WEBVTT 00:00:00.000 --> 00:00:02.330 The following content is provided under a Creative 00:00:02.330 --> 00:00:03.410 Commons license. 00:00:03.410 --> 00:00:05.770 Your support will help MIT OpenCourseWare 00:00:05.770 --> 00:00:09.930 continue to offer high quality educational resources for free. 00:00:09.930 --> 00:00:12.550 To make a donation or to view additional materials 00:00:12.550 --> 00:00:16.150 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.150 --> 00:00:23.170 at ocw.mit.edu. 00:00:23.170 --> 00:00:28.390 PROFESSOR: Today we're going to continue our discussion 00:00:28.390 --> 00:00:33.040 of methods of integration. 00:00:33.040 --> 00:00:35.560 The method that I'm going to describe today 00:00:35.560 --> 00:00:38.310 handles a whole class of functions 00:00:38.310 --> 00:00:40.510 of the following form. 00:00:40.510 --> 00:00:53.460 You take P(x) / Q(x) and this is known as a rational function. 00:00:53.460 --> 00:00:57.810 And all that means is that it's a ratio 00:00:57.810 --> 00:01:09.680 of two polynomials, which are these functions P(x) and Q(x). 00:01:13.300 --> 00:01:17.380 We'll handle all such functions by a method which 00:01:17.380 --> 00:01:26.250 is known as partial fractions. 00:01:26.250 --> 00:01:36.190 And what this does is, it splits P / Q into what 00:01:36.190 --> 00:01:45.260 you could call easier pieces. 00:01:45.260 --> 00:01:47.571 So that's going to be some kind of algebra. 00:01:47.571 --> 00:01:49.070 And that's what we're going to spend 00:01:49.070 --> 00:01:52.640 most of our time doing today. 00:01:52.640 --> 00:01:55.090 I'll start with an example. 00:01:55.090 --> 00:01:58.950 And all of my examples will be illustrating 00:01:58.950 --> 00:02:01.830 more general methods. 00:02:01.830 --> 00:02:08.200 The example is to integrate the function 1 1/(x-1) plus, say, 00:02:08.200 --> 00:02:15.240 3/(x+2), dx. 00:02:15.240 --> 00:02:17.400 That's easy to do. 00:02:17.400 --> 00:02:19.410 It's just, we already know the answer. 00:02:19.410 --> 00:02:23.170 It's ln x - ln(x-1) + 3 ln(x+3). 00:02:26.780 --> 00:02:28.340 Plus a constant. 00:02:28.340 --> 00:02:37.050 So that's done. 00:02:37.050 --> 00:02:41.330 So now, here's the difficulty that is going to arise. 00:02:41.330 --> 00:02:45.320 The difficulty is that I can start with this function, which 00:02:45.320 --> 00:02:49.210 is perfectly manageable. 00:02:49.210 --> 00:02:52.830 And than I can add these two functions together. 00:02:52.830 --> 00:02:55.380 The way I add fractions. 00:02:55.380 --> 00:03:00.750 So that's getting a common denominator. 00:03:00.750 --> 00:03:06.140 And so that gives me x+2 here, plus 3(x-1). 00:03:06.140 --> 00:03:11.090 And now if I combine together all of these terms, 00:03:11.090 --> 00:03:16.960 then altogether I have 4x + 2 - 3, that's -1. 00:03:16.960 --> 00:03:23.260 And if I multiply out the denominator that's x^2 plus-- 00:03:23.260 --> 00:03:28.060 That 2 turned into a 3, that's interesting. 00:03:28.060 --> 00:03:31.240 I hope there aren't too many more of those transformations. 00:03:31.240 --> 00:03:32.570 Is there another one here? 00:03:32.570 --> 00:03:36.830 STUDENT: [INAUDIBLE] 00:03:36.830 --> 00:03:39.150 PROFESSOR: Oh, it happened earlier on. 00:03:39.150 --> 00:03:42.620 Wow that's an interesting vibration there. 00:03:42.620 --> 00:03:45.510 OK. 00:03:45.510 --> 00:03:50.500 Thank you. 00:03:50.500 --> 00:03:56.770 So, I guess my 3's were speaking to my 2's. 00:03:56.770 --> 00:03:58.580 Somewhere in my past. 00:03:58.580 --> 00:04:07.200 OK, anyway, I think this is now correct. 00:04:07.200 --> 00:04:10.020 So the problem is the following. 00:04:10.020 --> 00:04:11.450 This is the problem with this. 00:04:11.450 --> 00:04:14.830 This integral was easy. 00:04:14.830 --> 00:04:18.500 I'm calling it easy, we already know how to do it. 00:04:18.500 --> 00:04:19.450 Over here. 00:04:19.450 --> 00:04:28.040 But now over here, it's disguised. 00:04:28.040 --> 00:04:31.947 It's the same function, but it's no longer clear 00:04:31.947 --> 00:04:32.780 how to integrate it. 00:04:32.780 --> 00:04:34.580 If you're faced with this one, you 00:04:34.580 --> 00:04:38.480 say, what am I supposed to do. 00:04:38.480 --> 00:04:41.790 And we have to get around that difficulty. 00:04:41.790 --> 00:04:43.940 And so what we're going to do is we're 00:04:43.940 --> 00:04:46.660 going to unwind this disguise. 00:04:46.660 --> 00:04:55.830 So we have the algebra problem that we have. 00:04:55.830 --> 00:04:57.950 Oh, wow. 00:04:57.950 --> 00:04:59.510 There must be something in the water. 00:04:59.510 --> 00:05:06.280 Impressive. 00:05:06.280 --> 00:05:06.920 Wow. 00:05:06.920 --> 00:05:08.350 OK, let's see. 00:05:08.350 --> 00:05:14.660 Is 2/3 = 3/2? 00:05:14.660 --> 00:05:17.960 Holy cow. 00:05:17.960 --> 00:05:18.760 Well that's good. 00:05:18.760 --> 00:05:21.340 Well, I'll keep you awake today with several other 00:05:21.340 --> 00:05:23.940 transpositions here. 00:05:23.940 --> 00:05:30.760 So our algebra problem is to detect the easy pieces which 00:05:30.760 --> 00:05:36.290 are inside. 00:05:36.290 --> 00:05:38.490 And the method that we're going to use, 00:05:38.490 --> 00:05:40.330 the one that we'll emphasize anyway, 00:05:40.330 --> 00:05:45.110 is one algebraic trick which is a shortcut, which 00:05:45.110 --> 00:05:49.320 is called the cover-up method. 00:05:49.320 --> 00:05:51.900 But we're going to talk about even more general things 00:05:51.900 --> 00:05:52.510 than that. 00:05:52.510 --> 00:05:54.300 But anyway, this is where we're headed. 00:05:54.300 --> 00:05:57.910 Is something called the cover-up method. 00:05:57.910 --> 00:05:59.490 Alright. 00:05:59.490 --> 00:06:02.640 So that's our intro. 00:06:02.640 --> 00:06:05.765 And I'll just have to remember that 2 is not 3. 00:06:05.765 --> 00:06:08.130 I'll keep on repeating that. 00:06:08.130 --> 00:06:10.280 So now here I'm going to describe to you 00:06:10.280 --> 00:06:13.880 how we unwind this disguise. 00:06:13.880 --> 00:06:17.660 The first step is, we write down the function 00:06:17.660 --> 00:06:18.690 we want to integrate. 00:06:18.690 --> 00:06:21.890 Which was this. 00:06:21.890 --> 00:06:26.310 And now we have to undo the first damage that we did. 00:06:26.310 --> 00:06:31.930 So the first step is to factor the denominator. 00:06:31.930 --> 00:06:34.010 And that factors, we happen to know the factors, 00:06:34.010 --> 00:06:36.150 so I'm not going to carry this out. 00:06:36.150 --> 00:06:38.510 But this can be a rather difficult step. 00:06:38.510 --> 00:06:41.110 But we're going to assume that it's done. 00:06:41.110 --> 00:06:44.030 For the purposes of illustration here. 00:06:44.030 --> 00:06:46.780 So I factor the denominator. 