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These are the user uploaded subtitles that are being translated: 1 00:00:00,510 --> 00:00:05,570 In this video we're talking about how to use tabular integration to evaluate an integral and tabular 2 00:00:05,570 --> 00:00:09,230 integration is similar to integration by parts. 3 00:00:09,230 --> 00:00:11,530 It's just an alternate method. 4 00:00:11,540 --> 00:00:14,080 The tricky thing is that integration by parts. 5 00:00:14,120 --> 00:00:17,440 If it's applicable to a certain integral it'll work every time. 6 00:00:17,540 --> 00:00:23,390 Tabular integration is a more specific case and it doesn't work every time like integration by parts 7 00:00:23,390 --> 00:00:23,700 well. 8 00:00:23,720 --> 00:00:26,260 So you can't always use it but where you can use it. 9 00:00:26,280 --> 00:00:29,520 Oftentimes it can be a lot faster than integration by parts. 10 00:00:29,750 --> 00:00:35,150 So what I want to do here is take this integral we have the integral of x squared either the X x. 11 00:00:35,330 --> 00:00:38,770 And we've been asked to use tabular integration to evaluate this integral. 12 00:00:38,900 --> 00:00:42,770 Now we could use traditional integration by parts which is what I've done here. 13 00:00:42,800 --> 00:00:47,960 What I want to show you is how to use the tabular integration method to get to the exact same answer 14 00:00:47,960 --> 00:00:50,330 and you'll see that we end up with the same answer. 15 00:00:50,330 --> 00:00:55,910 So before we get tabular integration Let's review really quickly what we would do if we used integration 16 00:00:55,910 --> 00:00:58,160 by parts to evaluate this integral. 17 00:00:58,190 --> 00:01:02,720 First of all we would notice that we have the product of two functions so we have x squared multiplied 18 00:01:02,720 --> 00:01:04,050 by either the X.. 19 00:01:04,070 --> 00:01:06,480 So those are two separate functions they're multiplied together. 20 00:01:06,620 --> 00:01:09,410 So this is a good candidate for integration by parts. 21 00:01:09,530 --> 00:01:14,630 And what we want to do first is identify you and divi inside of our integral. 22 00:01:14,630 --> 00:01:20,120 So in this case we would say you equal to x squared which means DV has to be everything else in the 23 00:01:20,120 --> 00:01:23,400 integral which would be either the X X so we say us x squared. 24 00:01:23,480 --> 00:01:29,070 So DV is either the X D x then we take the derivative of you get d you. 25 00:01:29,090 --> 00:01:34,940 So we would say D-New is equal to the derivative x squared which is 2 x and we have that x and then 26 00:01:34,940 --> 00:01:37,850 we take the integral of DV to get V. 27 00:01:37,850 --> 00:01:44,510 So the integral of each of the x x is just either the X then we use this formula here this is the integration 28 00:01:44,510 --> 00:01:50,440 by parts formula it tells us that if we have the integral of u times divi in a row we assigned you and 29 00:01:50,450 --> 00:01:52,200 divi to values and our integral. 30 00:01:52,250 --> 00:01:59,210 So we do have the integral of U time Stevie when we have that integral it's equal to u times V minus 31 00:01:59,210 --> 00:02:00,730 the integral of VDU. 32 00:02:00,860 --> 00:02:06,020 So when we Bilour and roll this on the left hand side here is our original integral on the right hand 33 00:02:06,020 --> 00:02:06,380 side. 34 00:02:06,380 --> 00:02:09,400 Is this right hand side of the integration by parts formula. 35 00:02:09,410 --> 00:02:16,220 So we take you Times V or x squared times either the X and we get X squared times the X minus the integral 36 00:02:16,580 --> 00:02:18,510 of VDU from our formula. 37 00:02:18,560 --> 00:02:24,760 So we take V E to the X and D u to X X and we put that inside our integral. 38 00:02:25,100 --> 00:02:30,770 But now we're at a point where we need to use integration by parts again to evaluate this integral. 