1 00:00:00,490 --> 00:00:04,230 In the first of the units on algebraic fractions, we looked 2 00:00:04,230 --> 00:00:07,970 at what happened when we had a proper fraction with linear 3 00:00:07,970 --> 00:00:11,030 factors in the denominator of proper fraction with repeated 4 00:00:11,030 --> 00:00:13,750 linear factors in the denominator, and what happened 5 00:00:13,750 --> 00:00:18,170 when we had improper fractions. what I want to do in this video 6 00:00:18,170 --> 00:00:21,910 is look at what happens when we get an irreducible quadratic 7 00:00:21,910 --> 00:00:24,630 factor when we get an irreducible quadratic factor 8 00:00:24,630 --> 00:00:28,710 will end up with an integral of something which looks like this 9 00:00:28,710 --> 00:00:29,730 X plus B. 10 00:00:30,870 --> 00:00:37,818 Over a X squared plus BX plus, see where the A&B are 11 00:00:37,818 --> 00:00:41,638 known constants. And this quadratic in the denominator 12 00:00:41,638 --> 00:00:44,422 cannot be factorized. Now there's various things that 13 00:00:44,422 --> 00:00:48,946 could happen. It's possible that a could turn out to be 0. Now, 14 00:00:48,946 --> 00:00:53,122 if it turns out to be 0, what would be left with? 15 00:00:53,790 --> 00:00:55,490 Is trying to integrate a 16 00:00:55,490 --> 00:00:57,526 constant. Over this quadratic 17 00:00:57,526 --> 00:01:04,446 factor. So we'll just end up with a B over AX squared plus BX 18 00:01:04,446 --> 00:01:09,041 plus C. Now the first example I'd like to show you is what 19 00:01:09,041 --> 00:01:12,293 we do when we get a situation where we've just got a 20 00:01:12,293 --> 00:01:15,003 constant on its own, no ex terms over the irreducible 21 00:01:15,003 --> 00:01:17,713 quadratic factor, so let's have a look at a specific 22 00:01:17,713 --> 00:01:17,984 example. 23 00:01:21,240 --> 00:01:24,544 Suppose we want to integrate a constant one. 24 00:01:25,590 --> 00:01:27,018 Over X squared. 25 00:01:28,220 --> 00:01:32,310 Plus X plus one. We want to integrate this with 26 00:01:32,310 --> 00:01:33,537 respect to X. 27 00:01:35,640 --> 00:01:38,619 This denominator will not factorize if it would factorize, 28 00:01:38,619 --> 00:01:40,605 would be back to expressing it 29 00:01:40,605 --> 00:01:44,840 in partial fractions. The way we proceed is to try to complete 30 00:01:44,840 --> 00:01:46,240 the square in the denominator. 31 00:01:47,020 --> 00:01:50,440 Let me remind you of how we complete the square for X 32 00:01:50,440 --> 00:01:51,865 squared plus X plus one. 33 00:01:52,850 --> 00:01:57,101 It's a complete the square we try to write the first 2 terms. 34 00:01:58,500 --> 00:02:00,339 As something squared. 35 00:02:01,260 --> 00:02:04,704 Well, what do we write in this bracket? We want an X 36 00:02:04,704 --> 00:02:07,861 and clearly when the brackets are all squared out, will get 37 00:02:07,861 --> 00:02:10,444 an X squared which is that term dealt with. 38 00:02:11,740 --> 00:02:15,510 To get an ex here, we need actually a term 1/2 here because 39 00:02:15,510 --> 00:02:19,280 you imagine when you square the brackets out you'll get a half X 40 00:02:19,280 --> 00:02:22,760 in another half X, which is the whole X which is that. 41 00:02:24,730 --> 00:02:27,293 We get something we don't want when these brackets are all 42 00:02:27,293 --> 00:02:28,691 squared out, we'll end it with 43 00:02:28,691 --> 