00:06:46.780 --> 00:06:50.630 And now, the second thing that I'm going to do 00:06:50.630 --> 00:06:54.770 is what I'm going to call the setup here. 00:06:54.770 --> 00:06:56.520 How I'm going to set things up. 00:06:56.520 --> 00:06:59.340 And I'll tell you what these things are more 00:06:59.340 --> 00:07:01.010 systematically in a second. 00:07:01.010 --> 00:07:03.720 And the setup is that I want to somehow detect 00:07:03.720 --> 00:07:05.240 what I did before. 00:07:05.240 --> 00:07:11.880 And I'm going to write some unknowns here. 00:07:11.880 --> 00:07:15.650 What I expect is that this will break up into two pieces. 00:07:15.650 --> 00:07:17.590 One with the denominator x - 1, and 00:07:17.590 --> 00:07:23.470 the other with the denominator x + 2. 00:07:23.470 --> 00:07:32.860 So now, my third step is going to be to solve for A and B. 00:07:32.860 --> 00:07:34.500 And then I'm done, if I do that. 00:07:34.500 --> 00:07:41.370 That's the complete unwinding of this disguise. 00:07:41.370 --> 00:07:43.940 And this is where the cover-up method comes in handy. 00:07:43.940 --> 00:07:46.800 This is this method that I'm about to describe. 00:07:46.800 --> 00:07:49.370 Now, you can do the algebra in a clumsy way, 00:07:49.370 --> 00:07:50.970 or you can do it in a quick way. 00:07:50.970 --> 00:07:54.240 And we'd like to get efficient about the algebra involved. 00:07:54.240 --> 00:07:58.590 And so let me show you what the first step in the trick is. 00:07:58.590 --> 00:08:12.470 We're going to solve for A by multiplying by (x - 1). 00:08:12.470 --> 00:08:16.045 Now, notice if you multiply by (x - 1) in that equation 2, 00:08:16.045 --> 00:08:16.920 what you get is this. 00:08:16.920 --> 00:08:21.840 You got (4x - 2) divided by-- The (x-1)'s cancel. 00:08:21.840 --> 00:08:23.920 You get this on the left-hand side. 00:08:23.920 --> 00:08:26.480 And on the right-hand side you get A. 00:08:26.480 --> 00:08:29.620 The (x-1)'s cancel again. 00:08:29.620 --> 00:08:31.420 And then we get this extra term. 00:08:31.420 --> 00:08:35.200 Which is B x+2 times x-1. 00:08:35.200 --> 00:08:39.200 Now, the trick here, and we're going to get even better trick 00:08:39.200 --> 00:08:40.370 in just a second. 00:08:40.370 --> 00:08:42.870 The trick here is that I didn't try to clear 00:08:42.870 --> 00:08:44.560 the denominators completely. 00:08:44.560 --> 00:08:46.820 I was very efficient about the way I did it. 00:08:46.820 --> 00:08:51.390 It just cleared one factor. 00:08:51.390 --> 00:08:54.780 And the result here is very useful. 00:08:54.780 --> 00:09:03.480 Namely, if I plug in now x = 1, this term drops out too. 00:09:03.480 --> 00:09:10.560 So what I'm going to do now is I'm going to plug in x = 1. 00:09:10.560 --> 00:09:15.500 And what I get on the left-hand side here is 4 - 1 and 1 + 2, 00:09:15.500 --> 00:09:19.740 and on the left-hand side I get A. That's the end. 00:09:19.740 --> 00:09:27.150 This is my formula for A. A happens to be equal to 1. 00:09:27.150 --> 00:09:29.100 And that's, of course, what I expect. 00:09:29.100 --> 00:09:33.370 A had better be 1, because the thing broke up into 1/(x-1) + 00:09:33.370 --> 00:09:33.870 3/(x+2). 00:09:37.100 --> 00:09:39.694 So this is the correct answer. 00:09:39.694 --> 00:09:41.610 There was a question out here, which I missed. 00:09:41.610 --> 00:09:46.110 STUDENT: Aren't polynomials defined 00:09:46.110 --> 00:09:56.210 as functions with whole powers, or could they be square roots? 00:09:56.210 --> 00:09:57.650 PROFESSOR: Are polynomials defined 00:09:57.650 --> 00:10:00.330 as functions with whole powers, or can they be square roots? 00:10:00.330 --> 00:10:01.350 That's the question. 00:10:01.350 --> 00:10:04.140 The answer is, they only have whole powers. 00:10:04.140 --> 00:10:06.950 So for instance here I only have the power 1 and 0. 00:10:06.950 --> 00:10:10.410 Here I have the powers 2, 1 and 0 in the denominator. 00:10:10.410 --> 00:10:16.890 Square roots are no good for this method. 00:10:16.890 --> 00:10:17.670 Another question. 00:10:17.670 --> 00:10:18.620 STUDENT: [INAUDIBLE] 00:10:18.620 --> 00:10:22.260 PROFESSOR: Why did I say x = 1? 00:10:22.260 --> 00:10:26.650 The reason why I said x = 1 was that it works really fast. 00:10:26.650 --> 00:10:30.400 You can't know this in advance, that's part of the method. 00:10:30.400 --> 00:10:32.810 It just turns out to be the best thing to do. 00:10:32.810 --> 00:10:36.130 The fastest way of getting at the coefficient A. Now 00:10:36.130 --> 00:10:38.510 the curious thing, let me just pause for a second 00:10:38.510 --> 00:10:39.320 before I do it. 00:10:39.320 --> 00:10:43.600 If I had plugged x = 1 into the original equation, 00:10:43.600 --> 00:10:45.070 I would have gotten nonsense. 00:10:45.070 --> 00:10:48.210 Because I would've gotten 0 in the denominator. 00:10:48.210 --> 00:10:50.940 And that seems like the most horrible thing to do. 00:10:50.940 --> 00:10:54.280 The worst possible thing to do, is to set x = 1. 00:10:54.280 --> 00:10:56.490 On the other hand, what we did is a trick. 00:10:56.490 --> 00:10:58.870 We multiplied by x - 1. 00:10:58.870 --> 00:11:01.100 And that turned the equation into this. 00:11:01.100 --> 00:11:05.810 So now, in disguise, I multiplied by 0. 00:11:05.810 --> 00:11:07.890 But that turns out to be legitimate. 00:11:07.890 --> 00:11:11.770 Because really this equation is true for all x except 1. 00:11:11.770 --> 00:11:13.480 And then instead of taking x = 1, 00:11:13.480 --> 00:11:15.580 I can really take x tends to 1. 00:11:15.580 --> 00:11:17.200 That's really what I need. 00:11:17.200 --> 00:11:18.530 The limit as x goes to one. 00:11:18.530 --> 00:11:20.610 The equation is still valid then. 00:11:20.610 --> 00:11:22.760 So I'm using the worst case, the case that 00:11:22.760 --> 00:11:24.360 looks like it's dividing by 0. 00:11:24.360 --> 00:11:26.760 And it's helping me because it's cancelling out all 00:11:26.760 --> 00:11:31.730 the information in terms of B. So the advantage here is this 00:11:31.730 --> 00:11:36.090 cancellation that occurs in this part. 00:11:36.090 --> 00:11:37.440 So that's the method. 00:11:37.440 --> 00:11:39.930 We're going to shorten it much, much more in a second. 00:11:39.930 --> 00:11:44.640 But let me carry it out for the other coefficient as well. 00:11:44.640 --> 00:11:51.550 So the other coefficient I'm going to solve for B, 00:11:51.550 --> 00:11:57.490 I'm going to multiply by x + 2. 00:11:57.490 --> 00:12:01.260 And when I do that, I get (4x-1)/(x-1), 00:12:01.260 --> 00:12:05.020 that's the left-hand side, the very top expression there. 00:12:05.020 --> 00:12:07.890 And then down below I get A/(x-1) ( (x+2). 00:12:10.700 --> 00:12:14.380 And then again the (x+2)'s cancel. 00:12:14.380 --> 00:12:15.960 So I get B sitting alone. 