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These are the user uploaded subtitles that are being translated: 1 00:00:00,690 --> 00:00:05,370 Today we're going to be talking about how to use integration by parts to evaluate an indefinite integral. 2 00:00:05,550 --> 00:00:11,400 And in this particular problem we've been given the integral of x cubed times natural log or Ellen of 3 00:00:11,400 --> 00:00:13,320 x x. 4 00:00:13,380 --> 00:00:17,280 Now we've already been told to use integration by parts to evaluate this integral. 5 00:00:17,460 --> 00:00:22,410 But if you're new to integrals you may not recognize that this is an integration by parts integral right 6 00:00:22,410 --> 00:00:23,130 away. 7 00:00:23,310 --> 00:00:28,050 The question you have to ask yourself is can you evaluate this integral as it is as it stands right 8 00:00:28,050 --> 00:00:28,670 now. 9 00:00:28,710 --> 00:00:30,280 And the answer is probably no. 10 00:00:30,290 --> 00:00:35,370 You're not quite sure how to evaluate this integral with just basic integration techniques. 11 00:00:35,370 --> 00:00:38,180 This isn't just a simple polynomial it's a little bit more complex. 12 00:00:38,190 --> 00:00:43,140 So you're going to have to manipulate the function a little bit before you can take its integral if 13 00:00:43,140 --> 00:00:48,360 you have to manipulate the function the first thing you want to try is use substitution if you can use 14 00:00:48,480 --> 00:00:49,250 substitution. 15 00:00:49,290 --> 00:00:53,610 That's the simplest way to manipulate an integral so that it's easier to evaluate. 16 00:00:53,790 --> 00:00:57,660 But in this case you substitution isn't really going to get us anywhere. 17 00:00:58,020 --> 00:01:04,320 Integration by parts should jump out at us as a good way to manipulate this integral because we have 18 00:01:04,440 --> 00:01:08,460 two components to our integral are two functions inside of our integral. 19 00:01:08,580 --> 00:01:15,240 One is x cubed here and the other one is natural log of X right here. 20 00:01:15,270 --> 00:01:20,280 Whenever we have two functions that are multiplied together inside of our integral we know that the 21 00:01:20,280 --> 00:01:25,830 integral is probably a good candidate for integration by parts because our integration by parts formula 22 00:01:26,220 --> 00:01:29,110 tells us that we have two functions that are multiplied together. 23 00:01:29,250 --> 00:01:33,410 One that we call you and the other one which we call DVH. 24 00:01:33,540 --> 00:01:39,420 Our only challenge when we use integration by parts is to identify which one will be you and which one 25 00:01:39,420 --> 00:01:40,630 will be DVH. 26 00:01:40,740 --> 00:01:47,580 Now when it comes to identifying you and DVH what I like to do is identify you first and then everything 27 00:01:47,580 --> 00:01:52,100 else in the integral just becomes divi by default and I've already picked you. 28 00:01:52,200 --> 00:01:54,340 When it comes to picking a value for you. 29 00:01:54,450 --> 00:02:00,390 A lot of people like to use what's called the least rule and the rule really become intuitive to you 30 00:02:00,390 --> 00:02:02,290 after you've had some progress doing this. 31 00:02:02,460 --> 00:02:08,790 But what the league rule tells you is that if you have a logarithmic function inside of your integral 32 00:02:09,060 --> 00:02:12,800 that's probably what you want to assign you to first. 33 00:02:12,810 --> 00:02:18,270 So if you have a logarithmic function like laga X or the natural log of X that's probably what you're 34 00:02:18,270 --> 00:02:24,060 going to want to use for you if you don't have a logarithmic function you go to the next letter and 35 00:02:24,060 --> 00:02:27,100 you look for an inverse trigonometric function. 36 00:02:27,180 --> 00:02:29,230 And if you have one of those you might want to use that. 37 00:02:29,430 --> 00:02:35,130 And then I think it's algebraic function triggered the metric function an exponential function in that 38 00:02:35,130 --> 00:02:36,520 particular order. 39 00:02:36,540 --> 00:02:41,510 So because we have the first letter the L we have a natural log of X here and are integral. 40 00:02:41,520 --> 00:02:43,790 That's probably what we want to use for you. 41 00:02:43,830 --> 00:02:49,620 The lead rule is not foolproof it's not perfect but it's a pretty good rule of thumb to go by. 42 00:02:49,620 --> 00:02:53,160 You should try to assign you to one of these functions in this order. 43 00:02:53,310 --> 00:03:00,690 So we're going to say that you is equal to natural log of X and that leaves everything else in integral 44 00:03:01,020 --> 00:03:02,250 x cubed times. 45 00:03:02,320 --> 00:03:09,480 Dyaks we're going to assign divi to x cubed d x remember that everything we have inside of our InterOil 46 00:03:09,480 --> 00:03:12,980 here has to be allocated either to you or to TV. 47 00:03:12,990 --> 00:03:18,780 And because we said that natural log of x is probably our candidate for you everything that's left over 48 00:03:19,050 --> 00:03:26,500 x cubed and DX has to go here to divi that DX will always go with DV. 49 00:03:26,760 --> 00:03:28,460 So we assigned you and DVH. 50 00:03:28,500 --> 00:03:36,310 Now at this point what we need to find are d u the derivative of u and v the integral of DVH. 51 00:03:36,330 --> 00:03:41,080 And that's why I wrote these here on two separate Rosing going to make a little mini table. 