00:02:32,681 1/2 squared. Which is 1/4 and we don't want a quarter, so I'm 44 00:02:32,681 --> 00:02:34,247 going to subtract it again here. 45 00:02:35,540 --> 00:02:38,988 So altogether, all those terms written down there 46 00:02:38,988 --> 00:02:42,867 are equivalent to the first 2 terms over here. 47 00:02:44,500 --> 00:02:47,580 And to make these equal, we still need the plus one. 48 00:02:50,590 --> 00:02:56,200 So tidying this up, we've actually got X plus 1/2 all 49 00:02:56,200 --> 00:03:02,320 squared, and one subtract 1/4 is 3/4. That is the process of 50 00:03:02,320 --> 00:03:03,850 completing the square. 51 00:03:12,040 --> 00:03:17,695 OK, how will that help us? Well, it means that what we want to do 52 00:03:17,695 --> 00:03:21,842 now is considered instead of the integral we started with. We 53 00:03:21,842 --> 00:03:26,366 want to consider this integral one over X plus 1/2 all squared. 54 00:03:29,320 --> 00:03:33,533 Plus 3/4 we want to integrate that with respect to X. 55 00:03:34,430 --> 00:03:37,634 Now, the way I'm going to proceed is going to make a 56 00:03:37,634 --> 00:03:40,838 substitution in. Here, I'm going to let you be X plus 1/2. 57 00:03:48,420 --> 00:03:51,876 When we do that, are integral will become the integral of one 58 00:03:51,876 --> 00:03:55,908 over X plus 1/2 will be just you, so will end up with you 59 00:03:55,908 --> 00:03:58,628 squared. We've got plus 3/4. 60 00:04:00,210 --> 00:04:02,106 We need to take care of the DX. 61 00:04:03,190 --> 00:04:07,450 Now remember that if we want the differential du, that's du DX 62 00:04:07,450 --> 00:04:12,775 DX. But in this case du DX is just one. This is just one. So 63 00:04:12,775 --> 00:04:18,100 do you is just DX. So this is nice and simple. The DX we have 64 00:04:18,100 --> 00:04:19,875 here just becomes a du. 65 00:04:22,490 --> 00:04:26,384 Now this integral is a standard form. There's a standard result 66 00:04:26,384 --> 00:04:30,632 which says that if you want to integrate one over a squared 67 00:04:30,632 --> 00:04:32,756 plus X squared with respect to 68 00:04:32,756 --> 00:04:38,416 X. That's equal to one over a inverse tangent that's 10 to the 69 00:04:38,416 --> 00:04:43,498 minus one of X over a plus a constant. Now we will use that 70 00:04:43,498 --> 00:04:47,491 result to write the answer down to this integral, because this 71 00:04:47,491 --> 00:04:49,669 is one of these where a. 72 00:04:50,750 --> 00:04:52,710 Is the square root of 3 over 2? 73 00:04:53,360 --> 00:04:55,080 That's a squared is 3/4. 74 00:04:57,320 --> 00:05:03,368 So A is the square root of 3 / 2, so we can write down the 75 00:05:03,368 --> 00:05:08,282 answer to this straight away and this will workout at one over a, 76 00:05:08,282 --> 00:05:12,818 which is one over root 3 over 210 to the minus one. 77 00:05:14,390 --> 00:05:18,342 Of X over A. In this case it will be U over a 78 00:05:18,342 --> 00:05:20,166 which is Route 3 over 2. 79 00:05:21,660 --> 00:05:22,428 Plus a constant. 80 00:05:24,940 --> 00:05:28,736 Just to tidy this up a little bit where dividing by a fraction 81 00:05:28,736 --> 00:05:32,240 here. So dividing by Route 3 over 2 is like multiplying by 82 00:05:32,240 --> 00:05:33,408 two over Route 3. 83 00:05:35,300 --> 00:05:37,030 We've attempted the minus one. 84 00:05:37,950 --> 00:05:41,430 You we can replace with X plus 1/2. 