39 00:02:30,770 --> 00:02:33,590 The remaining integral here is a little simpler than what we started with. 40 00:02:33,590 --> 00:02:38,690 Because instead of x squared we have a first degree x variable but we still have to use integration 41 00:02:38,690 --> 00:02:40,050 by parts a second time. 42 00:02:40,070 --> 00:02:42,560 So when we do that we say you is 2 x. 43 00:02:42,590 --> 00:02:49,340 So DV has to be everything else either the X so we have u equals 2 x D-B equals either the X X we take 44 00:02:49,340 --> 00:02:56,780 the derivative of you and we do is two times d x we take the integral of DV and we get v is equal to 45 00:02:57,260 --> 00:03:00,390 x and then we use these values here. 46 00:03:00,530 --> 00:03:05,850 And the right hand side of our integration by parts formula we replace just this integral here. 47 00:03:05,850 --> 00:03:13,090 So we replaced just this integral right here with the right hand side from our integration by parts 48 00:03:13,100 --> 00:03:13,790 formula. 49 00:03:13,850 --> 00:03:21,440 And that's where this comes in here because we left in our answer we left the X square either the X 50 00:03:21,440 --> 00:03:24,420 right here X square either the X we left or minus sign. 51 00:03:24,440 --> 00:03:29,490 But then this integral gets replaced by the right hand side of our integration by parts formula. 52 00:03:29,660 --> 00:03:35,660 So we have u times V or 2 x times each of the X and then minus the integral of the times. 53 00:03:35,660 --> 00:03:38,750 D u so V and D U. 54 00:03:38,840 --> 00:03:42,070 We get to either the X and X inside of our integral. 55 00:03:42,170 --> 00:03:44,990 Then we distribute this negative sign across these two terms. 56 00:03:44,990 --> 00:03:52,490 We distribute the negative sign here and we distribute the negative sign here and we end up with a minus 57 00:03:52,550 --> 00:03:53,600 2 x either the X. 58 00:03:53,630 --> 00:03:55,570 And then this negative cancels with this negative. 59 00:03:55,580 --> 00:03:56,720 We have a positive. 60 00:03:56,720 --> 00:03:58,880 We pull that two out in front of the integral. 61 00:03:58,910 --> 00:04:03,800 So we just have the integral of either the X X and then we just have to take this integral and the integral 62 00:04:03,800 --> 00:04:09,320 of each of the X is either the X so we end up with this plus two times either the X and we had C to 63 00:04:09,320 --> 00:04:10,760 account for our constant of integration. 64 00:04:10,760 --> 00:04:14,440 So that's our final answer using integration by parts. 65 00:04:14,570 --> 00:04:17,150 And it took us a little while and it was a little complicated. 66 00:04:17,180 --> 00:04:20,110 So this is a perfect candidate for tabular integration. 67 00:04:20,120 --> 00:04:24,920 You only want to use tabular integration when ever one of the functions inside of your integral So in 68 00:04:24,920 --> 00:04:30,680 this case we have x squared and you do the X when one of the functions if you take its derivative over 69 00:04:30,680 --> 00:04:33,800 and over and over again the derivative will eventually go to zero. 70 00:04:33,860 --> 00:04:38,060 And we can see that that would be the case with x squared because if we take the derivative x squared 71 00:04:38,060 --> 00:04:39,180 we get to X. 72 00:04:39,200 --> 00:04:41,760 If we take the derivative again we get two. 73 00:04:41,930 --> 00:04:44,540 If we take the derivative again we get zero. 74 00:04:44,540 --> 00:04:50,660 So the derivatives of x squared eventually go to zero which means that we could probably use tabular 75 00:04:50,660 --> 00:04:53,090 integration to evaluate this integral. 76 00:04:53,090 --> 00:04:58,590 So the first thing we want to do in the same way that with integration by parts we assign you and D.V. 77 00:04:58,620 --> 00:05:03,950 to values in our integral The first thing we want to do is assign f of x and g of x to the functions 78 00:05:03,950 --> 00:05:05,690 inside of our integral. 79 00:05:05,750 --> 00:05:12,440 So essentially with tabular integration instead of the integral of you DV we're looking at the integral 80 00:05:12,560 --> 00:05:17,710 of f of x x x. 81 00:05:17,720 --> 00:05:23,900 So what we want to do is figure out whether f of x is x squared or the X and then whether G of x is 82 00:05:23,960 --> 00:05:29,930 x squared or either the X you want to sign f of x to the function whose derivatives go to zero which 83 00:05:29,930 --> 00:05:35,750 means we would say that f of x is going to be equal to x squared. 84 00:05:35,750 --> 00:05:42,710 That means the other function is going to be g of X will say g of x is equal to E to the X and now from 85 00:05:42,710 --> 00:05:46,240 this point the steps might seem a little foreign but they're actually really simple. 