00:12:15.960 --> 00:12:18.150 And now I'm going to do the same trick. 00:12:18.150 --> 00:12:21.540 I'm going to set x = -2. 00:12:21.540 --> 00:12:28.530 That's the value which is going to knock out this A term here. 00:12:28.530 --> 00:12:30.770 So that cancels this term completely. 00:12:30.770 --> 00:12:37.800 And what we get here all told is - 8 - (-8 - 1)/(-2 - 1) = B. 00:12:37.800 --> 00:12:43.600 In other words, B = 3, which was also what it was supposed 00:12:43.600 --> 00:12:44.100 to be. 00:12:44.100 --> 00:12:48.460 B was this number 3, right here. 00:12:48.460 --> 00:12:50.820 Which I'm now not going to change to 2. 00:12:50.820 --> 00:12:52.496 Because I know that it's not 2. 00:12:52.496 --> 00:12:53.370 There was a question. 00:12:53.370 --> 00:12:59.540 STUDENT: [INAUDIBLE] 00:12:59.540 --> 00:13:01.420 PROFESSOR: All right. 00:13:01.420 --> 00:13:05.770 Now, this is the method which is called cover-up. 00:13:05.770 --> 00:13:09.490 But it's really carried out much, much faster than this. 00:13:09.490 --> 00:13:11.250 So I'm going to review the method 00:13:11.250 --> 00:13:14.300 and I'm going to show you what it is in general. 00:13:14.300 --> 00:13:23.030 So the first step is to factor the denominator, Q. That's 00:13:23.030 --> 00:13:24.590 what I labeled 1 over there. 00:13:24.590 --> 00:13:29.350 That was the factorization of the denominator up top. 00:13:29.350 --> 00:13:36.260 The second step is what I'm going to call the setup. 00:13:36.260 --> 00:13:37.250 That's step 2. 00:13:37.250 --> 00:13:40.915 And that's where I knew what I was aiming for in advance. 00:13:40.915 --> 00:13:42.540 And I'm going to have to explain to you 00:13:42.540 --> 00:13:45.780 in every instance exactly what this setup should be. 00:13:45.780 --> 00:13:49.070 That is, what the unknowns should be and what target, 00:13:49.070 --> 00:13:53.080 simplified expression, we're aiming for. 00:13:53.080 --> 00:13:54.950 So that's the setup. 00:13:54.950 --> 00:14:01.010 And then the third step is what I'll now call cover-up. 00:14:01.010 --> 00:14:03.660 Which is just a very fast way of doing 00:14:03.660 --> 00:14:05.720 what I did on this last board, which is solving 00:14:05.720 --> 00:14:09.470 for the unknown coefficients. 00:14:09.470 --> 00:14:12.890 So now, let me perform it for you again. 00:14:12.890 --> 00:14:14.170 Over here. 00:14:14.170 --> 00:14:18.990 So it's 4x - 1 divided by-- So this is to eliminate writing 00:14:18.990 --> 00:14:19.650 here. 00:14:19.650 --> 00:14:20.190 Handwriting. 00:14:20.190 --> 00:14:24.130 It makes it much faster. 00:14:24.130 --> 00:14:28.490 So this part just factoring the denominator, 00:14:28.490 --> 00:14:31.290 that was 1, that was step 1. 00:14:31.290 --> 00:14:34.050 And then step 2, again, is the setup, which 00:14:34.050 --> 00:14:39.700 is setting it up like this. 00:14:39.700 --> 00:14:42.580 Alright, that's the setup. 00:14:42.580 --> 00:14:46.150 And now I claim that without writing very much, 00:14:46.150 --> 00:14:49.350 I can figure out what A and B are. 00:14:49.350 --> 00:14:51.350 Just by staring at this. 00:14:51.350 --> 00:14:53.670 So now what I'm going to do is I'm just going 00:14:53.670 --> 00:14:55.720 to think what I did over there. 00:14:55.720 --> 00:14:57.570 And I'm just going to do it directly. 00:14:57.570 --> 00:15:02.680 So let me show you what the method consists of visually. 00:15:02.680 --> 00:15:08.950 I'm going to cover up, that is, knock out this factor, 00:15:08.950 --> 00:15:13.560 and focus on this number here. 00:15:13.560 --> 00:15:15.610 And I'm going to plug in the thing that 00:15:15.610 --> 00:15:17.580 makes this 0, which is x = 1. 00:15:17.580 --> 00:15:20.180 So I'm plugging in x = 1. 00:15:20.180 --> 00:15:21.670 To this left-hand side. 00:15:21.670 --> 00:15:28.195 And what I get is (4 - 1)/(1 + 2) = A. Now, 00:15:28.195 --> 00:15:29.820 that's the same thing I did over there. 00:15:29.820 --> 00:15:33.610 I just did it by skipping the intermediate algebra step, 00:15:33.610 --> 00:15:35.860 which is a lot of writing. 00:15:35.860 --> 00:15:38.580 So the cover-up method really amounts to the following thing. 00:15:38.580 --> 00:15:40.385 I'm thinking of multiplying this over here. 00:15:40.385 --> 00:15:43.240 It cancels this and it gets rid of everything else. 00:15:43.240 --> 00:15:45.960 And it just leaves me with A on the right-hand side. 00:15:45.960 --> 00:15:47.680 And I have to get rid of it on this side. 00:15:47.680 --> 00:15:50.030 So in other words, by eliminating this, 00:15:50.030 --> 00:15:53.090 I'm isolating a on the right-hand side. 00:15:53.090 --> 00:15:56.170 So the cover-up is that I'm covering this and getting 00:15:56.170 --> 00:15:57.670 A out of it. 00:15:57.670 --> 00:15:59.560 Similarly, I can do the same thing 00:15:59.560 --> 00:16:04.720 with B. It's focused on the value x = -2. 00:16:04.720 --> 00:16:07.010 And B is what I'm getting on the right-hand side. 00:16:07.010 --> 00:16:10.400 And then I have to cover up this. 00:16:10.400 --> 00:16:14.530 So if I cover up that, then what's left over, with x = -2, 00:16:14.530 --> 00:16:16.720 is again (-8 - 1)/(-2 - 1). 00:16:22.580 --> 00:16:26.970 So this is the way the method gets carried out in practice. 00:16:26.970 --> 00:16:32.070 Writing, essentially, the least you can. 00:16:32.070 --> 00:16:36.580 Now, when you get to several variables, 00:16:36.580 --> 00:16:42.170 it becomes just way more convenient to do this. 00:16:42.170 --> 00:16:45.660 So now, let me just review when cover-up works. 00:16:45.660 --> 00:16:55.980 So this cover-up method works if Q(x) 00:16:55.980 --> 00:17:06.060 has distinct linear factors. 00:17:06.060 --> 00:17:11.780 And, so you need two things here. 00:17:11.780 --> 00:17:13.840 It has to factor completely, the denominator 00:17:13.840 --> 00:17:15.180 has to factor completely. 00:17:15.180 --> 00:17:20.250 And the degree of the numerator has 00:17:20.250 --> 00:17:27.750 to be strictly less than the degree of the denominator. 00:17:27.750 --> 00:17:30.080 I'm going to give you an example here. 00:17:30.080 --> 00:17:36.010 So, for instance-- And this tells you the general pattern 00:17:36.010 --> 00:17:38.070 of the setup also. 00:17:38.070 --> 00:17:44.090 Say you had x^2 + 3x + 8, let's say. 00:17:44.090 --> 00:17:46.020 Over (x - 1) (x - 2) (x + 5). 00:17:50.380 --> 00:17:52.920 So here I'm going to tell you the setup. 00:17:52.920 --> 00:17:58.710 The setup is going to be - A/(x-1) + B/(x-2) + C/(x+5). 00:18:02.820 --> 00:18:04.620 And it will always break up into something. 00:18:04.620 --> 00:18:06.860 So however many factors you have, 00:18:06.860 --> 00:18:09.770 you'll have to put in a term for each of those. 00:18:09.770 --> 00:18:13.150 And then you can find each number here 00:18:13.150 --> 00:18:26.060 by this cover-up method. 