85 00:05:45,040 --> 00:05:49,360 And again, dividing by Route 3 over 2 is like multiplying by 86 00:05:49,360 --> 00:05:50,800 two over Route 3. 87 00:05:52,120 --> 00:05:53,576 And we have a constant at the end. 88 00:05:56,560 --> 00:05:59,730 And that's the answer. So In other words, to integrate. 89 00:06:01,000 --> 00:06:03,576 A constant over an irreducible quadratic factor. 90 00:06:03,576 --> 00:06:07,992 We can complete the square as we did here and then use 91 00:06:07,992 --> 00:06:10,936 integration by substitution to finish the problem off. 92 00:06:12,320 --> 00:06:15,807 So that's what happens when we get a constant over the 93 00:06:15,807 --> 00:06:16,441 quadratic factor. 94 00:06:20,970 --> 00:06:24,846 What else could happen? It may happen that we get a situation 95 00:06:24,846 --> 00:06:28,722 like this. We end up with a quadratic function at the bottom 96 00:06:28,722 --> 00:06:30,660 and it's derivative at the top. 97 00:06:33,570 --> 00:06:36,558 If that happens, it's very straightforward to finish the 98 00:06:36,558 --> 00:06:40,210 integration of because we know from a standard result that this 99 00:06:40,210 --> 00:06:43,530 evaluates to the logarithm of the modulus of the denominator 100 00:06:43,530 --> 00:06:47,182 plus a constant. So, for example, I'm thinking now of an 101 00:06:47,182 --> 00:06:48,510 example like this one. 102 00:06:52,960 --> 00:06:56,768 Again, irreducible quadratic factor in the denominator. 103 00:06:58,430 --> 00:07:01,436 Attorney X plus be constant times X plus another 104 00:07:01,436 --> 00:07:04,776 constant on the top and if you inspect this carefully, 105 00:07:04,776 --> 00:07:08,116 if you look at the bottom here and you differentiate 106 00:07:08,116 --> 00:07:10,120 it, you'll get 2X plus one. 107 00:07:11,320 --> 00:07:14,524 So we've got a situation where we've got a function at the 108 00:07:14,524 --> 00:07:17,728 bottom and it's derivative at the top so we can write this 109 00:07:17,728 --> 00:07:20,665 down straight away. The answer is going to be the natural 110 00:07:20,665 --> 00:07:23,068 logarithm of the modulus of what's at the bottom. 111 00:07:27,740 --> 00:07:31,160 Let's see and that's finished. That's nice and straightforward. 112 00:07:31,160 --> 00:07:35,340 If you get a situation where you've got something times X 113 00:07:35,340 --> 00:07:36,480 plus another constant. 114 00:07:37,070 --> 00:07:39,875 And this top line is not the derivative of the bottom 115 00:07:39,875 --> 00:07:43,445 line. Then you gotta do a bit more work on it as we'll see 116 00:07:43,445 --> 00:07:44,465 in the next example. 117 00:07:48,830 --> 00:07:53,010 Let's have a look at this example. Suppose we want to 118 00:07:53,010 --> 00:07:58,330 integrate X divided by X squared plus X Plus One, and we want to 119 00:07:58,330 --> 00:08:00,610 integrate it with respect to X. 120 00:08:01,940 --> 00:08:05,410 Still, if we differentiate, the bottom line will get 2X. 121 00:08:06,040 --> 00:08:08,999 Plus One, and that's not what we have at the top. 122 00:08:08,999 --> 00:08:12,227 However, what we can do is we can introduce it to at 123 00:08:12,227 --> 00:08:15,455 the top, so we have two X in this following way. By 124 00:08:15,455 --> 00:08:18,145 little trick we can put a two at the top. 125 00:08:23,160 --> 00:08:26,046 And in order to make this the same as the integral