, the extra information that I want to impart right now, 80 00:04:49,810 --> 00:04:56,450 is just this setup. 81 00:04:56,450 --> 00:04:58,990 Which I called Step 2 last time. 82 00:04:58,990 --> 00:05:05,260 And the setup is this. 83 00:05:05,260 --> 00:05:07,590 Now, it's going to take us a while to do this. 84 00:05:07,590 --> 00:05:10,810 We have this factor here. 85 00:05:10,810 --> 00:05:12,450 We have another factor. 86 00:05:12,450 --> 00:05:14,620 We have another term, with the square. 87 00:05:14,620 --> 00:05:18,220 We have another term with the cube. 88 00:05:18,220 --> 00:05:22,581 We have another term with the fourth power. 89 00:05:22,581 --> 00:05:24,330 So this is what's going to happen whenever 90 00:05:24,330 --> 00:05:25,530 you have linear factors. 91 00:05:25,530 --> 00:05:28,560 You'll have a collection of terms like this. 92 00:05:28,560 --> 00:05:31,030 So you have four constants to take care of. 93 00:05:31,030 --> 00:05:34,550 Now, with a quadratic in the denominator, 94 00:05:34,550 --> 00:05:36,840 you need a linear function in the numerator. 95 00:05:36,840 --> 00:05:41,230 So that's, if you like, B_0 x + C_0 divided 96 00:05:41,230 --> 00:05:49,370 by this quadratic term here. 97 00:05:49,370 --> 00:05:52,500 And what I didn't show you last time 98 00:05:52,500 --> 00:05:59,720 was how you deal with higher powers of quadratic terms. 99 00:05:59,720 --> 00:06:04,090 So when you have a quadratic term, what's going to happen 100 00:06:04,090 --> 00:06:07,370 is you're going to take that first factor here. 101 00:06:07,370 --> 00:06:11,870 Just the way you did in this case. 102 00:06:11,870 --> 00:06:15,500 But then you're going to have to do the same thing 103 00:06:15,500 --> 00:06:24,120 with the next power. 104 00:06:24,120 --> 00:06:27,880 Now notice, just as in the case of this top row, 105 00:06:27,880 --> 00:06:29,830 I have just a constant here. 106 00:06:29,830 --> 00:06:33,030 And even though I increased the degree of the denominator 107 00:06:33,030 --> 00:06:34,540 I'm not increasing the numerator. 108 00:06:34,540 --> 00:06:35,970 It's staying just a constant. 109 00:06:35,970 --> 00:06:38,230 It's not linear up here. 110 00:06:38,230 --> 00:06:39,850 It's better than that. 111 00:06:39,850 --> 00:06:41,990 It's just a constant. 112 00:06:41,990 --> 00:06:44,130 And here it stayed a constant. 113 00:06:44,130 --> 00:06:45,630 And here it stayed a constant. 114 00:06:45,630 --> 00:06:48,070 Similarly here, even though I'm increasing 115 00:06:48,070 --> 00:06:49,620 the degree of the denominator, I'm 116 00:06:49,620 --> 00:06:52,810 leaving the numerator, the form of the numerator, alone. 117 00:06:52,810 --> 00:06:55,150 It's just a linear factor and a linear factor. 118 00:06:55,150 --> 00:07:05,350 So that's the key to this pattern. 119 00:07:05,350 --> 00:07:09,850 I don't have quite as jazzy a song on mine. 120 00:07:09,850 --> 00:07:13,410 So this is so long that it runs off the blackboard here. 121 00:07:13,410 --> 00:07:15,820 So let's continue it on the next. 122 00:07:15,820 --> 00:07:20,200 We've got this B_2 x + C_2-- sorry, 123 00:07:20,200 --> 00:07:23,150 (B_3 x + C_3) / (x^2 + 4)^3. 124 00:07:26,590 --> 00:07:38,590 I guess I have room for it over here. 125 00:07:38,590 --> 00:07:41,120 I'm going to talk about this in just a second. 126 00:07:41,120 --> 00:07:43,590 Alright, so here's the pattern. 127 00:07:43,590 --> 00:07:51,270 Now, let me just do a count of the number of unknowns 128 00:07:51,270 --> 00:07:52,344 we have here. 129 00:07:52,344 --> 00:07:54,010 The number of unknowns that we have here 130 00:07:54,010 --> 00:07:58,750 is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. 131 00:07:58,750 --> 00:08:00,930 That 12 is no coincidence. 132 00:08:00,930 --> 00:08:03,815 That's the degree of the polynomial. 133 00:08:03,815 --> 00:08:05,690 And it's the number of unknowns that we have. 134 00:08:05,690 --> 00:08:08,060 And it's the number of degrees of freedom 135 00:08:08,060 --> 00:08:11,215 in a polynomial of degree 11. 136 00:08:11,215 --> 00:08:13,090 If you have all these free coefficients here, 137 00:08:13,090 --> 00:08:17,560 you have the coefficient x^0, x^1, all the way up to x^ 11. 138 00:08:17,560 --> 00:08:23,100 And 0 through 11 is 12 different coefficients. 139 00:08:23,100 --> 00:08:26,380 And so this is a very complicated system 140 00:08:26,380 --> 00:08:28,150 of equations for unknowns. 141 00:08:28,150 --> 00:08:33,259 This is twelve equations for twelve unknowns. 142 00:08:33,259 --> 00:08:34,800 So I'll get rid of this for a second. 143 00:08:34,800 --> 00:08:41,090 So we have twelve equations, twelve unknowns. 144 00:08:41,090 --> 00:08:43,830 So that's the other bad news. 145 00:08:43,830 --> 00:08:46,020 Machines handle this very well, but human beings 146 00:08:46,020 --> 00:08:47,630 have a little trouble with 12. 147 00:08:47,630 --> 00:08:51,570 Now, the cover-up method works very neatly 148 00:08:51,570 --> 00:08:53,730 and picks out this term here. 149 00:08:53,730 --> 00:08:54,470 But that's it. 150 00:08:54,470 --> 00:08:56,550 So it reduces it to an 11 by 11. 151 00:08:56,550 --> 00:09:00,170 You'll be able to evaluate this in no time. 152 00:09:00,170 --> 00:09:00,890 But that's it. 153 00:09:00,890 --> 00:09:04,240 That's the only simplification of your previous method. 154 00:09:04,240 --> 00:09:06,250 We don't have a method for this. 155 00:09:06,250 --> 00:09:08,510 So I'm just showing what the whole method looks 156 00:09:08,510 --> 00:09:10,343 like but really you'd have to have a machine 157 00:09:10,343 --> 00:09:14,760 to implement this once it gets to be any size at all. 158 00:09:14,760 --> 00:09:15,490 Yeah, question. 159 00:09:15,490 --> 00:09:18,060 STUDENT: [INAUDIBLE] 160 00:09:18,060 --> 00:09:22,510 PROFESSOR: It's one big equation, 161 00:09:22,510 --> 00:09:24,920 but it's a polynomial equation. 162 00:09:24,920 --> 00:09:32,530 So there's an equation, there's this function R(x) = a_11 x^11 163 00:09:32,530 --> 00:09:37,610 + a_10 x^10... 164 00:09:37,610 --> 00:09:41,350 and these things are known. 165 00:09:41,350 --> 00:09:43,495 This is a known expression here. 166 00:09:43,495 --> 00:09:46,620 And then when you cross-multiply on the other side, 167 00:09:46,620 --> 00:09:51,330 what you have is, well, it's A_1 times-- 168 00:09:51,330 --> 00:09:54,400 If you cancel this denominator with that, 169 00:09:54,400 --> 00:10:05,330 you're going to get (x + (x+2)^3 (x^2+2x+3) (x^2+4)^3 plus 170 00:10:05,330 --> 00:10:08,260 the term for A_2, etc. 171 00:10:08,260 --> 00:10:10,350 It's a monster equation. 