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These are the user uploaded subtitles that are being translated: 1 00:00:00,330 --> 00:00:05,270 Today we're going to talk about how to use integration by parts to find the antiderivative of function. 2 00:00:06,080 --> 00:00:11,830 To complete this problem well understand the integration by parts formula match that this is our function 3 00:00:11,860 --> 00:00:15,280 to the parts of our formula and then evaluate the integral. 4 00:00:15,460 --> 00:00:21,640 In this particular video we've been asked to evaluate the integral of each of the 7 x times cosign of 5 00:00:21,640 --> 00:00:23,100 2 x x. 6 00:00:23,190 --> 00:00:28,060 Now since this is an integration by parts problem the first thing that we need to do is identify in 7 00:00:28,060 --> 00:00:32,970 our function a value for you and a value for. 8 00:00:33,250 --> 00:00:39,070 If we look at the integration by parts formula here what we can see is that it tells us that if we're 9 00:00:39,070 --> 00:00:46,930 taking the integral of you Times DVH then we can set that equal to you at times V minus the integral 10 00:00:46,990 --> 00:00:49,340 of the times d u. 11 00:00:49,360 --> 00:00:57,100 So what we have to first identify is value for you and a value for divi because the integral over here 12 00:00:57,100 --> 00:01:03,100 on the left hand side of our formula is representing the integral that we're taking here. 13 00:01:03,100 --> 00:01:11,610 So we have to identify which part of this function here will be you and which part will be DVH. 14 00:01:11,620 --> 00:01:14,520 Notice that there are no other parts of this integral. 15 00:01:14,560 --> 00:01:22,230 So we have to assign this entire integral including the dx to you and divi somehow. 16 00:01:22,300 --> 00:01:27,790 Now for this particular problem I'll go ahead and say that whenever you have the exponential function 17 00:01:27,790 --> 00:01:32,290 whenever you have e here multiplied by a Trigon the metric function. 18 00:01:32,290 --> 00:01:39,220 Your best bet will be to assign you to the trick in a metric function and DVH the exponential function. 19 00:01:39,220 --> 00:01:49,270 So let's go ahead and do that will say that u is equal to cosign of 2 x and that means since it is equal 20 00:01:49,270 --> 00:01:53,340 to cosign to X that divi here. 21 00:01:53,410 --> 00:01:58,660 The other part of the integral has to include everything else in our integral which would be either 22 00:01:58,660 --> 00:02:01,200 the 7 x and the x. 23 00:02:01,270 --> 00:02:08,880 So we'll say that DV is equal to E to the 7 x dx. 24 00:02:08,920 --> 00:02:14,290 One thing to note here is that X will always be here with your DVH. 25 00:02:14,320 --> 00:02:21,370 So now we need to since we have you and DVH we need to take the derivative of you to find D u and we 26 00:02:21,370 --> 00:02:26,140 need to take the integral of DVM to find the next thing that we have to do. 27 00:02:26,140 --> 00:02:34,120 Now that we have you and DVH is take the derivative of you to get the view and the integral or the antiderivative 28 00:02:34,210 --> 00:02:36,370 of DVH defined V. 29 00:02:36,610 --> 00:02:39,370 So taking the derivative of you we'll call it. 30 00:02:39,450 --> 00:02:44,370 D'you remember that the derivative of cosign is negative sign. 31 00:02:44,500 --> 00:02:50,890 And we'll be using chain rule because cosigners our outside function and to X is our inside function. 32 00:02:50,920 --> 00:02:55,910 So we'll take the derivative of the outside function and get negative sign of 2 x. 33 00:02:56,020 --> 00:03:00,670 But then chainwheel tells us that we have to multiply by the derivative of the inside function so be 34 00:03:00,670 --> 00:03:02,260 multiplying that by 2. 35 00:03:02,320 --> 00:03:12,430 And what we end up with is negative to sign of 2 x this 2 out in front here in the coefficient coming 36 00:03:12,430 --> 00:03:17,530 as a result of the application of changeroom multiplying by the derivative of 2 x. 37 00:03:17,560 --> 00:03:18,450 So that's our derivative. 38 00:03:18,450 --> 00:03:23,710 And of course we need to remember to add the x here whenever we take the derivative. 