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These are the user uploaded subtitles that are being translated: 1 00:00:00,330 --> 00:00:04,350 In this video we're talking about partial fractions and how to use partial fractions to evaluate an 2 00:00:04,350 --> 00:00:09,090 integral and in this particular problem we're going to be dealing with distinct linear factors and we 3 00:00:09,090 --> 00:00:15,060 call it distinct linear factors because when you look at the denominator you have three linear factors 4 00:00:15,150 --> 00:00:21,270 x x plus two and X minus 1 So those are the three linear factors we call them linear because they all 5 00:00:21,270 --> 00:00:22,930 involve X to the first powers. 6 00:00:22,930 --> 00:00:29,520 So this is X to the first is X to the first power plus two is X to the first power minus one of x to 7 00:00:29,520 --> 00:00:30,670 the first power. 8 00:00:30,690 --> 00:00:31,930 It's a linear factor. 9 00:00:32,160 --> 00:00:35,330 And we call them distinct because they aren't equal to each other. 10 00:00:35,340 --> 00:00:41,160 So a factor of x is not the same as a factor of x plus 2 which is not the same as a factor of x minus 11 00:00:41,250 --> 00:00:41,860 1. 12 00:00:41,970 --> 00:00:47,340 This is the easiest kind of partial fractions problem distinct linear factors because linear factors 13 00:00:47,340 --> 00:00:53,280 are easier to handle than quadratic factors and distinct factors are easier to handle than repeated 14 00:00:53,280 --> 00:00:54,230 factors. 15 00:00:54,240 --> 00:00:59,190 So when you have distinct linear factors this is how you do the partial fractions decomposition you're 16 00:00:59,190 --> 00:01:04,200 going to keep the original function the original fraction here exactly as is on the left hand side so 17 00:01:04,200 --> 00:01:12,030 we're going to say for x squared minus 3 X minus 4 divided by the original denominator X times x plus 18 00:01:12,030 --> 00:01:14,870 two times X minus 1. 19 00:01:15,180 --> 00:01:16,640 We're going to set that equal to. 20 00:01:16,650 --> 00:01:19,630 And we're going to give each factor its own fraction. 21 00:01:19,680 --> 00:01:25,710 So this X is going to be in one fraction then we're going to put the X plus two in the next fraction 22 00:01:26,040 --> 00:01:28,830 and the X minus one in the next fraction. 23 00:01:29,040 --> 00:01:33,530 And then when you have linear factors the numerators are just a single constant. 24 00:01:33,600 --> 00:01:34,370 So a. 25 00:01:34,500 --> 00:01:39,570 And then because we already used a we used B and then because we've already used a and b we use. 26 00:01:39,600 --> 00:01:44,570 See this right hand side is what we're going to use to replace the original fraction here. 27 00:01:44,580 --> 00:01:49,320 So we're actually going to end up integrating this instead of the original fraction. 28 00:01:49,440 --> 00:01:54,340 All we have to do before we get to that step is find values for A B and C.. 29 00:01:54,360 --> 00:01:57,780 So how do we go about finding those values for the constants A B and C. 30 00:01:57,780 --> 00:02:04,180 Well we're going to multiply both sides of this equation by the denominator from the left hand sides 31 00:02:04,180 --> 00:02:10,770 are going multiply everything by x times x plus two times X minus 1. 32 00:02:10,830 --> 00:02:14,620 When we do that we're going to get the denominator from the left hand side to go away completely that 33 00:02:14,620 --> 00:02:16,530 that'll cancel with this value. 34 00:02:16,740 --> 00:02:21,110 So we're left with four x squared minus three X minus four. 35 00:02:21,240 --> 00:02:28,980 And then on the right hand side this X here will cancel with this X leaving us with just x plus 2 times 36 00:02:29,040 --> 00:02:29,990 X minus 1. 