172 00:10:10,350 --> 00:10:12,620 And then to separate it out into separate equations, 173 00:10:12,620 --> 00:10:19,225 you take the coefficient on x^11, x^10, ... 174 00:10:19,225 --> 00:10:21,670 all the way down to x^0. 175 00:10:21,670 --> 00:10:27,280 And all told, that means there are a total of 12 equations. 176 00:10:27,280 --> 00:10:31,030 11 through 0 is 12 equations. 177 00:10:31,030 --> 00:10:34,727 Yeah, another question. 178 00:10:34,727 --> 00:10:35,560 STUDENT: [INAUDIBLE] 179 00:10:35,560 --> 00:10:37,777 PROFESSOR: Should I write down rest of this? 180 00:10:37,777 --> 00:10:38,610 STUDENT: [INAUDIBLE] 181 00:10:38,610 --> 00:10:40,820 PROFESSOR: Should you write down all this stuff? 182 00:10:40,820 --> 00:10:43,950 Well, that's a good question. 183 00:10:43,950 --> 00:10:46,070 So you notice I didn't write it down. 184 00:10:46,070 --> 00:10:47,300 Why didn't I write it down? 185 00:10:47,300 --> 00:10:50,200 Because it's incredibly long. 186 00:10:50,200 --> 00:10:54,104 In fact, you probably-- So how many pages of writing 187 00:10:54,104 --> 00:10:54,770 would this take? 188 00:10:54,770 --> 00:10:56,210 This is about a page of writing. 189 00:10:56,210 --> 00:10:58,950 So just think of you're a machine, how much time 190 00:10:58,950 --> 00:11:01,660 you want to spend on this. 191 00:11:01,660 --> 00:11:05,430 So the answer is that you have to be realistic. 192 00:11:05,430 --> 00:11:07,420 You're a human being, not a machine. 193 00:11:07,420 --> 00:11:10,070 And so there's certain things that you can write down 194 00:11:10,070 --> 00:11:12,510 and other things you should not attempt to write down. 195 00:11:12,510 --> 00:11:17,770 So do not do this at home. 196 00:11:17,770 --> 00:11:21,180 So that's the first down-side to this method. 197 00:11:21,180 --> 00:11:24,350 It gets more and more complicated as time goes on. 198 00:11:24,350 --> 00:11:27,280 The second down-side, I want to point out to you, 199 00:11:27,280 --> 00:11:35,100 is what's happening with the pieces. 200 00:11:35,100 --> 00:11:42,830 So the pieces still need to be integrated. 201 00:11:42,830 --> 00:11:48,130 We simplified this problem, but we didn't get rid of it. 202 00:11:48,130 --> 00:11:50,890 We still have the problem of integrating the pieces. 203 00:11:50,890 --> 00:11:52,740 Now, some of the pieces are very easy. 204 00:11:52,740 --> 00:11:55,540 This top row here, the antiderivatives of these, 205 00:11:55,540 --> 00:11:59,150 you can just write down. 206 00:11:59,150 --> 00:12:01,390 By advanced guessing. 207 00:12:01,390 --> 00:12:04,300 I'm going to skip over to the most complicated one over here. 208 00:12:04,300 --> 00:12:06,220 For one second here. 209 00:12:06,220 --> 00:12:09,240 And what is it that you'd have to deal with for that one. 210 00:12:09,240 --> 00:12:11,810 You'd have to deal with, for example, 211 00:12:11,810 --> 00:12:21,660 so e.g., for example, I need to deal with this guy. 212 00:12:21,660 --> 00:12:26,590 I've got to get this antiderivative here. 213 00:12:26,590 --> 00:12:28,972 Now, this one you're supposed to be able to know. 214 00:12:28,972 --> 00:12:30,430 So this is why I'm mentioning this. 215 00:12:30,430 --> 00:12:33,270 Because this kind of ingredient is something 216 00:12:33,270 --> 00:12:34,860 you already covered. 217 00:12:34,860 --> 00:12:35,740 And what is it? 218 00:12:35,740 --> 00:12:39,060 Well, you do this one by advanced guessing, 219 00:12:39,060 --> 00:12:42,000 although you learned it as the method of substitution. 220 00:12:42,000 --> 00:12:47,900 You realize that it's going to be of the form (x^2 + 4)^(-2), 221 00:12:47,900 --> 00:12:49,400 roughly speaking. 222 00:12:49,400 --> 00:12:51,170 And now we're going to fix that. 223 00:12:51,170 --> 00:12:53,950 Because if you differentiate it you get 2x times the -2, 224 00:12:53,950 --> 00:12:56,410 that's -4 times x times this. 225 00:12:56,410 --> 00:12:58,370 There's an x in the numerator here. 226 00:12:58,370 --> 00:13:02,600 So it's -1/4 of that will fix the factor. 227 00:13:02,600 --> 00:13:06,550 And here's the answer for that one. 228 00:13:06,550 --> 00:13:10,560 So that's one you can do. 229 00:13:10,560 --> 00:13:19,020 The second piece is this guy. 230 00:13:19,020 --> 00:13:20,480 This is the other piece. 231 00:13:20,480 --> 00:13:25,680 Now, this was the piece that came from B_3. 232 00:13:25,680 --> 00:13:27,170 This is the one that came from B_3. 233 00:13:27,170 --> 00:13:30,430 And this is the one that's coming from C_3. 234 00:13:30,430 --> 00:13:32,360 This is coming from C_3. 235 00:13:32,360 --> 00:13:35,000 We still need to get this one out there. 236 00:13:35,000 --> 00:13:37,630 So C_3 times that will be the correct answer, 237 00:13:37,630 --> 00:13:40,960 once we've found these numbers. 238 00:13:40,960 --> 00:13:44,160 So how do we do this? 239 00:13:44,160 --> 00:13:45,490 How's this one integrated? 240 00:13:45,490 --> 00:13:49,690 STUDENT: Trig substitution? 241 00:13:49,690 --> 00:13:51,640 PROFESSOR: Trig substitution. 242 00:13:51,640 --> 00:13:57,630 So the trig substitution here is x = 2 tan u. 243 00:13:57,630 --> 00:14:00,800 Or 2 tan theta. 244 00:14:00,800 --> 00:14:03,860 And when you do that, there are a couple of simplifications. 245 00:14:03,860 --> 00:14:06,450 Well, I wouldn't call this a simplification. 246 00:14:06,450 --> 00:14:14,830 This is just the differentiation formula. dx = 2 sec^2 u du. 247 00:14:14,830 --> 00:14:19,030 And then you have to plug in, and you're using the fact that 248 00:14:19,030 --> 00:14:22,960 when you plug in the tan^2, 4 tan ^2 + 4, 249 00:14:22,960 --> 00:14:24,370 you'll get a secant squared. 250 00:14:24,370 --> 00:14:32,240 So altogether, this thing is, 2 sec^2 u du. 251 00:14:32,240 --> 00:14:40,300 And then there's a (4 sec^2 u)^3, in the denominator. 252 00:14:40,300 --> 00:14:44,340 So that's what happens when you change variables here. 253 00:14:44,340 --> 00:14:46,790 And now look, this keeps on going. 254 00:14:46,790 --> 00:14:49,120 This is not the end of the problem. 255 00:14:49,120 --> 00:14:50,790 Because what does that simplify to? 256 00:14:50,790 --> 00:14:57,860 That is, let's see, it's 2/64, the integral of sec^6 257 00:14:57,860 --> 00:14:58,620 and sec^2. 258 00:14:58,620 --> 00:15:00,090 That's the same as cos^4. 259 00:15:04,300 --> 00:15:06,370 And now, you did a trig substitution 260 00:15:06,370 --> 00:15:11,140 but you still have a trig integral. 