39 00:03:23,710 --> 00:03:27,640 Now we take the integral of DV to find v. 40 00:03:27,700 --> 00:03:33,040 So say that V is equal to when we take the integral of each to the 7 x. 41 00:03:33,040 --> 00:03:37,090 Remember that we take the the integral of an exponential here. 42 00:03:37,150 --> 00:03:46,060 We have to divide by the coefficient here on the X we're going to end up with one seventh e m to the 43 00:03:46,060 --> 00:03:49,810 7 x and that will be our value for V. 44 00:03:49,870 --> 00:03:56,290 So now that we found each of those four values we can go ahead and plug them into our integration by 45 00:03:56,290 --> 00:03:57,480 parts formula. 46 00:03:57,670 --> 00:04:01,390 So we'll say that the integral of our Aboriginal function here. 47 00:04:01,390 --> 00:04:11,740 So each of the 7 x cosign of 2 x x will be equal to the right hand side of our formula. 48 00:04:11,740 --> 00:04:14,960 So U-V minus the integral of the d u. 49 00:04:14,980 --> 00:04:19,500 And we're going to just be grabbing those values straight from the values we calculated here. 50 00:04:19,510 --> 00:04:29,170 So you at times V when we multiply you and we together we'll get one seventh to the 7 x which is the 51 00:04:29,380 --> 00:04:33,230 value for V times you cosign of 2 x. 52 00:04:33,700 --> 00:04:37,830 And I just reordered those terms so that we didn't have to simplify them later. 53 00:04:38,050 --> 00:04:44,060 So that's U times V minus the integral of v d u. 54 00:04:44,230 --> 00:04:47,700 So the 172 to the 7 x and D'you hear. 55 00:04:47,830 --> 00:04:52,270 So notice we have the coefficients negative 2 and positive 1 seventh. 56 00:04:52,270 --> 00:04:57,280 So when we multiply those together we'll get negative two negative two sevenths. 57 00:04:57,280 --> 00:05:10,440 So we'll say negative two sevenths times each to the 7 x sine of 2 x the X. 58 00:05:10,470 --> 00:05:13,410 So that's the application of our integration by parts formula. 59 00:05:13,410 --> 00:05:21,600 Now we need to go ahead and simplify this so what we'll do is we'll just pull the negative two sevenths 60 00:05:22,020 --> 00:05:24,300 out in front of our integral. 61 00:05:24,480 --> 00:05:28,650 And what we'll end up with we have a negative sign here and a negative sign here that will cancel. 62 00:05:28,680 --> 00:05:36,460 So we'll get positive to sevenths times the integral of each of the 7 x. 63 00:05:37,060 --> 00:05:44,910 Sign of 2 x x and your first thought might be that we haven't made very much progress. 64 00:05:44,920 --> 00:05:49,900 But whenever you have an exponential time to train the metric function of using integration by parts 65 00:05:49,900 --> 00:05:55,720 to take the integral what will often happen is you'll need to end up applying integration by parts twice 66 00:05:56,020 --> 00:05:57,550 and you'll see how this is going to work out. 67 00:05:57,550 --> 00:06:01,750 In the end but we're going to have to go ahead and apply integration by parts again and we're going 68 00:06:01,750 --> 00:06:07,990 to do the same thing we did before where we set you equal to the trick a metric function and divi equal 69 00:06:07,990 --> 00:06:09,520 to the exponential function. 70 00:06:09,730 --> 00:06:21,730 So here I will say that you is equal to sign of 2 x and therefore we know that divi must be equal to 71 00:06:21,820 --> 00:06:31,060 the rest of it which will be easy to the 7 x times the X when we take the derivative of you to get the 72 00:06:31,080 --> 00:06:31,770 U. 73 00:06:31,800 --> 00:06:35,150 We know that the derivative of sine is cosign. 74 00:06:35,230 --> 00:06:41,740 So we'll get cosign of 2 x but remember chain rule tells us that we have to multiply that by the derivative 75 00:06:41,740 --> 00:06:46,780 of the inside function inside function is to X the derivative of which is 2. 76 00:06:46,840 --> 00:06:56,970 So our derivative will be to cosign of 2 x times the X and we take the integral of d to find V. 77 00:06:57,000 --> 00:06:58,690 And again we've already done this. 78 00:06:58,690 --> 00:07:00,130 We did the same thing up here. 79 00:07:00,250 --> 00:07:06,100 We know that the integral is one seventh key to the 7 x. 80 00:07:06,130 --> 00:07:10,420 So now that we've assigned these values we can go ahead and plug in again. 