that we 126 00:08:26,046 --> 00:08:27,600 started with, I'm going to put a 127 00:08:27,600 --> 00:08:29,660 factor of 1/2 outside. Half and 128 00:08:29,660 --> 00:08:33,762 the two canceling. Will will leave the integral that we 129 00:08:33,762 --> 00:08:34,770 started with that. 130 00:08:36,010 --> 00:08:39,590 Now. If we differentiate the bottom you see, we get. 131 00:08:40,140 --> 00:08:44,700 2X. Which is what we've got at the top. But we also get a 132 00:08:44,700 --> 00:08:46,860 plus one from differentiating the extreme and we haven't 133 00:08:46,860 --> 00:08:49,500 got a plus one there, so we apply another little trick 134 00:08:49,500 --> 00:08:50,940 now, and we do the following. 135 00:08:54,460 --> 00:08:56,716 We'd like a plus one there. 136 00:09:00,700 --> 00:09:04,120 So that the derivative of the denominator occurs in 137 00:09:04,120 --> 00:09:04,880 the numerator. 138 00:09:05,960 --> 00:09:08,914 But this is no longer the same as that because I've added a one 139 00:09:08,914 --> 00:09:10,391 here. So I've got to take it 140 00:09:10,391 --> 00:09:13,440 away again. In order that were still with the same 141 00:09:13,440 --> 00:09:14,540 problem that we started with. 142 00:09:16,540 --> 00:09:21,258 Now what I can do is I can split this into two integrals. I've 143 00:09:21,258 --> 00:09:24,628 got a half the integral of these first 2 terms. 144 00:09:27,300 --> 00:09:29,946 Over X squared plus X plus one. 145 00:09:31,730 --> 00:09:34,580 DX and I've got a half. 146 00:09:35,820 --> 00:09:40,236 The integral of the second term, which is minus one over X 147 00:09:40,236 --> 00:09:45,020 squared plus X plus one DX so that little bit of trickery has 148 00:09:45,020 --> 00:09:49,068 allowed me to split the thing into two integrals. Now this 149 00:09:49,068 --> 00:09:52,748 first one we've already seen is straightforward to finish off, 150 00:09:52,748 --> 00:09:56,428 because the numerator now is the derivative of the denominator, 151 00:09:56,428 --> 00:09:59,004 so this is just a half the 152 00:09:59,004 --> 00:10:02,805 natural logarithm. Of the modulus of X squared plus 153 00:10:02,805 --> 00:10:03,960 X plus one. 154 00:10:06,620 --> 00:10:10,076 And then we've got minus 1/2. Take the minus sign out minus 155 00:10:10,076 --> 00:10:13,532 1/2, and this integral integral of one over X squared plus X 156 00:10:13,532 --> 00:10:17,276 Plus one is the one that we did right at the very beginning. 157 00:10:17,850 --> 00:10:22,335 And if we just look back, let's see the results of finding that 158 00:10:22,335 --> 00:10:26,475 integral. Was this one here, two over Route 3 inverse tan of 159 00:10:26,475 --> 00:10:28,890 twice X plus 1/2 over Route 3? 160 00:10:29,890 --> 00:10:34,579 So we've got two over Route 3 inverse tangent. 161 00:10:35,750 --> 00:10:37,450 Twice X plus 1/2. 162 00:10:38,540 --> 00:10:39,728 Over Route 3. 163 00:10:41,580 --> 00:10:43,010 Plus a constant of integration. 164 00:10:44,540 --> 00:10:49,340 I can just tidy this up so it's nice and neat to finish it off a 165 00:10:49,340 --> 00:10:52,340 half the logarithm of X squared plus X plus one. 166 00:10:54,020 --> 00:10:59,408 The Twos will counsel here, so I'm left with minus one over 167 00:10:59,408 --> 00:11:01,204 Route 3 inverse tangent. 