261 00:15:11,140 --> 00:15:15,540 The trig integral now, there's a method for this. 262 00:15:15,540 --> 00:15:18,730 The method for this is when it's an even power, 263 00:15:18,730 --> 00:15:22,280 you have to use the double angle formula. 264 00:15:22,280 --> 00:15:31,890 So that's this guy here. 265 00:15:31,890 --> 00:15:33,470 And you're still not done. 266 00:15:33,470 --> 00:15:35,040 You have to square this thing out. 267 00:15:35,040 --> 00:15:37,120 And then you'll still get a cos^2 (2u). 268 00:15:37,120 --> 00:15:38,160 And it keeps on going. 269 00:15:38,160 --> 00:15:41,737 So this thing goes on for a long time. 270 00:15:41,737 --> 00:15:43,320 But I'm not even going to finish this, 271 00:15:43,320 --> 00:15:44,780 but I just want to show you. 272 00:15:44,780 --> 00:15:46,450 The point is, we're not showing you how 273 00:15:46,450 --> 00:15:48,270 to do any complicated problem. 274 00:15:48,270 --> 00:15:50,550 We're just showing you all the little ingredients. 275 00:15:50,550 --> 00:15:52,050 And you have to string them together 276 00:15:52,050 --> 00:15:56,170 a long, long, long process to get to the final answer of one 277 00:15:56,170 --> 00:15:57,680 of these questions. 278 00:15:57,680 --> 00:16:07,300 So it always works, but maybe slowly. 279 00:16:07,300 --> 00:16:13,440 By the way, there's even another horrible thing that happens. 280 00:16:13,440 --> 00:16:22,669 Which is, if you handle this guy here, what's the technique. 281 00:16:22,669 --> 00:16:24,460 This is another technique that you learned, 282 00:16:24,460 --> 00:16:28,770 supposedly within the last few days. 283 00:16:28,770 --> 00:16:30,810 Completing the square. 284 00:16:30,810 --> 00:16:39,020 So this, it turns out, you have to rewrite it this way. 285 00:16:39,020 --> 00:16:42,530 And then the evaluation is going to be expressed in terms of, 286 00:16:42,530 --> 00:16:44,440 I'm going to jump to the end. 287 00:16:44,440 --> 00:16:49,310 It's going to turn out to be expressed in terms of this. 288 00:16:49,310 --> 00:16:53,940 That's what will eventually show up in the formula. 289 00:16:53,940 --> 00:16:56,370 And not only that, but if you deal 290 00:16:56,370 --> 00:16:59,890 with ones involving x as well, you'll 291 00:16:59,890 --> 00:17:07,420 also need to deal with something like log of this denominator 292 00:17:07,420 --> 00:17:09,560 here. 293 00:17:09,560 --> 00:17:13,120 So all of these things will be involved. 294 00:17:13,120 --> 00:17:16,700 So now, the last message that I have for you is just this. 295 00:17:16,700 --> 00:17:18,150 This thing is very complicated. 296 00:17:18,150 --> 00:17:20,150 We're certainly never going to ask you to do it. 297 00:17:20,150 --> 00:17:23,160 But you should just be aware that this level of complexity, 298 00:17:23,160 --> 00:17:26,690 we are absolutely stuck with in this problem. 299 00:17:26,690 --> 00:17:29,580 And the reason why we're stuck with it 300 00:17:29,580 --> 00:17:36,160 is that this is what the formulas look like in the end. 301 00:17:36,160 --> 00:17:39,130 If the answers look like this, the formulas 302 00:17:39,130 --> 00:17:41,045 have to be this complicated. 303 00:17:41,045 --> 00:17:43,170 If you differentiate this, you get your polynomial, 304 00:17:43,170 --> 00:17:44,254 your ratio of polynomials. 305 00:17:44,254 --> 00:17:46,794 If you differentiate this, you get some ratio of polynomials. 306 00:17:46,794 --> 00:17:48,480 These are the things that come up 307 00:17:48,480 --> 00:17:51,710 when you take antiderivatives of those rational functions. 308 00:17:51,710 --> 00:17:56,080 So we're just stuck with these guys. 309 00:17:56,080 --> 00:17:58,770 And so don't let it get to you too much. 310 00:17:58,770 --> 00:17:59,770 I mean, it's not so bad. 311 00:17:59,770 --> 00:18:01,510 In fact, there are computer programs 312 00:18:01,510 --> 00:18:03,510 that will do this for you anytime you want. 313 00:18:03,510 --> 00:18:05,800 And you just have to be not intimidated by them. 314 00:18:05,800 --> 00:18:10,260 They're like other functions. 315 00:18:10,260 --> 00:18:20,600 OK, that's it for the general comments on partial fractions. 316 00:18:20,600 --> 00:18:24,215 Now we're going to change subjects to our last technique. 317 00:18:24,215 --> 00:18:25,840 This is one more technical thing to get 318 00:18:25,840 --> 00:18:27,540 you familiar with functions. 319 00:18:27,540 --> 00:18:32,260 And this technique is called integration by parts. 320 00:18:32,260 --> 00:18:34,580 Please, just because its name sort 321 00:18:34,580 --> 00:18:35,957 of sounds like partial fractions, 322 00:18:35,957 --> 00:18:37,290 don't think it's the same thing. 323 00:18:37,290 --> 00:18:38,450 It's totally different. 324 00:18:38,450 --> 00:18:44,340 It's not the same. 325 00:18:44,340 --> 00:19:06,640 So this one is called integration by parts. 326 00:19:06,640 --> 00:19:09,570 Now, unlike the previous case, where I couldn't actually 327 00:19:09,570 --> 00:19:12,736 justify to you that the linear algebra always works. 328 00:19:12,736 --> 00:19:14,860 I claimed it worked, but I wasn't able to prove it. 329 00:19:14,860 --> 00:19:17,390 That's a complicated theorem which I'm not 330 00:19:17,390 --> 00:19:19,560 able to do in this class. 331 00:19:19,560 --> 00:19:22,270 Here I can explain to you what's going on 332 00:19:22,270 --> 00:19:24,200 with integration by parts. 333 00:19:24,200 --> 00:19:26,600 It's just the fundamental theorem of calculus, 334 00:19:26,600 --> 00:19:30,430 if you like, coupled with the product formula. 335 00:19:30,430 --> 00:19:33,740 Sort of unwound and read in reverse. 336 00:19:33,740 --> 00:19:35,610 And here's how that works. 337 00:19:35,610 --> 00:19:38,480 If you take the product of two functions and you differentiate 338 00:19:38,480 --> 00:19:41,910 them, then we know that the product rule says that this is 339 00:19:41,910 --> 00:19:45,790 u'v + uv'. 340 00:19:45,790 --> 00:19:50,400 And now I'm just going to rearrange in the following way. 341 00:19:50,400 --> 00:19:53,370 I'm going to solve for uv'. 342 00:19:53,370 --> 00:19:54,710 That is, this term here. 343 00:19:54,710 --> 00:19:56,370 So what is this term? 344 00:19:56,370 --> 00:19:59,990 It's this other term, (uv)'. 345 00:19:59,990 --> 00:20:04,520 Minus the other piece. 346 00:20:04,520 --> 00:20:08,360 So I just rewrote this equation. 347 00:20:08,360 --> 00:20:10,900 And now I'm going to integrate it. 348 00:20:10,900 --> 00:20:11,860 So here's the formula. 