86 00:05:46,250 --> 00:05:50,630 So we just write those functions right underneath these values here. 87 00:05:50,630 --> 00:05:55,850 We're going to create a little table so forever x we're always going to take successive derivatives 88 00:05:55,880 --> 00:05:57,130 until we get to zero. 89 00:05:57,260 --> 00:06:03,110 So we're going to say the derivative x squared is to X the derivative to x is to the derivative of two 90 00:06:03,110 --> 00:06:04,110 is zero. 91 00:06:04,280 --> 00:06:07,880 So we always take successive derivatives of f of x. 92 00:06:07,940 --> 00:06:13,970 We're going to take successive integrals of each of the X so what we have here we have either the X 93 00:06:13,970 --> 00:06:16,430 if we take the integral we get either the X.. 94 00:06:16,550 --> 00:06:18,850 If we take the integral again we get either the X.. 95 00:06:18,860 --> 00:06:24,560 If we take the integral again we get either the X and we just drop this down all the way until we hit 96 00:06:24,560 --> 00:06:28,780 this last row here where we go 0 4 F of X.. 97 00:06:28,820 --> 00:06:30,540 Now we're going to do something a little strange. 98 00:06:30,560 --> 00:06:36,410 We're just going to take the value from the first row in the 5 x column and we're going to hit with 99 00:06:36,410 --> 00:06:37,140 a line. 100 00:06:37,190 --> 00:06:42,830 Connect it to the value in the second row in the G of X column and then we're just going to draw parallel 101 00:06:42,830 --> 00:06:46,770 lines here so this 2 x is going to get connected to this value. 102 00:06:46,850 --> 00:06:50,180 The twos going to get connected to this value and then we're done. 103 00:06:50,180 --> 00:06:55,150 When you hit that zero term you don't have to connect that to anything in the G of X column. 104 00:06:55,160 --> 00:07:00,680 So you just start with the value in the first row in the X column and you connect to the next lowest 105 00:07:00,680 --> 00:07:03,670 value in the G of X column until you're done here. 106 00:07:03,830 --> 00:07:10,430 Then the last thing you do is you add signs to this f of x column and you always start with a positive 107 00:07:10,430 --> 00:07:11,680 sign of new alt.. 108 00:07:11,690 --> 00:07:16,460 So we're going to do a positive sign a negative sign and a positive sign and you want to keep doing 109 00:07:16,460 --> 00:07:18,880 that all the way down until you get here to zero. 110 00:07:18,890 --> 00:07:23,070 You don't have to assign a sign to zero just to the non-zero term. 111 00:07:23,060 --> 00:07:26,900 So you always start with positive and you just alternate positive negative positive negative positive 112 00:07:26,900 --> 00:07:27,420 negative. 113 00:07:27,500 --> 00:07:33,530 As long as it takes for you to get down to this zero value and now this is actually all we need to do 114 00:07:33,800 --> 00:07:39,770 to get directly to our final answer because here's what we do we start with this value in the first 115 00:07:39,770 --> 00:07:43,940 row for f of x and we just multiply it by the value it's connected to. 116 00:07:43,940 --> 00:07:47,510 So we say x squared times each of the X and the sign is positive. 117 00:07:47,510 --> 00:07:50,120 So in other words positive x squared either. 118 00:07:50,150 --> 00:07:55,180 So we start with positive x squared B to the X.. 119 00:07:55,340 --> 00:08:02,750 Then we go to the next row we have a negative 2 x either the X so minus 2 x either the X then we have 120 00:08:02,750 --> 00:08:08,900 a positive so plus two times either the X or plus two times either the X and then we can't forget to 121 00:08:08,900 --> 00:08:10,900 add our constant of integration. 122 00:08:10,920 --> 00:08:13,150 See but that's our final answer. 123 00:08:13,150 --> 00:08:19,600 And if you notice here this is exactly the same answer that we got when we used integration by parts. 124 00:08:19,640 --> 00:08:20,740 They match exactly. 125 00:08:20,900 --> 00:08:25,140 So tabular integration is going to work for you in place of integration by parts. 126 00:08:25,230 --> 00:08:31,280 Whenever you have one function inside of your integral whose derivatives go to zero if that's the case 127 00:08:31,520 --> 00:08:34,900 you can use tabular integration to find the correct answer. 128 00:08:34,970 --> 00:08:39,500 And especially for a problem like this one where you have to use integration by parts multiple times 129 00:08:39,500 --> 00:08:43,680 to get to the answer tabular integration can sometimes be a lot faster. 14306