00:18:26.060 --> 00:18:29.450 Now we're done with that. 00:18:29.450 --> 00:18:33.280 And now we have to go on to the algebraic complications. 00:18:33.280 --> 00:18:38.420 So would the first typical algebraic complication be. 00:18:38.420 --> 00:18:50.760 It would be repeated roots or repeated factors. 00:18:50.760 --> 00:18:56.850 Let me get one that doesn't come out to be extremely ugly here. 00:18:56.850 --> 00:19:02.930 So this is what we'll call Example 2. 00:19:02.930 --> 00:19:05.620 And this is going to work when the degree, 00:19:05.620 --> 00:19:07.650 you always need that the degree of the numerator 00:19:07.650 --> 00:19:12.060 is less than the degree of the denominator. 00:19:12.060 --> 00:19:22.980 And Q has now repeated linear factors. 00:19:22.980 --> 00:19:26.030 So let's see which example I wanted to show you. 00:19:26.030 --> 00:19:28.830 So let's just give this here. 00:19:28.830 --> 00:19:34.080 I'll just repeat the denominator. 00:19:34.080 --> 00:19:39.580 With an extra factor like this. 00:19:39.580 --> 00:19:41.440 Now, the main thing you need to know, 00:19:41.440 --> 00:19:44.370 since I've already performed the factorization for you. 00:19:44.370 --> 00:19:46.140 Already performed Step 1. 00:19:46.140 --> 00:19:49.177 This is Step 1 here. 00:19:49.177 --> 00:19:50.760 You have to factor things all the way, 00:19:50.760 --> 00:19:53.910 and that's already been done for you. 00:19:53.910 --> 00:19:56.840 And here's what this setup is. 00:19:56.840 --> 00:20:05.540 The setup is that it's of the form A/(x-1) plus B/(x-1)^2 - 00:20:05.540 --> 00:20:11.550 we need another term for the square here - plus C/(x+2). 00:20:15.590 --> 00:20:17.210 In general, if you have more powers 00:20:17.210 --> 00:20:19.310 you just need to keep on putting in those powers. 00:20:19.310 --> 00:20:24.140 You need one for each of the powers. 00:20:24.140 --> 00:20:26.340 Why does it have to be squared? 00:20:26.340 --> 00:20:27.570 OK. 00:20:27.570 --> 00:20:28.680 Good question. 00:20:28.680 --> 00:20:31.660 So why in the world am I doing this? 00:20:31.660 --> 00:20:37.070 Let me just give you one hint as to why I'm doing this. 00:20:37.070 --> 00:20:41.600 It's very, very much like the decimal expansion of a number 00:20:41.600 --> 00:20:44.820 or, say, the base 2 expansion of a number. 00:20:44.820 --> 00:21:00.410 So, for, example the number 7/16 is 0/2 + 1/2^2 + 1/2^3 plus, 00:21:00.410 --> 00:21:02.120 is that right? 00:21:02.120 --> 00:21:07.300 So it's 4/16 + 1/2^4. 00:21:07.300 --> 00:21:09.090 It's this sort of thing. 00:21:09.090 --> 00:21:11.880 And I'm getting this power and this power. 00:21:11.880 --> 00:21:13.455 If I have higher powers, I'm going 00:21:13.455 --> 00:21:14.705 to have to have more and more. 00:21:14.705 --> 00:21:17.310 So this is what happens when I have a 2^4. 00:21:17.310 --> 00:21:20.550 I have to represent things like this. 00:21:20.550 --> 00:21:24.430 That's what's coming out of this piece with the repetitious 00:21:24.430 --> 00:21:24.930 here. 00:21:24.930 --> 00:21:26.390 Of the powers. 00:21:26.390 --> 00:21:31.070 This is just an analogy. 00:21:31.070 --> 00:21:31.906 Of what we're doing. 00:21:31.906 --> 00:21:33.280 Yeah, another question over here. 00:21:33.280 --> 00:21:34.455 STUDENT: [INAUDIBLE] 00:21:34.455 --> 00:21:35.080 PROFESSOR: Yes. 00:21:35.080 --> 00:21:36.621 So this is an example, but it's meant 00:21:36.621 --> 00:21:38.860 to represent the general case and I will also 00:21:38.860 --> 00:21:41.210 give you a general picture. 00:21:41.210 --> 00:21:43.442 For sure, once you have the second power here, 00:21:43.442 --> 00:21:45.400 you'll need both the first and the second power 00:21:45.400 --> 00:21:46.504 mentioned over here. 00:21:46.504 --> 00:21:48.420 And since there's only a first power over here 00:21:48.420 --> 00:21:52.060 I only have to mention a first power over there. 00:21:52.060 --> 00:21:55.790 If this were a 3 here, there would be one more term which 00:21:55.790 --> 00:22:00.630 would be the one for (x-1)^3 in the denominator. 00:22:00.630 --> 00:22:03.940 That's what you just said. 00:22:03.940 --> 00:22:08.330 OK, now, what's different about this setup 00:22:08.330 --> 00:22:11.670 is that the cover-up method, although it works, 00:22:11.670 --> 00:22:13.120 it doesn't work so well. 00:22:13.120 --> 00:22:14.660 It doesn't work quite as well. 00:22:14.660 --> 00:22:29.790 The cover-up works for the coefficients B and C, 00:22:29.790 --> 00:22:38.370 not A. We'll have a quick method for the numbers B and C. 00:22:38.370 --> 00:22:40.530 To figure out what they are. 00:22:40.530 --> 00:22:42.630 But it will be a little slower to get 00:22:42.630 --> 00:22:47.910 to A, which we will do last. 00:22:47.910 --> 00:22:56.780 Let me show you how it works. 00:22:56.780 --> 00:23:01.610 First of all, I'm going to do the ordinary cover-up with C. 00:23:01.610 --> 00:23:05.610 So for C, I just want to do the same old thing 00:23:05.610 --> 00:23:06.360 that I did before. 00:23:06.360 --> 00:23:09.370 I cover up this, and that's going 00:23:09.370 --> 00:23:12.300 to get rid of all the junk except for the C term. 00:23:12.300 --> 00:23:16.800 So I have to plug in x = -2. 00:23:16.800 --> 00:23:21.810 And I get x-- sorry, I get (-2)^2 + 2 in the numerator. 00:23:21.810 --> 00:23:25.660 And I get (-2 - 1)^2 in the denominator. 00:23:25.660 --> 00:23:28.690 Remember I'm covering this up. 00:23:28.690 --> 00:23:30.790 So that's all there is on the left-hand side. 00:23:30.790 --> 00:23:37.780 And on the right-hand side all there is C. 00:23:37.780 --> 00:23:39.980 Everything else got killed off, because it was x - 00:23:39.980 --> 00:23:40.580 2 times that. 00:23:40.580 --> 00:23:42.570 That's 0 times all that other stuff. 00:23:42.570 --> 00:23:45.240 And the x - 2 over here canceled. 00:23:45.240 --> 00:23:47.330 This is the shortcut that I just described, 00:23:47.330 --> 00:23:50.920 and this is much faster than doing all that arithmetic. 00:23:50.920 --> 00:23:52.620 And algebra. 00:23:52.620 --> 00:23:57.160 So all together this is a 6/9, right? 00:23:57.160 --> 00:24:09.140 So it's C = 6/9, which is 2/3. 00:24:09.140 --> 00:24:12.640 Now, the other one which is easy to do, 00:24:12.640 --> 00:24:14.730 I'm going to do by the slow method first. 00:24:14.730 --> 00:24:17.350 But you omit a term. 00:24:17.350 --> 00:24:23.380 The idea is to cover up the other bad factor. 00:24:23.380 --> 00:24:27.277 Cover-ups, I'll do it both the fast way and the slow way. 00:24:27.277 --> 00:24:29.110 I'll do it the fast way first, and then I'll 00:24:29.110 --> 00:24:30.440 show you the slow way. 00:24:30.440 --> 00:24:32.467 The fast way is to cover this up. 00:24:32.467 --> 00:24:34.300 And then I have to cover up everything else. 00:24:34.300 --> 00:24:35.740 That gets eliminated. 