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These are the user uploaded subtitles that are being translated: 1 00:00:00,400 --> 00:00:04,050 In this video we're doing another partial fractions problem and we've been asked to find the integral 2 00:00:04,050 --> 00:00:07,190 of one divided by x cubed plus x. 3 00:00:07,230 --> 00:00:11,340 Now at first glance this might not look like a partial fractions problem because it doesn't look like 4 00:00:11,340 --> 00:00:14,190 we have any factors in our denominator. 5 00:00:14,190 --> 00:00:17,930 But what we have to realize is that this denominator can actually be factored. 6 00:00:18,060 --> 00:00:24,900 So we can actually call this the integral of one divided by here in the denominator if we factor out 7 00:00:24,900 --> 00:00:30,910 an x we get X times x squared plus 1 D X.. 8 00:00:30,990 --> 00:00:37,440 So now we have factors in our denominator we have one linear factor this x to the first power factor 9 00:00:37,440 --> 00:00:41,490 here and we have one quadratic factor the x squared plus 1. 10 00:00:41,490 --> 00:00:45,570 Remember when you you have X to the First it's a linear factor when you have x squared. 11 00:00:45,570 --> 00:00:51,150 It's a quadratic factor so to set up our partial fraction decomposition will take this original fraction 12 00:00:51,150 --> 00:00:55,830 here with the denominator factored and we'll put that on the left hand side of this new equation. 13 00:00:55,830 --> 00:01:00,400 So X times quantity x squared plus 1 on the right hand side. 14 00:01:00,430 --> 00:01:05,990 Remember whenever you have a linear factor you just have a single constant in the numerator and then 15 00:01:06,010 --> 00:01:07,920 just add these fractions together. 16 00:01:07,920 --> 00:01:11,000 We have our quadratic factor X squared plus 1. 17 00:01:11,130 --> 00:01:17,090 And whenever you have a quadratic factor your numerator needs to look like and B X plus C. 18 00:01:17,220 --> 00:01:20,210 We start with B because A is already taken. 19 00:01:20,220 --> 00:01:23,190 So now this is going to be our partial fractions decomposition. 20 00:01:23,310 --> 00:01:26,400 This value over here on the right hand side. 21 00:01:26,550 --> 00:01:30,980 This is the value that we're going to use to replace the original function. 22 00:01:31,020 --> 00:01:35,900 So we're going to eventually be taking the integral of this instead of the original fraction. 23 00:01:35,910 --> 00:01:39,690 All we need to do first is find values for A B and C.. 24 00:01:39,720 --> 00:01:45,900 So how are we going to find values for A B and C Well as always we want to multiply both sides by the 25 00:01:45,900 --> 00:01:47,850 denominator from the left hand side. 26 00:01:48,120 --> 00:01:53,310 So that denominator is X times quantity x squared plus 1. 27 00:01:53,550 --> 00:01:57,130 So we want to multiply every term by these two factors. 28 00:01:57,300 --> 00:02:02,370 When we do that on the left hand side over here we're going to get this X to cancel with this X and 29 00:02:02,370 --> 00:02:06,150 we're going to get X squared plus 1 to cancel this x squared plus 1. 30 00:02:06,180 --> 00:02:10,280 So we're just going to be left with one on the left hand side on the right hand side. 31 00:02:10,290 --> 00:02:17,010 When we multiply x 10 x squared plus 1 by a divided by X we're going to get this X to cancel with this 32 00:02:17,010 --> 00:02:20,940 factor of x leaving us with just this x squared plus 1. 33 00:02:21,090 --> 00:02:24,770 So we get a times quantity x squared plus 1. 34 00:02:24,870 --> 00:02:30,360 And then here when we multiply these two factors by this be X possi fraction we're going to get the 35 00:02:30,450 --> 00:02:34,810 x squared plus 1 in the denominator to cancel with this x squared plus 1. 36 00:02:34,860 --> 00:02:36,750 Leaving us with just that X there. 37 00:02:36,750 --> 00:02:43,440 So we're going to get plus B X plus C multiplied by x. 38 00:02:43,440 --> 00:02:46,470 Now what we want to do is multiply out the right hand side. 39 00:02:46,710 --> 00:02:55,700 So we're going get one is equal to a x squared plus a plus b x squared plus C x. 40 00:02:55,830 --> 00:02:59,940 Now keep in mind that over here on the left hand side we just have one which is a constant. 41 00:03:00,120 --> 00:03:03,800 So we could really change this left hand side and we could write 0. 42 00:03:03,840 --> 00:03:07,830 X squared plus 0 x plus 1. 43 00:03:07,860 --> 00:03:13,450 That original Cosson we haven't changed anything by adding 0 X because 0 times X is 0. 44 00:03:13,680 --> 00:03:19,130 We also haven't changed anything by adding 0 x squared because zero times x square it is still 0. 45 00:03:19,260 --> 00:03:21,630 So we've just added 0 to the left hand side. 46 00:03:21,750 --> 00:03:26,280 But this will help us visualize our method of undetermined coefficients that we're going to go to in 47 00:03:26,280 --> 00:03:26,940 a second. 48 00:03:26,970 --> 00:03:31,520 So we have that left hand side over here on the right hand side we want to group together like terms 49 00:03:31,540 --> 00:03:34,490 we're going to pull all of our x squared terms together. 50 00:03:34,650 --> 00:03:42,240 So we're going to say a x squared plus B X squared to take care of these 2 x squared terms. 51 00:03:42,330 --> 00:03:44,190 Then we want all of our x terms together. 52 00:03:44,190 --> 00:03:48,740 In this case let's just see X right there and then we want all of our constants together. 