37 00:02:30,000 --> 00:02:37,710 So we'll get a times quantity x plus two times quantity x minus 1 for this fraction here involving b. 38 00:02:37,800 --> 00:02:40,700 We'll get the X plus two to cancel with this X plus two. 39 00:02:40,830 --> 00:02:48,650 Leaving us with just X and X minus once we'll say plus B times x times quantity x minus 1. 40 00:02:48,750 --> 00:02:54,210 And then for our last fraction involving C We'll get the X Minus One to cancel with this X Minus One 41 00:02:54,540 --> 00:02:59,620 leaving us with just c times x times x plus 2. 42 00:02:59,790 --> 00:03:05,610 Now we want to do is go ahead and expand the right hand side we'll leave the left hand side alone for 43 00:03:05,610 --> 00:03:11,100 right now on the left hand side here we're going to get a time's quantity x plus two times quantity 44 00:03:11,100 --> 00:03:18,560 x minus one is going to give us X squared minus X plus 2 x or just Plus X and then minus 2. 45 00:03:18,870 --> 00:03:23,290 Here we're going to get B X squared minus B X.. 46 00:03:23,310 --> 00:03:26,530 When we distribute the B X across the X minus one. 47 00:03:26,670 --> 00:03:31,590 And then here we're going to get c x squared plus to see X.. 48 00:03:31,710 --> 00:03:38,740 Lastly we'll distribute the a across the quantity x squared plus X minus 2. 49 00:03:38,790 --> 00:03:45,570 And so will get a x squared plus a X minus 2 A and the rest will be the same. 50 00:03:45,570 --> 00:03:51,570 Now we want to grouped together like terms so again we'll leave the left hand side as is for right now 51 00:03:51,870 --> 00:03:55,350 on the right hand side we want to put all of our x squared terms together. 52 00:03:55,410 --> 00:04:02,820 So that's going to look like a x squared plus and B X squared plus C x squared. 53 00:04:02,820 --> 00:04:08,490 That's going to cover this term this term and this term then we want to put all of our first degree 54 00:04:08,520 --> 00:04:09,670 x terms together. 55 00:04:09,690 --> 00:04:17,640 So we'll put these in parentheses here then we'll say plus a X minus B X plus 2. 56 00:04:17,640 --> 00:04:18,930 C x. 57 00:04:18,960 --> 00:04:22,140 So that's going to cover this this and this. 58 00:04:22,140 --> 00:04:24,560 And then lastly we'll put all of our constants together. 59 00:04:24,570 --> 00:04:30,180 So we're going to end up with a minus to a and I'll cover the last her there. 60 00:04:30,210 --> 00:04:36,720 Next we'll factor out the x variable from each of these sets here of parentheses. 61 00:04:36,720 --> 00:04:41,180 So again we leave the left hand side and on the right hand side we'll pull out an x squared. 62 00:04:41,220 --> 00:04:48,390 So that'll just leave us with quantity A plus B plus C when we factor in x squared out of X squared 63 00:04:48,390 --> 00:04:50,570 plus square pussy x squared. 64 00:04:50,850 --> 00:04:52,430 And then here will factor out an x. 65 00:04:52,440 --> 00:05:00,320 So we'll be left with a minus B plus 2 C and then we pull that X out and then we have the Today. 66 00:05:00,510 --> 00:05:04,830 Now the reason that we do it like this is because we want to equate coefficients from the left and the 67 00:05:04,830 --> 00:05:06,750 right hand side of the equal sign. 68 00:05:07,050 --> 00:05:11,550 So you can do is draw boxes around these coefficients to make it even more clear for yourself. 69 00:05:11,550 --> 00:05:17,700 So what we see is that the coefficient on the x squared term on the right hand side is this a plus b 70 00:05:17,700 --> 00:05:20,580 plus C that we created the coefficient on x squared. 71 00:05:20,580 --> 00:05:23,500 On the left hand side is this value here. 72 00:05:24,240 --> 00:05:29,590 So we're going to be able to do is say four is equal to a plus b plus C. 73 00:05:29,850 --> 00:05:38,190 We can also say that a minus B plus 2 C is going to be equal to negative 3 because those are the coefficients 74 00:05:38,250 --> 00:05:40,390 on the first degree x variable. 75 00:05:40,680 --> 00:05:45,840 And then lastly the constants were going to be able to say negative 2A is going to be equal to negative 76 00:05:45,840 --> 00:05:46,510 4. 