00:24:35.740 --> 00:24:40.950 And that includes everything but B. So I get B on this side. 00:24:40.950 --> 00:24:42.970 And I get 1 on that side. 00:24:42.970 --> 00:24:45.790 So that's (1^2 + 2) / (1 + 2). 00:24:48.340 --> 00:24:56.250 So in other words, B = 1. 00:24:56.250 --> 00:24:58.710 That was pretty fast, so let me show you what arithmetic 00:24:58.710 --> 00:25:00.060 was hiding behind that. 00:25:00.060 --> 00:25:01.690 What algebra was hiding behind it. 00:25:01.690 --> 00:25:06.900 What I was really doing is this. 00:25:06.900 --> 00:25:16.390 And multiplying through by (x-1)^2, so I got this. 00:25:16.390 --> 00:25:19.910 So this canceled here, so this C just stands alone. 00:25:19.910 --> 00:25:21.790 And then I have here C/(x+2) (x-1)^2. 00:25:26.290 --> 00:25:30.000 Notice again, I cleared out that one-- 00:25:30.000 --> 00:25:32.040 this term from the denominator and sent it over 00:25:32.040 --> 00:25:33.600 to the other side. 00:25:33.600 --> 00:25:38.790 Now, what's happening is that when I set x = 1 00:25:38.790 --> 00:25:43.050 here, this term is dying. 00:25:43.050 --> 00:25:44.730 This term is going away, because there's 00:25:44.730 --> 00:25:47.840 more powers in the numerator than in the denominator. 00:25:47.840 --> 00:25:50.070 This is still 0. 00:25:50.070 --> 00:25:52.330 And this one is gone also. 00:25:52.330 --> 00:25:58.220 So all that's left is B. Now, I cannot pull that off with 00:25:58.220 --> 00:25:59.820 a single power of x - 1. 00:25:59.820 --> 00:26:02.250 I can't expose the A term. 00:26:02.250 --> 00:26:03.962 It's the B term that I can expose. 00:26:03.962 --> 00:26:06.170 Because I can multiply through by this thing squared. 00:26:06.170 --> 00:26:09.310 If I multiply through by just x - 1, 00:26:09.310 --> 00:26:12.520 what'll happen here is I won't have canceled this (x-1)^2. 00:26:12.520 --> 00:26:13.160 It's useless. 00:26:13.160 --> 00:26:15.020 I still have a 0 in the denominator. 00:26:15.020 --> 00:26:17.150 I'll have B / 0 when I plug in x = 1. 00:26:17.150 --> 00:26:22.060 Which I can't use. 00:26:22.060 --> 00:26:32.950 Again, the cover-up method is giving us B and C, not A. 00:26:32.950 --> 00:26:36.510 Now, for the last term, for A, I'm 00:26:36.510 --> 00:26:40.380 going to just have to be straightforward about it. 00:26:40.380 --> 00:26:51.260 And so I'll just suggest for A, plug in your favorite number. 00:26:51.260 --> 00:26:59.510 So plug in my favorite number. 00:26:59.510 --> 00:27:01.430 Which is x = 0. 00:27:01.430 --> 00:27:04.617 And you won't be able to plug in x = 0 00:27:04.617 --> 00:27:05.700 if you've already used it. 00:27:05.700 --> 00:27:09.770 Here the two numbers we've already used are x = 1 00:27:09.770 --> 00:27:13.820 and x = -2. 00:27:13.820 --> 00:27:18.300 But we haven't used x = 0 yet, so that's good. 00:27:18.300 --> 00:27:21.400 I'm going to plug in now x = 0 into the equation. 00:27:21.400 --> 00:27:22.450 What do I get? 00:27:22.450 --> 00:27:35.070 I get 0 (0^2 + 2) / ((-1)^2 * 2) is equal to, let's see. 00:27:35.070 --> 00:27:36.800 A is the thing that I don't know. 00:27:36.800 --> 00:27:48.190 So it's A A/(-1) + B/(x-1)^2, so B = 1, so that's 1 1/(-1)^2. 00:27:48.190 --> 00:27:51.760 And then C was 2/3. 00:27:51.760 --> 00:27:55.490 2/3 / (x+2). 00:27:55.490 --> 00:28:00.900 So that's 0 + 2. 00:28:00.900 --> 00:28:02.690 Don't give up at this point. 00:28:02.690 --> 00:28:03.900 This is a lot of algebra. 00:28:03.900 --> 00:28:06.290 You really have to plug in all these numbers. 00:28:06.290 --> 00:28:08.400 You make one arithmetic mistake and you're always 00:28:08.400 --> 00:28:09.649 going to get the wrong answer. 00:28:09.649 --> 00:28:16.790 This is very arithmetically intensive. 00:28:16.790 --> 00:28:18.980 However, it does simplify at this point. 00:28:18.980 --> 00:28:22.680 We have 2/2, that's 1. 00:28:22.680 --> 00:28:27.790 Is equal to -A + 1 + 1/3. 00:28:27.790 --> 00:28:29.050 So let's see. 00:28:29.050 --> 00:28:34.920 A on the other side, this becomes A = 1/3. 00:28:34.920 --> 00:28:35.520 And that's it. 00:28:35.520 --> 00:28:36.600 This is the end. 00:28:36.600 --> 00:28:40.020 We've we've simplified our function. 00:28:40.020 --> 00:28:48.280 And now it's easy to integrate. 00:28:48.280 --> 00:28:48.780 Question. 00:28:48.780 --> 00:28:49.488 Another question. 00:28:49.488 --> 00:29:01.927 STUDENT: [INAUDIBLE] 00:29:01.927 --> 00:29:04.510 PROFESSOR: So the question is, if x = 0 has already been used, 00:29:04.510 --> 00:29:05.590 what do I do? 00:29:05.590 --> 00:29:10.024 And the answer is, pick something else. 00:29:10.024 --> 00:29:11.440 And you said pick a random number. 00:29:11.440 --> 00:29:13.400 And that's right, except that if you really 00:29:13.400 --> 00:29:18.020 picked a random number it would be 4.12567843, 00:29:18.020 --> 00:29:20.000 which would be difficult. What you want to pick 00:29:20.000 --> 00:29:24.240 is the easiest possible number you can think of. 00:29:24.240 --> 00:29:24.740 Yeah. 00:29:24.740 --> 00:29:32.710 STUDENT: [INAUDIBLE] 00:29:32.710 --> 00:29:37.050 PROFESSOR: If you had, as in this sort of situation here, 00:29:37.050 --> 00:29:39.690 more powers. 00:29:39.690 --> 00:29:41.610 Wouldn't you have more variables. 00:29:41.610 --> 00:29:42.900 Very good question. 00:29:42.900 --> 00:29:44.490 That's absolutely right. 00:29:44.490 --> 00:29:47.860 This was a 3 by 3 system in disguise, 00:29:47.860 --> 00:29:50.080 for these three unknowns, A, B and C. 00:29:50.080 --> 00:29:52.770 What we started with in the previous problem was 00:29:52.770 --> 00:29:53.950 two variables. 00:29:53.950 --> 00:29:56.190 It's over here, the variables A and B. 00:29:56.190 --> 00:29:59.310 And as the degree of the denominator goes up, 00:29:59.310 --> 00:30:02.900 the number of variables goes up. 00:30:02.900 --> 00:30:05.040 It gets more and more and more complicated. 00:30:05.040 --> 00:30:06.820 More and more arithmetically intensive. 00:30:06.820 --> 00:30:09.077 STUDENT: [INAUDIBLE] 00:30:09.077 --> 00:30:09.910 PROFESSOR: Well, so. 00:30:09.910 --> 00:30:11.535 The question is, how would you solve it 00:30:11.535 --> 00:30:12.880 if you have two unknowns. 00:30:12.880 --> 00:30:16.610 That's exactly the point here. 00:30:16.610 --> 00:30:19.560 This is a system of simultaneous equations for unknowns. 00:30:19.560 --> 00:30:23.180 And we have little tricks for isolating single variables. 00:30:23.180 --> 00:30:25.880 Otherwise we're stuck with solving the whole system. 00:30:25.880 --> 00:30:28.560 And you'd have to solve the whole system by elimination, 00:30:28.560 --> 00:30:35.980 various other tricks. 00:30:35.980 --> 00:30:38.950 I'll say a little more about that later. 