53 00:03:48,750 --> 00:03:50,480 And in this case that's just a. 54 00:03:50,490 --> 00:03:51,970 So that deals with that term. 55 00:03:52,050 --> 00:03:57,930 Then on the right hand side we want to factor out the x value wherever we can. 56 00:03:57,930 --> 00:04:03,600 So here we're going to pull x squared out of X squared plus B x square which is just going to leave 57 00:04:03,600 --> 00:04:07,890 us with quantity a plus be multiplied by x squared. 58 00:04:08,160 --> 00:04:09,330 Then we're going to have this. 59 00:04:09,420 --> 00:04:13,110 See times X and then plus a. 60 00:04:13,110 --> 00:04:19,080 Now the reason we pulled out the X variables is because at this point we can go ahead and equate coefficients 61 00:04:19,080 --> 00:04:21,000 from the left in the right hand side. 62 00:04:21,000 --> 00:04:28,230 In other words we know that the coefficient on x squared on the right hand side is a plus b right here. 63 00:04:28,230 --> 00:04:29,540 The coefficient on x squared. 64 00:04:29,540 --> 00:04:35,570 On the left hand side is zero which means we can say A plus B is equal to zero. 65 00:04:35,580 --> 00:04:40,800 We also know here that the coefficient on X on the right is c and of the coefficient on X on the left 66 00:04:40,830 --> 00:04:41,420 is zero. 67 00:04:41,430 --> 00:04:47,480 So we can say C equals zero and we have a constant of a on the right and a constant of one on the left. 68 00:04:47,490 --> 00:04:50,380 So we can say A is equal to 1. 69 00:04:50,460 --> 00:04:53,190 So when we set up those three equations Here's what we get. 70 00:04:53,280 --> 00:04:56,760 WE HAVE A plus B equals zero. 71 00:04:56,760 --> 00:05:03,690 We have C equals zero and we have a equals one our goal in doing this is to solve for the three values 72 00:05:03,690 --> 00:05:11,310 here a b and c so we can take this value in the purple box and evaluate the integral of this function 73 00:05:11,340 --> 00:05:14,180 instead of the integral of the original function. 74 00:05:14,400 --> 00:05:19,860 So we already have a in C we know A's equal to one in C's equal to zero to find a B. 75 00:05:19,860 --> 00:05:25,380 We're just going to take this equation plug in equals one and instead of A plus B equals zero we get 76 00:05:25,470 --> 00:05:28,200 1 plus B is equal to zero. 77 00:05:28,230 --> 00:05:32,180 Subtracting one from both sides we get B is equal to negative 1. 78 00:05:32,190 --> 00:05:38,100 So now we plug those three values into this right hand side here instead of a will put in 1 instead 79 00:05:38,100 --> 00:05:41,230 of B we'll put in negative 1. 80 00:05:41,400 --> 00:05:47,120 And instead of C we'll put in zero and now we want to rewrite our integral. 81 00:05:47,580 --> 00:05:55,410 So our integral looks like this instead of the original integral we'll say 1 over X so 1 over X we have 82 00:05:55,440 --> 00:06:02,400 a plus negative 1 times X we'll go ahead and say plus negative x and then we have plus 0 so don't need 83 00:06:02,390 --> 00:06:08,390 to write that so we have negative X divided by x squared plus 1 D x. 84 00:06:08,400 --> 00:06:14,970 Now we can split this into two integrals we can say the integral of one over x d x will pull this minus 85 00:06:14,970 --> 00:06:22,360 sign out and we'll say minus the integral of x over x squared plus 1 D x. 86 00:06:22,380 --> 00:06:28,010 Now for this second integral here this first one is easy but for the second one we need to use substitution. 87 00:06:28,050 --> 00:06:31,860 So I will say you is equal to the denominator x square plus 1. 88 00:06:32,010 --> 00:06:39,810 We'll say d u the derivative of you is equal to 2 x d x and then we'll solve for x by dividing both 89 00:06:39,810 --> 00:06:45,470 sides by 2 x and we'll get x is equal to do you over to x. 90 00:06:45,480 --> 00:06:50,760 So now will evaluate this first integral we know that the integral of 1 divided by X is the natural 91 00:06:50,760 --> 00:06:56,370 log of x are going to get natural laga of the absolute value of x according to this formula here. 92 00:06:56,580 --> 00:07:03,330 Then for our second integral I will say minus the integral leave this X in the numerator the denominator 93 00:07:03,330 --> 00:07:06,380 x squared plus 1 we said equal to use we have you there. 94 00:07:06,630 --> 00:07:12,600 And then we know dx is d u over to X so we have do you over to x. 95 00:07:12,690 --> 00:07:16,690 Now we know that we can get X to cancel from the numerator and denominator. 96 00:07:16,710 --> 00:07:23,700 So what we end up with is natural log the absolute value of X minus we'll pull this two from the denominator 97 00:07:23,790 --> 00:07:30,010 out in front so we'll say minus 1 1/2 times the integral of 1 over u d u. 98 00:07:30,210 --> 00:07:35,440 Well we know that the integral of one over you is natural log of the absolute value of use we get natural 99 00:07:35,440 --> 00:07:41,630 log absolute value of X minus 1 1/2 times the natural log of the absolute value of you. 100 00:07:41,790 --> 00:07:44,800 And then we add c to account for the constant of integration. 101 00:07:45,090 --> 00:07:49,650 We know that you is x squared plus once we just have the back substitute and then we can say that our 102 00:07:49,650 --> 00:07:56,040 final answer is the natural log of the absolute value of X minus 1 1/2 times the natural log of the 103 00:07:56,040 --> 00:08:02,590 absolute value of x squared plus 1 plus C and this value here is our final answer. 104 00:08:02,700 --> 00:08:08,760 It's the value of the original integral which we found using partial fractions and distinct quadratic 105 00:08:08,760 --> 00:08:09,640 factors. 12012