52 00:03:41,310 --> 00:03:43,880 But I like to have my original functions here. 53 00:03:44,000 --> 00:03:51,200 And V in my top row and my derivative functions d u and DV in my bottom row. 54 00:03:51,420 --> 00:03:55,470 We're going to start with you and take the derivative to get you. 55 00:03:55,620 --> 00:03:59,330 Then we're going to start with divi and take the integral to get V. 56 00:03:59,370 --> 00:04:04,390 These are the four values we're going to need to plug into our integration by parts formula. 57 00:04:04,440 --> 00:04:11,250 So taking the derivative of you I take the derivative of natural log of x which is 1 over X and because 58 00:04:11,250 --> 00:04:15,440 I'm taking the derivative here I have to then multiply by DX. 59 00:04:15,510 --> 00:04:16,680 I move over here to dvh. 60 00:04:16,680 --> 00:04:23,030 I want to take the integral of x cubed x y when I take the integral the DX will drop away. 61 00:04:23,190 --> 00:04:28,000 And the integral of x cubed is 1 fourth X to the four. 62 00:04:28,410 --> 00:04:32,610 So those are my four values that I'm going to be plugging into my integration by parts formula. 63 00:04:32,610 --> 00:04:38,370 I'm going to be plugging them in to the right hand side over here on plugging them in to this part right 64 00:04:38,370 --> 00:04:39,060 here. 65 00:04:39,150 --> 00:04:44,400 The left hand side of our integration by parts formula where it says the integral of you times the that 66 00:04:44,400 --> 00:04:49,710 matches up with our original integral the integral of x cubed Ellen of X because we said that Ellen 67 00:04:49,760 --> 00:04:53,760 X was you and that x cubed T X was DVH. 68 00:04:53,910 --> 00:04:58,740 So we're going to say the integral we're going to make substitutions here and say that the integral 69 00:04:59,100 --> 00:05:08,210 of cubed L.N. of x dx our original integral is equal to now using our formula you Times V. 70 00:05:08,220 --> 00:05:12,810 So we're going to multiply you and V here and when we combine these together we're going to get one 71 00:05:12,810 --> 00:05:21,180 fourth X to the four times natural log of X. Then according to our formula we're going to subtract the 72 00:05:21,210 --> 00:05:30,220 integral of V times d you will see here as one fourth X to the fore and D U is 1 over x dx. 73 00:05:30,360 --> 00:05:35,190 When we multiply these two values together if we look at it over here we're going to get one fourth 74 00:05:35,610 --> 00:05:36,740 X to the four. 75 00:05:36,810 --> 00:05:41,810 We're going to multiply that by 1 over x x when we do that. 76 00:05:41,850 --> 00:05:47,950 We're going to get X's to cancel here and to get this X to cancel this X the fourth will become x cubed. 77 00:05:47,970 --> 00:05:54,960 And so what we're going to have inside of our integral here is one fourth x cubed dx. 78 00:05:54,960 --> 00:06:00,120 Now before we evaluate this integral one thing we can do is move this one fourth right here in front 79 00:06:00,210 --> 00:06:04,620 of our integral because it's a constant coefficient on this x cube term. 80 00:06:04,620 --> 00:06:12,000 So what we're going to do is move that one forth there one fourth out in front of our integral and all 81 00:06:12,000 --> 00:06:17,790 we're left with now is the integral of x cubed dx and we can start to see why the integration by parts 82 00:06:17,790 --> 00:06:19,790 formula is so useful. 83 00:06:19,860 --> 00:06:26,240 We started with an integral for execute natural log of x dx which we didn't know how to evaluate. 84 00:06:26,580 --> 00:06:32,460 All we have left now is the integral of x cubed which is extremely easy for us to evaluate in fact we 85 00:06:32,460 --> 00:06:38,310 already found it here when we took the integral of divi which was exactly x cubed dx we already know 86 00:06:38,310 --> 00:06:41,270 that the integral is 1:48 X to the fore. 87 00:06:41,310 --> 00:06:47,370 So given that we're going to evaluate the integral in here we'll get one fourth X to the four times 88 00:06:47,400 --> 00:06:52,070 natural log of X Minus One fourth and here's where we have our integral. 89 00:06:52,140 --> 00:06:57,040 We know that the integral of x cubed is one fourth X to the four. 90 00:06:57,090 --> 00:07:03,300 And now we can't forget to add c to account for the constant of integration because we took this integral 91 00:07:03,520 --> 00:07:08,250 all we have to do at this point to get our final answer is simplify this as much as we can. 92 00:07:08,250 --> 00:07:13,620 One way that we can simplify it is by factoring out one fourth X to the four because we have a one fourth 93 00:07:13,620 --> 00:07:16,140 X to the four involved in both of our terms here. 94 00:07:16,410 --> 00:07:20,150 So we'll factor out one fourth X to the four. 95 00:07:20,370 --> 00:07:24,180 And what's left of our first term is just this natural log of X here. 96 00:07:24,210 --> 00:07:27,300 So we'll get natural log of X minus. 97 00:07:27,300 --> 00:07:30,010 We took out this whole one fourth X of the force. 98 00:07:30,030 --> 00:07:36,320 All we have left is this negative one force we get minus one fourth and then we add here. 99 00:07:36,330 --> 00:07:40,610 See we keep that C to account for the constant integration and that's it. 100 00:07:40,620 --> 00:07:41,730 That's our final answer. 101 00:07:41,820 --> 00:07:45,690 And that's how you use integration by parts to evaluate an indefinite integral. 11715