349 00:20:11,860 --> 00:20:15,160 The integral of the left-hand side 350 00:20:15,160 --> 00:20:17,200 is equal to the integral of the right-hand side. 351 00:20:17,200 --> 00:20:18,670 Well when I integrate a derivative, 352 00:20:18,670 --> 00:20:21,070 of I get back the function itself. 353 00:20:21,070 --> 00:20:27,010 That's the fundamental theorem. 354 00:20:27,010 --> 00:20:27,600 So this is it. 355 00:20:27,600 --> 00:20:30,500 Sorry, I missed the dx, which is important. 356 00:20:30,500 --> 00:20:32,460 I apologize. 357 00:20:32,460 --> 00:20:35,410 Let's put that in there. 358 00:20:35,410 --> 00:20:41,540 So this is the integration by parts formula. 359 00:20:41,540 --> 00:20:46,760 I'm going to write it one more time with the limits stuck in. 360 00:20:46,760 --> 00:21:02,170 It's also written this way, when you have a definite integral. 361 00:21:02,170 --> 00:21:13,260 Just the same formula, written twice. 362 00:21:13,260 --> 00:21:14,910 Alright, now I'm going to show you 363 00:21:14,910 --> 00:21:24,360 how it works on a few examples. 364 00:21:24,360 --> 00:21:29,470 And I have to give you a flavor for how it works. 365 00:21:29,470 --> 00:21:34,230 But it'll grow as we get more and more experience. 366 00:21:34,230 --> 00:21:38,360 The first example that I'm going to take 367 00:21:38,360 --> 00:21:43,040 is one that looks intractable on the face of it. 368 00:21:43,040 --> 00:21:49,740 Which is the integral of ln x dx. 369 00:21:49,740 --> 00:21:52,560 Now, it looks like there's sort of nothing we can do with this. 370 00:21:52,560 --> 00:21:55,310 And we don't know what the solution is. 371 00:21:55,310 --> 00:21:59,480 However, I claim that if we fit it into this form, 372 00:21:59,480 --> 00:22:03,100 we can figure out what the integral is relatively easily. 373 00:22:03,100 --> 00:22:07,160 By some little magic of cancellation, it happens. 374 00:22:07,160 --> 00:22:08,960 The idea is the following. 375 00:22:08,960 --> 00:22:13,130 If I consider this function to be u, 376 00:22:13,130 --> 00:22:15,680 then what's going to appear on the other side 377 00:22:15,680 --> 00:22:19,540 in the integrated form is the function u', which 378 00:22:19,540 --> 00:22:22,680 is-- so, if you like, u = ln x. 379 00:22:22,680 --> 00:22:25,620 So u' = 1 / x. 380 00:22:25,620 --> 00:22:28,680 Now, 1 / x is a more manageable function than ln x. 381 00:22:28,680 --> 00:22:31,370 What we're using is that when we differentiate the function, 382 00:22:31,370 --> 00:22:33,100 it's getting nicer. 383 00:22:33,100 --> 00:22:36,830 It's getting more tractable for us. 384 00:22:36,830 --> 00:22:38,860 In order for this to fit into this pattern, 385 00:22:38,860 --> 00:22:45,410 however, I need a function v. So what in the world 386 00:22:45,410 --> 00:22:48,220 am I going to put here for v? 387 00:22:48,220 --> 00:22:51,920 The answer is, well, dx is almost the right answer. 388 00:22:51,920 --> 00:22:53,860 The answer turns out to be x. 389 00:22:53,860 --> 00:23:01,260 And the reason is that that makes v' = 1. 390 00:23:01,260 --> 00:23:02,630 It makes v' = 1. 391 00:23:02,630 --> 00:23:05,560 So that means that this is u, but it's also uv'. 392 00:23:05,560 --> 00:23:11,240 Which was what I had on the left-hand side. 393 00:23:11,240 --> 00:23:13,470 So it's both u and uv'. 394 00:23:13,470 --> 00:23:14,710 So this is the setup. 395 00:23:14,710 --> 00:23:19,300 And now all I'm going to do is read off what the formula says. 396 00:23:19,300 --> 00:23:24,190 What it says is, this is equal to u times v. So u 397 00:23:24,190 --> 00:23:25,270 is this and v is that. 398 00:23:25,270 --> 00:23:32,440 So it's x ln x minus, so that again, this is uv. 399 00:23:32,440 --> 00:23:37,170 Except in the other order, vu. 400 00:23:37,170 --> 00:23:40,510 And then I'm integrating, and what do I have to integrate? 401 00:23:40,510 --> 00:23:42,540 u'v. So look up there. 402 00:23:42,540 --> 00:23:47,760 u'v with a minus sign here. u' = 1 / x, and v = x. 403 00:23:47,760 --> 00:23:50,890 So it's 1 / x, that's u'. 404 00:23:50,890 --> 00:23:56,070 And here is x, that's v, dx. 405 00:23:56,070 --> 00:23:58,270 Now, that one is easy to integrate. 406 00:23:58,270 --> 00:24:00,870 Because (1/x) x = 1. 407 00:24:00,870 --> 00:24:07,630 And the integral of 1 dx is x, plus c, if you like. 408 00:24:07,630 --> 00:24:10,510 So the antiderivative of 1 is x. 409 00:24:10,510 --> 00:24:11,610 And so here's our answer. 410 00:24:11,610 --> 00:24:34,480 Our answer is that this is x ln x - x + c. 411 00:24:34,480 --> 00:24:37,610 I'm going to do two more slightly more 412 00:24:37,610 --> 00:24:39,380 complicated examples. 413 00:24:39,380 --> 00:24:42,590 And then really, the main thing is 414 00:24:42,590 --> 00:24:44,580 to get yourself used to this method. 415 00:24:44,580 --> 00:24:47,930 And there's no one way of doing that. 416 00:24:47,930 --> 00:24:49,970 Just practice makes perfect. 417 00:24:49,970 --> 00:24:53,070 And so we'll just do a few more examples. 418 00:24:53,070 --> 00:24:55,870 And illustrate them. 419 00:24:55,870 --> 00:24:59,960 The second example that I'm going to use is the integral 420 00:24:59,960 --> 00:25:03,590 of (ln x)^2 dx. 421 00:25:03,590 --> 00:25:08,510 And this is just slightly more recalcitrant. 422 00:25:08,510 --> 00:25:13,270 Namely, I'm going to let u be (ln x)^2. 423 00:25:17,740 --> 00:25:20,440 And again, v = x. 424 00:25:20,440 --> 00:25:21,890 So that matches up here. 425 00:25:21,890 --> 00:25:23,730 That is, v' = 1. 426 00:25:23,730 --> 00:25:28,390 So this is uv'. 427 00:25:28,390 --> 00:25:31,440 So this thing is uv'. 428 00:25:31,440 --> 00:25:33,510 And then we'll just see what happens. 429 00:25:33,510 --> 00:25:38,060 Now, the game that we get is that when I differentiate 430 00:25:38,060 --> 00:25:42,870 the logarithm squared, I'm going to to get something simpler. 431 00:25:42,870 --> 00:25:46,850 It's not going to win us the whole battle, 432 00:25:46,850 --> 00:25:49,860 but it will get us started. 433 00:25:49,860 --> 00:25:51,770 So here we get u'. 434 00:25:51,770 --> 00:25:56,650 And that's 2 ln x times 1/x. 435 00:25:56,650 --> 00:26:00,470 Applying the chain rule. 436 00:26:00,470 --> 00:26:06,020 And so the formula is that this is x (ln x)^2, 437 00:26:06,020 --> 00:26:11,710 minus the integral of, well it's u'v, right, 438 00:26:11,710 --> 00:26:13,210 that's what I have to put over here. 439 00:26:13,210 --> 00:26:22,190 So u' = 2 ln x 1/x and v = x. 440 00:26:22,190 --> 00:26:25,880 And so now, you notice something interesting happening here. 441 00:26:25,880 --> 00:26:28,960 So let me just demarcate this a little bit. 442 00:26:28,960 --> 00:26:34,680 And let you see what it is that I'm doing here. 443 00:26:34,680 --> 00:26:36,820 So notice, this is the same integral. 