168 00:11:01,920 --> 00:11:05,196 And it might be nice just to multiply these brackets out to 169 00:11:05,196 --> 00:11:08,472 finish it off, so I'll have two X and 2 * 1/2. 170 00:11:09,110 --> 00:11:09,820 Is one. 171 00:11:10,890 --> 00:11:12,178 All over Route 3. 172 00:11:13,800 --> 00:11:15,370 Plus a constant of integration. 173 00:11:17,670 --> 00:11:19,050 And that's the problem solved. 174 00:11:26,100 --> 00:11:30,280 Let's have a look at one final example where we can 175 00:11:30,280 --> 00:11:33,700 draw some of these threads together. Supposing we want 176 00:11:33,700 --> 00:11:37,500 to integrate 1 divided by XX squared plus one DX. 177 00:11:39,400 --> 00:11:42,700 What if we got? In this case, it's a proper fraction. 178 00:11:44,030 --> 00:11:45,482 And we've got a linear factor 179 00:11:45,482 --> 00:11:47,880 here. And a quadratic factor here. 180 00:11:48,900 --> 00:11:52,376 You can try, but you'll find that this quadratic factor will 181 00:11:52,376 --> 00:11:54,272 not factorize, so this is an 182 00:11:54,272 --> 00:11:57,754 irreducible quadratic factor. So what we're going to do is 183 00:11:57,754 --> 00:11:59,932 we're going to, first of all, express the integrand. 184 00:12:02,580 --> 00:12:06,050 As the sum of its partial fractions and the appropriate 185 00:12:06,050 --> 00:12:09,867 form of partial fractions are going to be a constant over 186 00:12:09,867 --> 00:12:10,908 the linear factor. 187 00:12:12,870 --> 00:12:15,586 And then we'll need BX plus C. 188 00:12:16,850 --> 00:12:20,378 Over the irreducible quadratic factor X squared plus one. 189 00:12:22,450 --> 00:12:28,255 We now have to find abian. See, we do that in the usual way by 190 00:12:28,255 --> 00:12:31,738 adding these together, the common denominator will be XX 191 00:12:31,738 --> 00:12:32,899 squared plus one. 192 00:12:35,200 --> 00:12:39,208 Will need to multiply top and bottom here by X squared plus 193 00:12:39,208 --> 00:12:42,882 one to achieve the correct denominator so we'll have an AX 194 00:12:42,882 --> 00:12:43,884 squared plus one. 195 00:12:45,530 --> 00:12:51,004 And we need to multiply top and bottom here by X to achieve that 196 00:12:51,004 --> 00:12:54,914 denominator. So we'll have VX plus C4 multiplied by X. 197 00:12:57,330 --> 00:12:59,160 This quantity is equal to that 198 00:12:59,160 --> 00:13:03,143 quantity. The denominators are already the same, so we can 199 00:13:03,143 --> 00:13:07,303 equate the numerators. If we just look at the numerators 200 00:13:07,303 --> 00:13:11,252 will have one is equal to a X squared plus one. 201 00:13:13,650 --> 00:13:15,518 Plus BX Plus C. 202 00:13:16,820 --> 00:13:18,410 Multiplied by X. 203 00:13:20,400 --> 00:13:23,777 What's a sensible value to substitute for X so we can 204 00:13:23,777 --> 00:13:27,154 find abian? See while a sensible value is clearly X is 205 00:13:27,154 --> 00:13:30,838 zero, whi is that sensible? Well, if X is zero, both of 206 00:13:30,838 --> 00:13:32,987 these terms at the end will disappear. 207 00:13:34,520 --> 00:13:38,304 So X being zero will have one is equal to A. 208 00:13:39,080 --> 00:13:41,078 0 squared is 0 + 1. 