00:30:38.950 --> 00:30:47.610 Now, I have to get one step more complicated 00:30:47.610 --> 00:30:53.560 with my next example. 00:30:53.560 --> 00:31:02.860 My next example is going to have a quadratic factor. 00:31:02.860 --> 00:31:05.760 So still I'm sticking to the degree of the polynomial 00:31:05.760 --> 00:31:08.520 and the numerator is less than the degree of the polynomial 00:31:08.520 --> 00:31:09.660 in the denominator. 00:31:09.660 --> 00:31:21.700 And I'm going to take the case where Q has a quadratic factor. 00:31:21.700 --> 00:31:26.140 Let me just again illustrate this by example. 00:31:26.140 --> 00:31:30.470 I have here (x - 1) (x^2 + 1). 00:31:30.470 --> 00:31:34.630 I'll make it about as easy as they come. 00:31:34.630 --> 00:31:38.830 Now, the setup will be slightly different here. 00:31:38.830 --> 00:31:40.510 Here's the setup. 00:31:40.510 --> 00:31:42.570 It's already factored. 00:31:42.570 --> 00:31:44.360 I've already done as much as I can do. 00:31:44.360 --> 00:31:48.820 I can't factor this x^2 + 1 into linear factors unless you know 00:31:48.820 --> 00:31:49.940 about complex numbers. 00:31:49.940 --> 00:31:51.600 If you know about complex numbers 00:31:51.600 --> 00:31:53.170 this method becomes much easier. 00:31:53.170 --> 00:31:55.400 And it comes back to the cover-up method. 00:31:55.400 --> 00:31:58.370 Which is the way that the cover-up method was originally 00:31:58.370 --> 00:32:01.060 conceived by Heaviside. 00:32:01.060 --> 00:32:04.280 But you won't get to that until 18.03. 00:32:04.280 --> 00:32:05.610 So we'll wait. 00:32:05.610 --> 00:32:08.360 This, by the way, is a method which is used for integration. 00:32:08.360 --> 00:32:10.980 But it was invented to do something 00:32:10.980 --> 00:32:13.565 with Laplace transforms and inversion 00:32:13.565 --> 00:32:15.600 of certain kinds of differential equations. 00:32:15.600 --> 00:32:17.020 By Heaviside. 00:32:17.020 --> 00:32:21.530 And so it came much later than integration. 00:32:21.530 --> 00:32:26.360 But anyway, it's a very convenient method. 00:32:26.360 --> 00:32:30.210 So here's the setup with this one. 00:32:30.210 --> 00:32:34.440 Again, we want a term for this (x - 1) factor. 00:32:34.440 --> 00:32:37.880 And now we're going to also have a term with the denominator x^2 00:32:37.880 --> 00:32:39.120 + 1. 00:32:39.120 --> 00:32:40.760 But this is the difference. 00:32:40.760 --> 00:32:44.590 It's now going to be a first degree polynomial. 00:32:44.590 --> 00:32:53.040 One degree down from the quadratic here. 00:32:53.040 --> 00:32:55.620 So this is what I keep on calling the setup, 00:32:55.620 --> 00:32:57.250 this is number 2. 00:32:57.250 --> 00:32:59.870 You have to know that in advance based on the pattern 00:32:59.870 --> 00:33:03.160 that you see on the left-hand side. 00:33:03.160 --> 00:33:03.660 Yes. 00:33:03.660 --> 00:33:12.630 STUDENT: [INAUDIBLE] 00:33:12.630 --> 00:33:15.130 PROFESSOR: The question is, if the degree of the numerator-- 00:33:15.130 --> 00:33:18.290 So in this case, if this were cubed, 00:33:18.290 --> 00:33:20.510 and this is matching with the denominator, which 00:33:20.510 --> 00:33:23.590 is total of degree 3. 00:33:23.590 --> 00:33:26.910 The answer is that this does not work. 00:33:26.910 --> 00:33:29.150 STUDENT: [INAUDIBLE] 00:33:29.150 --> 00:33:30.995 PROFESSOR: It definitely doesn't work. 00:33:30.995 --> 00:33:33.370 And we're going to have to do something totally different 00:33:33.370 --> 00:33:34.050 to handle it. 00:33:34.050 --> 00:33:37.180 Which turns out, fortunately, to be much easier than this. 00:33:37.180 --> 00:33:41.850 But we'll deal with that at the end. 00:33:41.850 --> 00:33:43.227 Keep this in mind. 00:33:43.227 --> 00:33:45.310 This is an easy way to make a mistake if you start 00:33:45.310 --> 00:33:47.230 with a higher degree numerator. 00:33:47.230 --> 00:33:51.300 You'll never get the right answer. 00:33:51.300 --> 00:33:54.870 So now, so I have my setup now. 00:33:54.870 --> 00:33:56.840 And now what can I do? 00:33:56.840 --> 00:33:58.960 Well, I claim that I can still do cover-up 00:33:58.960 --> 00:34:05.790 for A. It's the same idea. 00:34:05.790 --> 00:34:07.680 I cover this guy up. 00:34:07.680 --> 00:34:10.460 And if I really multiply by it it would knock everything out 00:34:10.460 --> 00:34:14.550 but A. So I cover this up and I plug in x = 1. 00:34:14.550 --> 00:34:25.460 So I get here 1^2 / (1^2 +1) = A. In other words, A = 1/2. 00:34:25.460 --> 00:34:28.560 Again cover-up is pretty fast, as you can see. 00:34:28.560 --> 00:34:32.030 It's not too bad. 00:34:32.030 --> 00:34:41.040 Now, at this next stage, I want to find B and C. 00:34:41.040 --> 00:34:45.990 And the best idea is the slow way. 00:34:45.990 --> 00:34:48.120 Here, it's not too terrible. 00:34:48.120 --> 00:34:50.010 But it's just what we're going to do. 00:34:50.010 --> 00:34:54.270 Which is to clear the denominators completely. 00:34:54.270 --> 00:35:05.520 So for B and C, just clear the denominator. 00:35:05.520 --> 00:35:07.645 That means multiply through by that whole business. 00:35:07.645 --> 00:35:09.978 Now, when you do that on the left-hand side you're going 00:35:09.978 --> 00:35:10.870 to get x^2. 00:35:10.870 --> 00:35:12.940 Because I got rid of the whole denominator. 00:35:12.940 --> 00:35:16.850 On the right-hand side when I bring this up, the x - 00:35:16.850 --> 00:35:18.600 1 will cancel with this. 00:35:18.600 --> 00:35:23.160 So the A term will be A(x^2 + 1). 00:35:23.160 --> 00:35:29.230 And the Bx + C term will have a remaining factor of x - 1. 00:35:29.230 --> 00:35:33.300 Because the x^2 + 1 will cancel. 00:35:33.300 --> 00:35:38.670 Again, the arithmetic here is not too terrible. 00:35:38.670 --> 00:35:41.000 Now I'm going to do the following. 00:35:41.000 --> 00:35:46.550 I'm going to look at the x^2 term. 00:35:46.550 --> 00:35:49.580 On the left-hand side and the right-hand side. 00:35:49.580 --> 00:35:51.970 And that will give me one equation for B and C. 00:35:51.970 --> 00:35:54.490 And then I'm going to do the same thing with another term. 00:35:54.490 --> 00:35:57.350 The x^2 term on the left-hand side, the coefficient is 1. 00:35:57.350 --> 00:35:59.740 It's 1 times x^2. 00:35:59.740 --> 00:36:06.030 On the other side, it's A. Remember I actually have A. 00:36:06.030 --> 00:36:08.870 So I'm going to put it in, it's 1/2. 00:36:08.870 --> 00:36:11.250 So this is the A term. 00:36:11.250 --> 00:36:14.020 And so I get 1/2 x^2. 00:36:14.020 --> 00:36:18.640 And then the only other x^2 is when this Bx multiplies this x. 00:36:18.640 --> 00:36:24.440 So Bx * x is Bx^2, so this is-- the other coefficient on x^2 is 00:36:24.440 --> 00:36:36.060 B. And that forces B to be 1/2. 