444 00:26:36,820 --> 00:26:38,650 So here we have x (ln x)^2. 445 00:26:38,650 --> 00:26:41,050 We've already solved that part. 446 00:26:41,050 --> 00:26:43,870 But now know notice that the 1/x and the x cancel. 447 00:26:43,870 --> 00:26:46,890 So we're back to the previous case. 448 00:26:46,890 --> 00:26:49,570 We didn't win all the way, but actually we reduced ourselves 449 00:26:49,570 --> 00:26:51,350 to this integral. 450 00:26:51,350 --> 00:26:56,630 To the integral of ln x, which we already know. 451 00:26:56,630 --> 00:26:58,700 So here, I can copy that down. 452 00:26:58,700 --> 00:27:04,450 That's - -2(x ln x - x), and then I have to throw 453 00:27:04,450 --> 00:27:05,460 in a constant, c. 454 00:27:05,460 --> 00:27:07,270 And that's the end of the problem here. 455 00:27:07,270 --> 00:27:10,260 That's it. 456 00:27:10,260 --> 00:27:26,400 So this piece, I got from Example 1. 457 00:27:26,400 --> 00:27:34,350 Now, this illustrates a principle 458 00:27:34,350 --> 00:27:36,200 which is a little bit more complicated 459 00:27:36,200 --> 00:27:40,390 than just the one of integration by parts. 460 00:27:40,390 --> 00:27:43,630 Which is a sort of a general principle which 461 00:27:43,630 --> 00:27:48,150 I'll call my Example 3, which is something which 462 00:27:48,150 --> 00:27:56,520 is called a reduction formula. 463 00:27:56,520 --> 00:28:00,540 A reduction formula is a case where we apply some rule 464 00:28:00,540 --> 00:28:03,220 and we figure out one of these integrals 465 00:28:03,220 --> 00:28:05,440 in terms of something else. 466 00:28:05,440 --> 00:28:07,320 Which is a little bit simpler. 467 00:28:07,320 --> 00:28:09,070 And eventually we'll get down to the end, 468 00:28:09,070 --> 00:28:12,810 but it may take us n steps from the beginning. 469 00:28:12,810 --> 00:28:17,541 So the example is (ln x)^n dx. 470 00:28:17,541 --> 00:28:18,040 . 471 00:28:18,040 --> 00:28:21,710 And the claim is that if I do what I did in Example 2, 472 00:28:21,710 --> 00:28:26,500 to this case, I'll get a simpler one which will involve 473 00:28:26,500 --> 00:28:28,224 the (n-1)st power. 474 00:28:28,224 --> 00:28:29,640 And that way I can get all the way 475 00:28:29,640 --> 00:28:32,440 back down to the final answer. 476 00:28:32,440 --> 00:28:34,160 So here's what happens. 477 00:28:34,160 --> 00:28:37,090 We take u as (ln x)^n. 478 00:28:37,090 --> 00:28:40,980 This is the same discussion as before, v = x. 479 00:28:40,980 --> 00:28:44,850 And then u' is n n (ln x)^(n-1) 1/x. 480 00:28:47,440 --> 00:28:50,020 And v' is 1. 481 00:28:50,020 --> 00:28:52,800 And so the setup is similar. 482 00:28:52,800 --> 00:28:59,320 We have here x (ln x)^n minus the integral. 483 00:28:59,320 --> 00:29:05,020 And there's n times, it turns out to be (ln x)^(n-1). 484 00:29:05,020 --> 00:29:26,410 And then there's a 1/x and an x, which cancel. 485 00:29:26,410 --> 00:29:31,010 So I'm going to explain this also abstractly 486 00:29:31,010 --> 00:29:35,450 a little bit just to show you what's happening here. 487 00:29:35,450 --> 00:29:44,360 If you use the notation F_n(x) is the integral of (ln x)^n dx, 488 00:29:44,360 --> 00:29:46,780 and we're going to forget the constant here. 489 00:29:46,780 --> 00:29:51,710 Then the relationship that we have here is that F_n(x) is 490 00:29:51,710 --> 00:29:56,590 equal to n ln-- I'm sorry, x (ln x)^n. 491 00:29:56,590 --> 00:29:59,490 That's the first term over here. 492 00:29:59,490 --> 00:30:03,650 Minus n times the preceding one. 493 00:30:03,650 --> 00:30:07,670 This one here. 494 00:30:07,670 --> 00:30:11,060 And the idea is that eventually we can get down. 495 00:30:11,060 --> 00:30:15,010 If we start with the nth one, we have a formula that includes-- 496 00:30:15,010 --> 00:30:17,440 So the reduction is to the n (n-1)st. 497 00:30:17,440 --> 00:30:21,280 Then we can reduce to the (n-2)nd and so on. 498 00:30:21,280 --> 00:30:23,610 Until we reduce to the 1, the first one. 499 00:30:23,610 --> 00:30:29,390 And then in fact we can even go down to the 0th one. 500 00:30:29,390 --> 00:30:32,510 So this is the idea of a reduction formula. 501 00:30:32,510 --> 00:30:37,320 And let me illustrate it exactly in the context of Examples 1 502 00:30:37,320 --> 00:30:38,870 and 2. 503 00:30:38,870 --> 00:30:44,670 So the first step would be to evaluate the first one. 504 00:30:44,670 --> 00:30:48,190 Which is, if you like, (ln x)^0 dx. 505 00:30:48,190 --> 00:30:52,370 That's very easy, that's x. 506 00:30:52,370 --> 00:31:01,000 And then F_1(x) = x ln x - F_0(x). 507 00:31:01,000 --> 00:31:03,240 Now, that's applying this rule. 508 00:31:03,240 --> 00:31:06,830 So let me just put it in a box here. 509 00:31:06,830 --> 00:31:09,380 This is the method of induction. 510 00:31:09,380 --> 00:31:13,510 Here's the rule. 511 00:31:13,510 --> 00:31:21,930 And I'm applying it for n = 1. 512 00:31:21,930 --> 00:31:23,810 I plugged in n = 1 here. 513 00:31:23,810 --> 00:31:26,720 So here, I have x (ln x)^1 - 1*F_0(x). 514 00:31:32,430 --> 00:31:39,240 And that's what I put right here, on the right-hand side. 515 00:31:39,240 --> 00:31:42,440 And that's going to generate for me the formula that I want, 516 00:31:42,440 --> 00:31:44,920 which is x ln x - x. 517 00:31:44,920 --> 00:31:49,160 That's the answer to this problem over here. 518 00:31:49,160 --> 00:31:51,230 This was Example 1. 519 00:31:51,230 --> 00:31:52,980 Notice I dropped the constants because I 520 00:31:52,980 --> 00:31:54,880 can add them in at the end. 521 00:31:54,880 --> 00:31:57,590 So I'll put in parentheses here, plus c. 522 00:31:57,590 --> 00:32:01,850 That's what would happen at the end of the problem. 523 00:32:01,850 --> 00:32:10,320 The next step, so that was Example 1, and now Example 2 524 00:32:10,320 --> 00:32:12,190 works more or less the same way. 525 00:32:12,190 --> 00:32:14,590 I'm just summarizing what I did on that blackboard 526 00:32:14,590 --> 00:32:16,640 right up here. 527 00:32:16,640 --> 00:32:21,030 The same thing, but in much more compact notation. 528 00:32:21,030 --> 00:32:29,950 If I take F_2(x), that's going to be equal to x (ln x)^2 - 529 00:32:29,950 --> 00:32:31,820 2 F_1(x). 530 00:32:31,820 --> 00:32:41,550 Again, this is box for n = 2. 