209 00:13:42,080 --> 00:13:45,032 Is still one, so we'll have one a. So a is one. 210 00:13:46,830 --> 00:13:48,090 That's our value for a. 211 00:13:49,410 --> 00:13:54,170 What can we do to find B and see what I'm going to do now is I'm 212 00:13:54,170 --> 00:13:56,690 going to equate some coefficients and let's start by 213 00:13:56,690 --> 00:13:59,490 looking at the coefficients of X squared on both side. 214 00:14:01,560 --> 00:14:04,730 On the left hand side there are no X squared's. 215 00:14:06,350 --> 00:14:09,626 What about on the right hand side? There's clearly 216 00:14:09,626 --> 00:14:10,718 AX squared here. 217 00:14:12,990 --> 00:14:15,339 And when we multiply the brackets out here, that 218 00:14:15,339 --> 00:14:16,383 would be X squared. 219 00:14:19,960 --> 00:14:24,202 There are no more X squares, so A plus B must be zero. That 220 00:14:24,202 --> 00:14:28,444 means that B must be the minus negative of a must be the minor 221 00:14:28,444 --> 00:14:32,080 say, but a is already one, so be must be minus one. 222 00:14:34,150 --> 00:14:38,518 We still need to find C and will do that by equating 223 00:14:38,518 --> 00:14:39,610 coefficients of X. 224 00:14:41,740 --> 00:14:43,484 There are no ex terms on the left. 225 00:14:45,240 --> 00:14:46,780 There are no ex terms in here. 226 00:14:48,460 --> 00:14:53,248 There's an X squared term there, and the only ex term is CX, so 227 00:14:53,248 --> 00:14:54,616 see must be 0. 228 00:14:56,310 --> 00:14:59,946 So when we express this in its partial fractions, will end up 229 00:14:59,946 --> 00:15:01,158 with a being one. 230 00:15:02,720 --> 00:15:06,704 Be being minus one and see being 0. 231 00:15:09,020 --> 00:15:14,987 So we'll be left with trying to integrate one over X minus X 232 00:15:14,987 --> 00:15:17,282 over X squared plus one. 233 00:15:18,990 --> 00:15:19,640 DX 234 00:15:21,610 --> 00:15:25,294 so we've used partial fractions to split this up into two terms, 235 00:15:25,294 --> 00:15:28,978 and all we have to do now is completely integration. Let me 236 00:15:28,978 --> 00:15:30,206 write that down again. 237 00:15:31,750 --> 00:15:38,386 We want to integrate one over X minus X over X squared plus one 238 00:15:38,386 --> 00:15:43,600 and all that wants to be integrated with respect to X. 239 00:15:46,660 --> 00:15:48,430 First term straightforward. 240 00:15:49,260 --> 00:15:52,388 The integral of one over X is the 241 00:15:52,388 --> 00:15:54,734 logarithm of the modulus of X. 242 00:15:56,570 --> 00:15:58,320 To integrate the second term. 243 00:15:59,040 --> 00:16:03,506 We notice that the numerator is almost the derivative of the 244 00:16:03,506 --> 00:16:05,942 denominator. If we differentiate, the denominator 245 00:16:05,942 --> 00:16:07,160 will get 2X. 246 00:16:08,110 --> 00:16:11,230 There is really only want 1X now. We fiddle that by putting 247 00:16:11,230 --> 00:16:13,310 it to at the top and a half 248 00:16:13,310 --> 00:16:17,598 outside like that. So this integral is going to workout 249 00:16:17,598 --> 00:16:21,538 to be minus 1/2 the logarithm of the modulus of 250 00:16:21,538 --> 00:16:23,114 X squared plus one. 251 00:16:24,710 --> 00:16:27,356 And there's a constant of integration at the end. 252 00:16:29,400 --> 00:16:32,611 And I'll leave the answer like that if you wanted to do. We 253 00:16:32,611 --> 00:16:34,587 could combine these using the laws of logarithms. 254 00:16:35,670 --> 00:16:37,861 And that's integration of algebraic fractions. You 255 00:16:37,861 --> 00:16:41,304 need a lot of practice at that, and there are more 256 00:16:41,304 --> 00:16:43,182 practice exercises in the accompanying text.