00:36:36.060 --> 00:36:41.370 And last of all, I'm going to do the x^0 term. 00:36:41.370 --> 00:36:44.530 Or, otherwise known as the constant term. 00:36:44.530 --> 00:36:48.650 And on the left-hand side, the constant term is 0. 00:36:48.650 --> 00:36:51.270 There is no constant term. 00:36:51.270 --> 00:36:57.310 On the right-hand side there's a constant term, 1/2 * 1. 00:36:57.310 --> 00:36:58.520 That's 1/2 here. 00:36:58.520 --> 00:37:00.760 And then there's another constant term, 00:37:00.760 --> 00:37:09.430 which is this constant times this -1 Which is -C. 00:37:09.430 --> 00:37:17.952 And so the conclusion here is that C = 1/2. 00:37:17.952 --> 00:37:18.660 Another question. 00:37:18.660 --> 00:37:19.160 Yeah. 00:37:19.160 --> 00:37:33.130 STUDENT: [INAUDIBLE] 00:37:33.130 --> 00:37:37.970 PROFESSOR: There's also an x^0 hidden in here. 00:37:37.970 --> 00:37:40.890 Sorry, an x^1, that's what you were asking about, sorry. 00:37:40.890 --> 00:37:42.400 There's also an x^1. 00:37:42.400 --> 00:37:46.050 The only reason why I didn't go to the x^1 is that it turns out 00:37:46.050 --> 00:37:49.080 with these two I didn't need it. 00:37:49.080 --> 00:37:51.150 The other thing is that by experience, I 00:37:51.150 --> 00:37:53.320 know that the extreme ends of the multiplication 00:37:53.320 --> 00:37:54.690 are the easiest ends. 00:37:54.690 --> 00:37:57.150 And the middle terms have tons of cross terms. 00:37:57.150 --> 00:37:59.750 And so I don't like the middle term as much because it always 00:37:59.750 --> 00:38:01.130 involves more arithmetic. 00:38:01.130 --> 00:38:05.530 So I stick to the lowest and the highest terms if I can. 00:38:05.530 --> 00:38:07.390 So that was really a sneaky thing. 00:38:07.390 --> 00:38:10.460 I did that without saying anything. 00:38:10.460 --> 00:38:10.960 Yes. 00:38:10.960 --> 00:38:14.095 STUDENT: [INAUDIBLE] 00:38:14.095 --> 00:38:15.470 PROFESSOR: Another good question. 00:38:15.470 --> 00:38:17.120 Could I just set x equals 0? 00:38:17.120 --> 00:38:17.910 Absolutely. 00:38:17.910 --> 00:38:22.370 In fact, that's equivalent to picking out the x^0 term. 00:38:22.370 --> 00:38:24.150 And you could plug in numbers. 00:38:24.150 --> 00:38:25.140 If you wanted. 00:38:25.140 --> 00:38:27.820 That's another way of doing this besides doing that. 00:38:27.820 --> 00:38:30.930 So you can also plug in numbers. 00:38:30.930 --> 00:38:38.060 Can plug in numbers. x = 0. 00:38:38.060 --> 00:38:42.210 Actually, not x = 1, right? 00:38:42.210 --> 00:38:44.292 -1, 2, etc. 00:38:44.292 --> 00:38:46.000 Not 1 just because we've already used it. 00:38:46.000 --> 00:38:48.270 We won't get interesting information out. 00:38:48.270 --> 00:38:48.770 Yes. 00:38:48.770 --> 00:38:56.347 STUDENT: [INAUDIBLE] 00:38:56.347 --> 00:38:57.930 PROFESSOR: So the question is, could I 00:38:57.930 --> 00:38:59.900 have done it this other way. 00:38:59.900 --> 00:39:02.880 With the polynomial, with this other one. 00:39:02.880 --> 00:39:04.020 Yes, absolutely. 00:39:04.020 --> 00:39:05.436 So in other words what I've taught 00:39:05.436 --> 00:39:09.466 you now is two choices which are equally reasonable. 00:39:09.466 --> 00:39:10.840 The one that I picked was the one 00:39:10.840 --> 00:39:14.105 that was the fastest for this guy and the one 00:39:14.105 --> 00:39:15.980 that was fastest for this one, but I could've 00:39:15.980 --> 00:39:19.040 done the other way around. 00:39:19.040 --> 00:39:22.490 There are a lot of ways of solving simultaneous equations. 00:39:22.490 --> 00:39:23.777 Yeah, another question. 00:39:23.777 --> 00:39:24.610 STUDENT: [INAUDIBLE] 00:39:24.610 --> 00:39:26.318 PROFESSOR: The question is the following. 00:39:26.318 --> 00:39:28.610 So now everybody can understand the question. 00:39:28.610 --> 00:39:33.640 If this, instead of being x^2 + 1, this were x^3 + 1. 00:39:33.640 --> 00:39:36.360 So that's an important case to understand. 00:39:36.360 --> 00:39:39.420 That's a case in which this denominator is not 00:39:39.420 --> 00:39:41.020 fully factored. 00:39:41.020 --> 00:39:46.830 So it's an x^3 + 1, you would have to factor out an x + 1. 00:39:46.830 --> 00:39:49.630 So that would be a situation like this, you have an x^3 + 1, 00:39:49.630 --> 00:39:57.740 but that's (x+1) (x^2 +x + 1), this kind of thing. 00:39:57.740 --> 00:40:01.830 If that's the right, there must be a minus sign in here maybe. 00:40:01.830 --> 00:40:03.240 OK, something like this. 00:40:03.240 --> 00:40:07.274 Right? 00:40:07.274 --> 00:40:08.190 Isn't that what it is? 00:40:08.190 --> 00:40:11.561 STUDENT: [INAUDIBLE] 00:40:11.561 --> 00:40:12.810 PROFESSOR: I think it's right. 00:40:12.810 --> 00:40:15.930 But anyway, the point is that you have to factor it. 00:40:15.930 --> 00:40:17.920 And then you have a linear and a quadratic. 00:40:17.920 --> 00:40:20.880 So you're always going to be faced eventually 00:40:20.880 --> 00:40:23.484 with linear factors and quadratic factors. 00:40:23.484 --> 00:40:25.650 If you have a cubic, that means you haven't factored 00:40:25.650 --> 00:40:28.620 sufficiently. 00:40:28.620 --> 00:40:31.947 So you're still back in Step 1. 00:40:31.947 --> 00:40:32.780 STUDENT: [INAUDIBLE] 00:40:32.780 --> 00:40:34.500 PROFESSOR: In the x^3 + 1 case? 00:40:34.500 --> 00:40:36.610 STUDENT: [INAUDIBLE] 00:40:36.610 --> 00:40:39.610 PROFESSOR: In the x^3 + 1 case, we are out of luck until 00:40:39.610 --> 00:40:41.700 we've completed the factorization. 00:40:41.700 --> 00:40:43.470 Once we've completed the factorization, 00:40:43.470 --> 00:40:46.370 we're going to have to deal with these two factors 00:40:46.370 --> 00:40:48.920 as denominators. 00:40:48.920 --> 00:40:52.730 So it'll be this plus something over x + 1 00:40:52.730 --> 00:40:58.160 plus a Bx + C type of thing over this thing here. 00:40:58.160 --> 00:41:01.730 That's what's eventually going to happen. 00:41:01.730 --> 00:41:03.940 But hold on to that idea. 00:41:03.940 --> 00:41:14.340 Let me carry out one more example here. 00:41:14.340 --> 00:41:17.870 So I've figured out what all the values are. 00:41:17.870 --> 00:41:21.420 But I think it's also worth it to remember now 00:41:21.420 --> 00:41:29.140 that we also have to carry out the integration. 00:41:29.140 --> 00:41:38.220 What I've just shown you is that the integral of x^2 dx over (x 00:41:38.220 --> 00:41:43.180 - 1)(x^2 + 1) is equal to, and I've split up into these 00:41:43.180 --> 00:41:43.970 pieces. 00:41:43.970 --> 00:41:45.070 So what are the pieces? 00:41:45.070 --> 00:42:01.560 The pieces are, 1/2, x - 1, plus 1/2 x / (x^2 + 1). 00:42:01.560 --> 00:42:02.990 This is the A term. 00:42:02.990 --> 00:42:04.400 This is the B term. 00:42:04.400 --> 00:42:12.460 And then there's the C term. 00:42:12.460 --> 00:42:16.340 So we'd better remember that we know how to antidifferentiate 00:42:16.340 --> 00:42:18.300 these things. 