531 00:32:41,550 --> 00:32:46,730 And if I plug it in, what I'm getting here is x (ln x)^2 532 00:32:46,730 --> 00:32:49,570 minus twice this stuff here. 533 00:32:49,570 --> 00:32:55,780 Which is right here. x ln x - x. 534 00:32:55,780 --> 00:32:58,580 If you like, plus c. 535 00:32:58,580 --> 00:33:07,360 So I'll leave the c off. 536 00:33:07,360 --> 00:33:12,170 So this is how reduction formulas work in general. 537 00:33:12,170 --> 00:33:22,620 I'm going to give you one more example of a reduction formula. 538 00:33:22,620 --> 00:33:30,560 So I guess we have to call this Example 4. 539 00:33:30,560 --> 00:33:34,050 Let's be fancy, let's make it the sine. 540 00:33:34,050 --> 00:33:35,950 No no, no, let's be fancier still. 541 00:33:35,950 --> 00:33:48,790 Let's make it e^x So this would also work for cos x and sin x. 542 00:33:48,790 --> 00:33:50,110 The same sort of thing. 543 00:33:50,110 --> 00:33:52,840 And I should mention that on your homework, 544 00:33:52,840 --> 00:33:54,300 you have to do it for cos x. 545 00:33:54,300 --> 00:33:56,550 I decided to change my mind on the spur of the moment. 546 00:33:56,550 --> 00:33:57,924 I'm not going to do it for cosine 547 00:33:57,924 --> 00:34:00,530 because you have to work it out on your homework for cosine. 548 00:34:00,530 --> 00:34:03,100 In a later homework you'll even do this case. 549 00:34:03,100 --> 00:34:05,190 So it's fine. 550 00:34:05,190 --> 00:34:07,400 You need the practice. 551 00:34:07,400 --> 00:34:10,240 OK, so how am I going to do it this time. 552 00:34:10,240 --> 00:34:13,970 This is again, a reduction formula. 553 00:34:13,970 --> 00:34:19,420 And the trick here is to pick u to be this function here. 554 00:34:19,420 --> 00:34:20,780 And the reason is the following. 555 00:34:20,780 --> 00:34:23,029 So it's very important to pick which function is the u 556 00:34:23,029 --> 00:34:26,450 and which function is the v. That's the only decision you 557 00:34:26,450 --> 00:34:30,020 have to make if you're going to apply integration by parts. 558 00:34:30,020 --> 00:34:34,420 When I pick this function as the u, the advantage that I have 559 00:34:34,420 --> 00:34:38,150 is that u' is simpler. 560 00:34:38,150 --> 00:34:39,630 How is it simpler? 561 00:34:39,630 --> 00:34:42,820 It's simpler because it's one degree down. 562 00:34:42,820 --> 00:34:45,420 So that's making progress for us. 563 00:34:45,420 --> 00:34:48,820 On the other hand, this function here 564 00:34:48,820 --> 00:34:52,530 is going to be what I'll use for v. 565 00:34:52,530 --> 00:34:55,279 And if I differentiated that, if I did it the other way around 566 00:34:55,279 --> 00:34:57,070 and I differentiated that, I would just get 567 00:34:57,070 --> 00:34:58,900 the same level of complexity. 568 00:34:58,900 --> 00:35:01,120 Differentiating e^x just gives you back e^x. 569 00:35:01,120 --> 00:35:02,000 So that's boring. 570 00:35:02,000 --> 00:35:05,750 It doesn't make any progress in this process. 571 00:35:05,750 --> 00:35:11,460 And so I'm going to instead let v = e^x and-- Sorry, 572 00:35:11,460 --> 00:35:12,830 this is v'. 573 00:35:12,830 --> 00:35:14,380 Make it v' = e^x. 574 00:35:14,380 --> 00:35:15,950 And then v = e^x. 575 00:35:15,950 --> 00:35:20,640 At least it isn't any worse when I went backwards like that. 576 00:35:20,640 --> 00:35:28,150 So now, I have u and v', and now I get x^n e^x. 577 00:35:28,150 --> 00:35:31,490 This again is u, and this is v. So it happens that v is equal 578 00:35:31,490 --> 00:35:34,200 to v ' so it's a little confusing here. 579 00:35:34,200 --> 00:35:37,640 But this is the one we're calling v'. 580 00:35:37,640 --> 00:35:41,510 And here's v. And now minus the integral and I have here 581 00:35:41,510 --> 00:35:43,760 nx^(n-1). 582 00:35:43,760 --> 00:35:45,120 And I have here e^x. 583 00:35:45,120 --> 00:35:52,060 So this is u' and this is v dx. 584 00:35:52,060 --> 00:35:55,180 So this recurrence is a new recurrence. 585 00:35:55,180 --> 00:35:57,050 And let me summarize it here. 586 00:35:57,050 --> 00:36:02,270 It's saying that G_n(x) should be the integral of x^n e^x dx. 587 00:36:05,210 --> 00:36:06,810 Again, I'm dropping the c. 588 00:36:06,810 --> 00:36:17,060 And then the reduction formula is that G_n(x) is equal to this 589 00:36:17,060 --> 00:36:25,300 expression here: x^n e^x - n*G_(n-1)(x). 590 00:36:25,300 --> 00:36:32,830 So here's our reduction formula. 591 00:36:32,830 --> 00:36:37,912 And to illustrate this, if I take G_0(x), 592 00:36:37,912 --> 00:36:39,870 if you think about it for a second that's just, 593 00:36:39,870 --> 00:36:40,744 there's nothing here. 594 00:36:40,744 --> 00:36:44,680 The antiderivative of e^x, that's going to be e^x, 595 00:36:44,680 --> 00:36:48,220 that's getting started at the real basement here. 596 00:36:48,220 --> 00:36:52,000 Again, as always, 0 is my favorite number. 597 00:36:52,000 --> 00:36:52,820 Not 1. 598 00:36:52,820 --> 00:36:55,850 I always start with the easiest one, if possible. 599 00:36:55,850 --> 00:37:00,150 And now G_1, applying this formula, 600 00:37:00,150 --> 00:37:06,830 is going to be equal to x e^x - G_0(x). 601 00:37:06,830 --> 00:37:11,180 Which is just-- Right, because n is 1 and n - 1 is 0. 602 00:37:11,180 --> 00:37:13,970 And so that's just ^ x e^x - e^x. 603 00:37:17,220 --> 00:37:19,770 So this is a very, very fancy way 604 00:37:19,770 --> 00:37:22,620 of saying the following fact. 605 00:37:22,620 --> 00:37:32,210 I'll put it over on this other board. 606 00:37:32,210 --> 00:37:38,270 Which is that the integral of x e^x dx is equal to x e^x - 607 00:37:38,270 --> 00:37:44,600 x + c. 608 00:37:44,600 --> 00:37:45,270 Yeah, question. 609 00:37:45,270 --> 00:37:50,950 STUDENT: [INAUDIBLE] 610 00:37:50,950 --> 00:37:53,020 PROFESSOR: The question is, why is this true. 611 00:37:53,020 --> 00:37:54,830 Why is this statement true. 612 00:37:54,830 --> 00:37:56,420 Why is G_0 equal to e^x. 613 00:37:56,420 --> 00:37:58,410 I did that in my head. 614 00:37:58,410 --> 00:38:02,910 What I did was, I first wrote down the formula for G_0. 615 00:38:02,910 --> 00:38:08,100 Which was G_0 is equal to the integral of e^x dx. 616 00:38:11,076 --> 00:38:12,950 Because there's an x to the 0 power in there, 617 00:38:12,950 --> 00:38:15,010 which is just 1. 618 00:38:15,010 --> 00:38:17,780 And then I know the antiderivative of e^x. 619 00:38:17,780 --> 00:38:23,230 It's e^x. 620 00:38:23,230 --> 00:38:30,780 STUDENT: [INAUDIBLE] 621 00:38:30,780 --> 00:38:33,030 PROFESSOR: How do you know when this method will work? 622 00:38:33,030 --> 00:38:37,370 The answer is only by experience. 