77 00:05:46,590 --> 00:05:49,350 Since those are the constants on each side. 78 00:05:49,680 --> 00:05:59,190 So we want to set up those equations for equals a plus b plus C negative 3 is equal to a minus B plus 79 00:05:59,250 --> 00:06:05,220 2 C and then a negative 4 is equal to negative to a. 80 00:06:05,310 --> 00:06:09,610 Well the easiest one to solve obviously is this negative 4 equals negative 2. 81 00:06:09,780 --> 00:06:14,770 So we'll divide both sides by negative two and we'll get positive two is equal to a. 82 00:06:14,940 --> 00:06:16,080 So a is equal to two. 83 00:06:16,080 --> 00:06:19,230 We'll put that up here and then give ourselves more room. 84 00:06:19,230 --> 00:06:25,380 So if a is equal to to what we can say then when we plug a was 2 into both these equations is instead 85 00:06:25,380 --> 00:06:31,890 of four equals a plus be Plessey we'll get four equals two plus B plus see if we subtract two from both 86 00:06:31,890 --> 00:06:36,730 sides then we'll end up with two is equal to B plus C. 87 00:06:36,930 --> 00:06:43,230 Same thing here with this equation will get negative three equals two minus B plus to see if we subtract 88 00:06:43,230 --> 00:06:49,500 two from both sides we're going to get negative three minus two is a negative five equals negative B 89 00:06:49,560 --> 00:06:50,580 plus 2. 90 00:06:50,610 --> 00:06:57,570 See now we can do is add these equations together because if we add them together we're going to get 91 00:06:57,600 --> 00:07:00,620 B plus a negative b which is zero. 92 00:07:00,720 --> 00:07:09,390 So to minus five or two plus a negative five gives us a negative three equals C plus 2 C gives us 3 93 00:07:09,390 --> 00:07:09,920 C. 94 00:07:10,100 --> 00:07:14,130 And if you divide both sides by three you get C is equal to negative 1. 95 00:07:14,130 --> 00:07:19,470 Now if we take that value for C and the plug it back into this first equation here instead of two equals 96 00:07:19,500 --> 00:07:24,110 B plus C we get two equals B minus 1. 97 00:07:24,180 --> 00:07:28,090 And then if we add 1 to both sides we add B is equal to 3. 98 00:07:28,350 --> 00:07:34,930 So now we have the values equal to be equal 3 and C equal negative 1. 99 00:07:35,040 --> 00:07:39,000 We're going to be taking this original partial fractions decomposition this right hand side that we 100 00:07:39,000 --> 00:07:44,720 found here and we're going to be plugging that in for the original fraction that we started with. 101 00:07:44,720 --> 00:07:49,890 So we're going to be replacing this original fraction with the partial fractions the composition except 102 00:07:49,950 --> 00:07:56,640 that instead of a B and C we're going to have two three and negative one. 103 00:07:56,640 --> 00:08:02,730 So instead of taking the integral of this original fraction we're going to say the integral of two over 104 00:08:02,910 --> 00:08:13,500 X plus three over X plus two and then we have a negative one so minus 1 over X minus 1 D X and since 105 00:08:13,500 --> 00:08:21,660 we know that the integral of one over x is equal to natural log of the absolute value of x plus C we 106 00:08:21,660 --> 00:08:27,150 can go ahead and say then that our integral is to we take this from the numerator times. 107 00:08:27,150 --> 00:08:32,610 Now we have two times one over X so two times the natural log of the absolute value of x. 108 00:08:32,640 --> 00:08:38,290 Here we're going to have plus three times the natural log of the absolute value of x plus 2. 109 00:08:38,490 --> 00:08:44,460 And here we're going to have minus 1 so minus natural log of the absolute value of X minus 1. 110 00:08:44,610 --> 00:08:50,550 And then we add c to account for our cost of integration and that's how you use partial fractions to 111 00:08:50,550 --> 00:08:53,930 evaluate an integral that has distinct linear factors. 12711