00:42:18.300 --> 00:42:20.256 In other words, I want to finish the problem. 00:42:20.256 --> 00:42:21.630 The others were pretty easy, so I 00:42:21.630 --> 00:42:23.310 didn't bother to finish my sentence, 00:42:23.310 --> 00:42:25.570 but here I want to be careful and have 00:42:25.570 --> 00:42:28.850 you realize that there's something a little more to do. 00:42:28.850 --> 00:42:31.340 First of all we have the, the first one is no problem. 00:42:31.340 --> 00:42:35.240 That's this. 00:42:35.240 --> 00:42:39.910 The second one actually is not too bad either. 00:42:39.910 --> 00:42:45.340 This is, by the advanced guessing method, 00:42:45.340 --> 00:42:47.696 my favorite method, something like the logarithm, 00:42:47.696 --> 00:42:50.070 because that's what's going to appear in the denominator. 00:42:50.070 --> 00:42:51.770 And then, if you differentiate this, 00:42:51.770 --> 00:42:53.620 you're going to get 2x over this. 00:42:53.620 --> 00:42:55.030 But here we have 1/2. 00:42:55.030 --> 00:42:59.470 So altogether it's 1/4 of this. 00:42:59.470 --> 00:43:02.470 So I fixed the coefficient here. 00:43:02.470 --> 00:43:06.180 And then the last one, you have to think back 00:43:06.180 --> 00:43:09.060 to some level of memorization here and remember that this 00:43:09.060 --> 00:43:15.610 is 1/2 the arc tangent. 00:43:15.610 --> 00:43:20.870 STUDENT: [INAUDIBLE] 00:43:20.870 --> 00:43:22.200 PROFESSOR: Why did I go to 1/4? 00:43:22.200 --> 00:43:24.570 Because in disguise, for this guy, 00:43:24.570 --> 00:43:35.140 I was thinking d/dx of ln(x^2 + 1) is 2x / (x^2 + 1). 00:43:35.140 --> 00:43:39.880 Because it's the derivative of this divided by itself. 00:43:39.880 --> 00:43:43.300 This is-- The derivative of ln u is u' / u. 00:43:46.620 --> 00:43:50.030 (ln u)' = u' / u. 00:43:50.030 --> 00:43:54.350 That was what I applied. 00:43:54.350 --> 00:43:58.210 And what I had was 1/2, so I need a total of 1/4 to cancel. 00:43:58.210 --> 00:44:06.640 So 2/4 is 1/2. 00:44:06.640 --> 00:44:09.990 Now I've got to get you out of one more deep hole. 00:44:09.990 --> 00:44:12.900 And I'm going to save the general pattern for next time. 00:44:12.900 --> 00:44:27.860 But I do want to clarify one thing. 00:44:27.860 --> 00:44:30.040 So let's handle this thing. 00:44:30.040 --> 00:44:33.650 What if the degree of P is bigger than 00:44:33.650 --> 00:44:41.570 or equal to the degree of Q. That's the thing that I claimed 00:44:41.570 --> 00:44:42.810 was easier. 00:44:42.810 --> 00:44:45.880 And I'm going to describe to you how it's done. 00:44:45.880 --> 00:44:48.940 Now, in analogy, with numbers you 00:44:48.940 --> 00:44:55.070 might call this an improper fraction. 00:44:55.070 --> 00:44:59.850 That's the thing that should echo in your mind 00:44:59.850 --> 00:45:01.970 when you're thinking about this. 00:45:01.970 --> 00:45:04.600 And I'm just going to do it by example here. 00:45:04.600 --> 00:45:06.800 Let's see., I cooked up an example 00:45:06.800 --> 00:45:09.730 so that I don't make an arithmetic 00:45:09.730 --> 00:45:11.220 mistake along the way. 00:45:11.220 --> 00:45:16.990 So there are two or three steps that I need to explain. 00:45:16.990 --> 00:45:18.030 So here's an example. 00:45:18.030 --> 00:45:21.320 The denominator's degree 2, the numerator is degree 3. 00:45:21.320 --> 00:45:24.610 It well exceeds, so there's a problem here. 00:45:24.610 --> 00:45:27.880 Our method is not going to work. 00:45:27.880 --> 00:45:32.260 And the first step that I want to carry out 00:45:32.260 --> 00:45:36.370 is to reverse Step 1. 00:45:36.370 --> 00:45:38.371 That is, I don't want the factorization 00:45:38.371 --> 00:45:39.620 for what I'm going to do next. 00:45:39.620 --> 00:45:42.040 I want it multiplied out. 00:45:42.040 --> 00:45:48.670 That means I have to multiply through, so I get x^2 + x - 2. 00:45:48.670 --> 00:45:52.980 I'm back to the starting place here. 00:45:52.980 --> 00:45:56.510 And now, the next thing that I'm going to do 00:45:56.510 --> 00:46:01.130 is, I'm going to use long division. 00:46:01.130 --> 00:46:04.270 That's how you convert an improper fraction 00:46:04.270 --> 00:46:07.432 to a proper fraction. 00:46:07.432 --> 00:46:08.890 This is something you were supposed 00:46:08.890 --> 00:46:16.780 to learn in, I don't know, Grade 4, I know. 00:46:16.780 --> 00:46:23.970 Grade 3, Grade 4, Grade 5, Grade 6, etc. 00:46:23.970 --> 00:46:27.560 So here's how it works in the case of polynomials. 00:46:27.560 --> 00:46:31.030 It's kind of amusing. 00:46:31.030 --> 00:46:37.760 So we're dividing this polynomial into that one. 00:46:37.760 --> 00:46:41.950 And so the quotient to first order here is x. 00:46:41.950 --> 00:46:44.300 That is, that's going to match the top terms. 00:46:44.300 --> 00:46:48.500 So I get x^3 + x^2 - 2x. 00:46:48.500 --> 00:46:50.310 That's the product. 00:46:50.310 --> 00:46:51.890 And now I subtract. 00:46:51.890 --> 00:46:53.990 And it cancels. 00:46:53.990 --> 00:46:58.440 So we get here -x^2 + 2x. 00:46:58.440 --> 00:47:01.110 That's the difference. 00:47:01.110 --> 00:47:04.260 And now I need to divide this next term in. 00:47:04.260 --> 00:47:08.190 And I need a -1. 00:47:08.190 --> 00:47:14.700 So -1 times this is -x^2 - x + 2. 00:47:14.700 --> 00:47:16.260 And I subtract. 00:47:16.260 --> 00:47:17.800 And the x^2's cancel. 00:47:17.800 --> 00:47:24.210 And here I get +3x - 2. 00:47:24.210 --> 00:47:27.090 Now, this thing has a name. 00:47:27.090 --> 00:47:30.880 This is called the quotient. 00:47:30.880 --> 00:47:33.090 And this thing also has a name. 00:47:33.090 --> 00:47:39.070 This is called the remainder. 00:47:39.070 --> 00:47:43.530 And now I'll show you how it works by sticking it 00:47:43.530 --> 00:47:44.560 into the answer here. 00:47:44.560 --> 00:47:47.250 The quotient is x - 1. 00:47:47.250 --> 00:47:52.010 And the remainder is, let's get down there. 00:47:52.010 --> 00:47:53.810 (3x - 2) / (x^2 + x - 2). 00:47:58.820 --> 00:48:03.300 So the punchline here is that this thing 00:48:03.300 --> 00:48:05.450 is easy to integrate. 00:48:05.450 --> 00:48:08.190 This is easy. 00:48:08.190 --> 00:48:13.350 And this one, you can use, now you 00:48:13.350 --> 00:48:18.292 can use cover-up, The method that we had before. 00:48:18.292 --> 00:48:19.750 Because the degree of the numerator 00:48:19.750 --> 00:48:22.700 is now below the degree of the denominator. 00:48:22.700 --> 00:48:25.920 It's now first degree and this is second degree. 00:48:25.920 --> 00:48:27.870 What you can't do is use cover-up 00:48:27.870 --> 00:48:29.950 to start out with here. 00:48:29.950 --> 00:48:32.630 That will give you the wrong answer. 00:48:32.630 --> 00:48:36.952 So that's the end for today, and see you next time.