623 00:38:37,370 --> 00:38:40,091 You must get practice doing this. 624 00:38:40,091 --> 00:38:41,590 If you look in your textbook, you'll 625 00:38:41,590 --> 00:38:44,430 see hints as to what to do. 626 00:38:44,430 --> 00:38:46,050 The other hint that I want to say 627 00:38:46,050 --> 00:38:48,180 is that if you find that you have 628 00:38:48,180 --> 00:38:51,000 one factor in your expression which when you differentiate 629 00:38:51,000 --> 00:38:52,590 it, it gets easier. 630 00:38:52,590 --> 00:38:55,140 And when you antidifferentiate the other half, 631 00:38:55,140 --> 00:38:57,780 it doesn't get any worse, then that's 632 00:38:57,780 --> 00:39:01,790 when this method has a chance of helping. 633 00:39:01,790 --> 00:39:04,430 And there is-- there's no general thing. 634 00:39:04,430 --> 00:39:09,330 The thing is, though, if you do it with x^n e^x, x^n cos x, 635 00:39:09,330 --> 00:39:11,970 x^n sin x, those are examples where it works. 636 00:39:11,970 --> 00:39:15,600 This power of the log. 637 00:39:15,600 --> 00:39:19,150 I'll give you one more example here. 638 00:39:19,150 --> 00:39:26,519 So this was G_1(x), right. 639 00:39:26,519 --> 00:39:28,310 I'll give you one more example in a second. 640 00:39:28,310 --> 00:39:29,330 Yeah. 641 00:39:29,330 --> 00:39:33,220 STUDENT: [INAUDIBLE] 642 00:39:33,220 --> 00:39:35,680 PROFESSOR: Thank you. 643 00:39:35,680 --> 00:39:38,490 There's a mistake here. 644 00:39:38,490 --> 00:39:39,240 That's bad. 645 00:39:39,240 --> 00:39:45,910 I was thinking in the back of my head of the following formula. 646 00:39:45,910 --> 00:39:51,159 Which is another one which we've just done. 647 00:39:51,159 --> 00:39:53,450 So these are the types of formulas that you can get out 648 00:39:53,450 --> 00:39:57,620 of integration by parts. 649 00:39:57,620 --> 00:40:00,650 There's also another way of getting these, which I'm not 650 00:40:00,650 --> 00:40:02,240 going to say anything about. 651 00:40:02,240 --> 00:40:04,282 Which is called advance guessing. 652 00:40:04,282 --> 00:40:06,740 You guess in advance what the form is, you differentiate it 653 00:40:06,740 --> 00:40:08,160 and you check. 654 00:40:08,160 --> 00:40:14,250 That does work too, with many of these cases. 655 00:40:14,250 --> 00:40:21,580 I want to give you an illustration. 656 00:40:21,580 --> 00:40:30,760 Just because, you know, these formulas are somewhat dry. 657 00:40:30,760 --> 00:40:34,570 So I want to give you just at least one application. 658 00:40:34,570 --> 00:40:42,230 We're almost done with the idea of these formulas. 659 00:40:42,230 --> 00:40:44,880 And we're going to get back now to being 660 00:40:44,880 --> 00:40:47,990 able to handle lots more integrals than we could before. 661 00:40:47,990 --> 00:40:49,810 And what's satisfying is that now we 662 00:40:49,810 --> 00:40:53,830 can get numbers out instead of being stuck and hamstrung 663 00:40:53,830 --> 00:40:55,120 with only a few techniques. 664 00:40:55,120 --> 00:40:57,680 Now we have all of the techniques of integration 665 00:40:57,680 --> 00:40:59,250 that anybody has. 666 00:40:59,250 --> 00:41:01,820 And so we can do pretty much anything 667 00:41:01,820 --> 00:41:04,430 we want that's possible to do. 668 00:41:04,430 --> 00:41:14,250 So here's, if you like, an application that 669 00:41:14,250 --> 00:41:18,890 illustrates how integration by parts can be helpful. 670 00:41:18,890 --> 00:41:27,030 And we're going to find the volume of an exponential wine 671 00:41:27,030 --> 00:41:34,290 glass here. 672 00:41:34,290 --> 00:41:38,350 Again, don't try this at home, but. 673 00:41:38,350 --> 00:41:40,500 So let's see. 674 00:41:40,500 --> 00:41:44,660 It's going to be this beautiful guy here. 675 00:41:44,660 --> 00:41:46,930 I think. 676 00:41:46,930 --> 00:41:49,060 OK, so what's it going to be. 677 00:41:49,060 --> 00:41:52,780 This graph is going to be y = e^x. 678 00:41:52,780 --> 00:42:04,030 Then we're going to rotate it around the y-axis. 679 00:42:04,030 --> 00:42:10,290 And this level here is the height y = 1. 680 00:42:10,290 --> 00:42:12,990 And the top, let's say, is y = e. 681 00:42:12,990 --> 00:42:22,160 So that the horizontal here, coming down, is x = 1. 682 00:42:22,160 --> 00:42:35,050 Now, there are two ways to set up this problem. 683 00:42:35,050 --> 00:42:40,050 And so there are two methods. 684 00:42:40,050 --> 00:42:44,110 And this is also a good review because, of course, 685 00:42:44,110 --> 00:42:46,330 we did this in the last unit. 686 00:42:46,330 --> 00:42:58,480 The two methods are horizontal and vertical slices. 687 00:42:58,480 --> 00:43:00,660 Those are the two ways we can do this. 688 00:43:00,660 --> 00:43:03,710 Now, if we do it with-- So let's start out 689 00:43:03,710 --> 00:43:09,370 with the horizontal ones. 690 00:43:09,370 --> 00:43:12,370 That's this shape here. 691 00:43:12,370 --> 00:43:15,370 And we're going like that. 692 00:43:15,370 --> 00:43:19,900 And the horizontal slices mean that this little bit here 693 00:43:19,900 --> 00:43:22,842 is of thickness dy. 694 00:43:22,842 --> 00:43:24,550 And then we're going to wrap that around. 695 00:43:24,550 --> 00:43:30,810 So this is going to become a disk. 696 00:43:30,810 --> 00:43:34,070 This is the method of disks. 697 00:43:34,070 --> 00:43:35,830 And what's this distance here? 698 00:43:35,830 --> 00:43:37,770 Well, this place is x. 699 00:43:37,770 --> 00:43:40,680 And so the disk has area pi x^2. 700 00:43:40,680 --> 00:43:42,840 And we're going to add up the thickness 701 00:43:42,840 --> 00:43:45,730 of the disks and we're going to integrate from 1 to e. 702 00:43:45,730 --> 00:43:51,930 So here's our volume. 703 00:43:51,930 --> 00:43:54,510 And now we have one last little item of business 704 00:43:54,510 --> 00:43:56,230 before we can evaluate this integral. 705 00:43:56,230 --> 00:43:58,480 And that is that we need to know the relationship here 706 00:43:58,480 --> 00:44:01,360 on the curve, that y = e^x. 707 00:44:01,360 --> 00:44:07,490 So that means x = ln y. 708 00:44:07,490 --> 00:44:09,180 And in order to evaluate this integral, 709 00:44:09,180 --> 00:44:13,050 we have to evaluate x correctly as a function of y. 710 00:44:13,050 --> 00:44:26,200 So that's the integral from 1 to e of (ln y)^2, times pi, dy. 711 00:44:26,200 --> 00:44:27,860 So now you see that this is an integral 712 00:44:27,860 --> 00:44:30,140 that we did calculate already. 713 00:44:30,140 --> 00:44:34,280 And in fact, it's sitting right here. 714 00:44:34,280 --> 00:44:37,030 Except with the variable x instead of the variable y. 715 00:44:37,030 --> 00:44:44,830 So the answer, which we already had, is this F_2(y) here. 716 00:44:44,830 --> 00:44:47,820 So maybe I'll write it that way. 717 00:44:47,820 --> 00:44:52,010 So this is F_2(y) between 1 and e. 718 00:44:52,010 --> 00:45:00,040 And now let's figure out what it is. 719 00:45:00,040 --> 00:45:02,060 It's written over there. 720 00:45:02,060 --> 00:45:15,100 It's y (ln y)^2 - 2(y ln y - y). 721 00:45:15,100 --> 00:45:24,460 The whole thing evaluated at 1, e. 722 00:45:24,460 --> 00:45:29,130 And that is, if I plug in e here, I get e. 723 00:45:29,130 --> 00:45:32,150 Except there's a factor of pi there, sorry. 724 00:45:32,150 --> 00:45:36,360 Missed the pi factor. 725 00:45:36,360 --> 00:45:38,780 So there's an e here. 726 00:45:38,780 --> 00:45:43,050 And then I subtract off, well, at 1 this is e - e. 727 00:45:43,050 --> 00:45:44,330 So it cancels. 728 00:45:44,330 --> 00:45:45,440 There's nothing left. 729 00:45:45,440 --> 00:45:50,200 And then at 1, I get ln 1 is 0, ln 1 730 00:45:50,200 --> 00:45:53,500 is 0, there's only one term left, which is 2. 731 00:45:53,500 --> 00:45:55,720 So it's -2. 732 00:45:55,720 --> 00:46:03,790 That's the answer. 733 00:46:03,790 --> 00:46:10,830 Now we get to compare that with what happens 734 00:46:10,830 --> 00:46:15,950 if we do it the other way. 735 00:46:15,950 --> 00:46:19,870 So what's the vertical? 736 00:46:19,870 --> 00:46:31,830 So by vertical slicing, we get shells. 737 00:46:31,830 --> 00:46:38,530 And that starts-- That's in the x variable. 738 00:46:38,530 --> 00:46:43,480 It starts at 0 and ends at 1 and it's dx. 739 00:46:43,480 --> 00:46:46,300 And what are the shells? 740 00:46:46,300 --> 00:46:51,750 Well, the shells are, if I can draw the picture again, 741 00:46:51,750 --> 00:46:55,120 they start-- the top value is e. 742 00:46:55,120 --> 00:47:02,530 And the bottom value is, I need a little bit of room for this. 743 00:47:02,530 --> 00:47:06,810 The bottom value is y. 744 00:47:06,810 --> 00:47:12,670 And then we have 2 pi x is the circumference, 745 00:47:12,670 --> 00:47:15,970 as we sweep it around dx. 746 00:47:15,970 --> 00:47:18,260 So here's our new volume. 747 00:47:18,260 --> 00:47:23,600 Expressed in this different way. 748 00:47:23,600 --> 00:47:26,220 So now I'm going to plug in what this is. 749 00:47:26,220 --> 00:47:30,380 It's the integral from 0 to 1 of e minus e^x, 750 00:47:30,380 --> 00:47:36,530 that's the formula for y, 2 pi x dx. 751 00:47:36,530 --> 00:47:39,670 And what you see is that you get the integral 752 00:47:39,670 --> 00:47:45,150 from 0 to 1 of 2 pi e x dx. 753 00:47:45,150 --> 00:47:46,540 That's easy, right? 754 00:47:46,540 --> 00:47:51,940 That's just 2 pi e times 1/2. 755 00:47:51,940 --> 00:47:54,820 This one is just the area of a triangle. 756 00:47:54,820 --> 00:47:56,890 If I factor out the 2 pi e. 757 00:47:56,890 --> 00:48:03,230 And then the other piece is the integral of 2 pi x e^x dx. 758 00:48:03,230 --> 00:48:08,610 From 0 to 1. 759 00:48:08,610 --> 00:48:12,480 STUDENT: [INAUDIBLE] PROFESSOR: Are you asking me whether I 760 00:48:12,480 --> 00:48:14,370 need an x^2 here? 761 00:48:14,370 --> 00:48:15,960 I just evaluated the integral. 762 00:48:15,960 --> 00:48:17,290 I just did it geometrically. 763 00:48:17,290 --> 00:48:19,570 I said, this is the area of a triangle. 764 00:48:19,570 --> 00:48:21,850 I didn't antidifferentiate and evaluate it, 765 00:48:21,850 --> 00:48:23,880 I just told you the number. 766 00:48:23,880 --> 00:48:27,580 Because it's a definite integral. 767 00:48:27,580 --> 00:48:31,650 So now, this one here, I can read off from right up 768 00:48:31,650 --> 00:48:33,980 here, above it. 769 00:48:33,980 --> 00:48:37,050 This is G_1. 770 00:48:37,050 --> 00:48:42,060 So this is equal to, let's check it out here. 771 00:48:42,060 --> 00:48:49,170 So this is pi e, right, minus 2 pi G_1(x), 772 00:48:49,170 --> 00:48:52,280 evaluated at 0 and 1. 773 00:48:52,280 --> 00:48:54,990 So let's make sure that it's the same as what we had before. 774 00:48:54,990 --> 00:48:59,860 It's pi e minus 2 pi times-- here's G_1. 775 00:48:59,860 --> 00:49:03,230 So it's x e^x - e^x. 776 00:49:03,230 --> 00:49:05,720 So at x = 1, that cancels. 777 00:49:05,720 --> 00:49:08,260 But at the bottom end, it's e^0. 778 00:49:08,260 --> 00:49:12,040 So it's -1 here. 779 00:49:12,040 --> 00:49:13,130 Is that right? 780 00:49:13,130 --> 00:49:13,710 Yep. 781 00:49:13,710 --> 00:49:17,060 So it's pi e - 2. 782 00:49:17,060 --> 00:49:21,650 It's the same. 783 00:49:21,650 --> 00:49:22,150 Question. 784 00:49:22,150 --> 00:49:28,030 STUDENT: [INAUDIBLE] 785 00:49:28,030 --> 00:49:33,380 PROFESSOR: From here to here, is that the question? 786 00:49:33,380 --> 00:49:39,710 STUDENT: [INAUDIBLE] 787 00:49:39,710 --> 00:49:43,730 PROFESSOR: So the step here is just the distributive law. 788 00:49:43,730 --> 00:49:46,810 This is e 2 pi x, that's this term. 789 00:49:46,810 --> 00:49:49,550 And the other terms, the minus sign is outside. 790 00:49:49,550 --> 00:49:51,320 The 2 pi I factored out. 791 00:49:51,320 --> 00:49:56,980 And the x and the e^x stayed inside the integral sign. 792 00:49:56,980 --> 00:49:59,140 Thank you. 793 00:49:59,140 --> 00:50:01,670 The correction is that there was a missing minus 794 00:50:01,670 --> 00:50:03,780 sign, last time. 795 00:50:03,780 --> 00:50:13,100 When I integrated from 0 to 1, x e^x dx, I had a x e^x - e^x. 796 00:50:13,100 --> 00:50:15,090 Evaluated at 0 and 1. 797 00:50:15,090 --> 00:50:18,350 And that's equal to +1. 798 00:50:18,350 --> 00:50:21,580 I was missing this minus sign. 799 00:50:21,580 --> 00:50:30,840 The place where it came in was in this wineglass example. 800 00:50:30,840 --> 00:50:36,210 We had the integral of 2 pi x (e - e^x) dx. 801 00:50:39,340 --> 00:50:48,900 And that was 2 pi e integral of x dx, from 0 to 1, -2 pi, 802 00:50:48,900 --> 00:50:52,810 integral from 0 to 1 of x e^x dx. 803 00:50:52,810 --> 00:50:58,400 And then I worked this out and it was pi e. 804 00:50:58,400 --> 00:51:03,030 And then this one was -2 pi, and what I wrote down was -1. 805 00:51:03,030 --> 00:51:05,410 But there should have been an extra minus sign there. 806 00:51:05,410 --> 00:51:08,430 So it's this. 807 00:51:08,430 --> 00:51:11,930 The final answer was correct, but this minus sign 808 00:51:11,930 --> 00:51:13,590 was missing. 809 00:51:13,590 --> 00:51:16,930 Right there. 810 00:51:16,930 --> 00:51:20,460 So just, right there.