81 00:07:10,420 --> 00:07:16,420 So what's going to be interesting about this we'll go ahead and rewrite the integral over here on our 82 00:07:16,420 --> 00:07:17,480 left hand side. 83 00:07:17,650 --> 00:07:25,170 So cosign to x dx is equal to what we're going to be plugging in for. 84 00:07:25,390 --> 00:07:29,380 Is the is just the integral above's. 85 00:07:29,440 --> 00:07:33,980 Is 7 x cosign of 2 x plus 2 7. 86 00:07:34,450 --> 00:07:39,760 And now the substitution we're making is just for the integral here. 87 00:07:39,760 --> 00:07:40,690 This whole thing right. 88 00:07:40,690 --> 00:07:48,040 So we're going to plug in the values of u d u v and DVH just for the integral here. 89 00:07:48,190 --> 00:07:49,120 Using this formula. 90 00:07:49,120 --> 00:07:55,940 So we're going to basically make a substitution here like this so we'll say to sevenths and drop in 91 00:07:55,950 --> 00:07:56,590 parentheses. 92 00:07:56,620 --> 00:08:01,030 And now we're going to plug in UVM minus the integral of VDU. 93 00:08:01,330 --> 00:08:07,720 So you at times V will give us one seventh E to the 7 x. 94 00:08:07,870 --> 00:08:15,120 Sign of 2 x minus the integral of v d u. 95 00:08:15,250 --> 00:08:21,300 So what we hear at times c you notice we have the coefficient 2 and 1 7. 96 00:08:21,430 --> 00:08:22,560 So that's a 2 7. 97 00:08:22,630 --> 00:08:28,030 And we can actually pull that out in front of the integral here as a coefficient on the inside what 98 00:08:28,030 --> 00:08:30,770 we left with is the rest of the. 99 00:08:30,770 --> 00:08:41,800 Here you the 7 x and the rest of the D U which is cosigned 2 x 6 so times cosign of 2 x dx and then 100 00:08:41,800 --> 00:08:44,850 we can go ahead and close our privacy. 101 00:08:44,850 --> 00:08:51,120 We notice that we have to multiply 2 7 through this whole formula here. 102 00:08:51,160 --> 00:08:57,250 Now as you can see I just copy the left hand side here and the beginning of the right hand side and 103 00:08:57,250 --> 00:09:01,800 now we're going to be distributing this to sevenths across both of these values here. 104 00:09:02,020 --> 00:09:07,790 So we have two seven times one seventh which will give us two over forty nine. 105 00:09:07,930 --> 00:09:18,910 So two over forty nine times each of the seven x sign of two x and then we have two seven times and 106 00:09:18,910 --> 00:09:25,900 negative two servants will give us a negative for over forty nine times. 107 00:09:25,900 --> 00:09:36,630 The integral of each to the seven x cosign of 2 x x and here's why this works out so well. 108 00:09:36,790 --> 00:09:41,290 What you have to notice now you might be discouraged because you think we ended up right back where 109 00:09:41,290 --> 00:09:41,860 we started. 110 00:09:41,860 --> 00:09:47,650 We have even less of the integral of each of the seven x cosign of 2 x x which is exactly the integral 111 00:09:47,650 --> 00:09:52,540 that we started with so it seems like we've made no progress and we've almost made this more complicated 112 00:09:53,020 --> 00:09:59,320 when in fact because we have the same integral over here on the right side that we do on the left hand 113 00:09:59,320 --> 00:10:07,310 side we can go ahead and add this entire negative for 49 times the integral here. 114 00:10:07,330 --> 00:10:10,560 We can go ahead and add that over the left hand side. 115 00:10:10,570 --> 00:10:13,580 So imagine you've got four or forty nine here. 116 00:10:13,600 --> 00:10:18,170 Let's make a common denominator and multiply the left hand side over here. 117 00:10:18,330 --> 00:10:20,650 I 49 over forty. 118 00:10:20,650 --> 00:10:23,630 It's the same as multiplying by 1 right. 119 00:10:23,910 --> 00:10:31,150 And now we can add this entire value to both sides so when we add four forty nine times the integral 120 00:10:31,150 --> 00:10:37,220 here both sides the value over here on the right hand side will go away. 121 00:10:37,380 --> 00:10:43,200 This whole value will go away when we add it to both sides and on the left hand side here will be left 122 00:10:43,200 --> 00:10:55,500 with 49 plus 4 which gives us 53 over forty nine times the integral of each of the seven acts cosign 123 00:10:56,320 --> 00:11:10,940 of two x dx and on the right hand side we're just left with everything besides the integral. 124 00:11:10,960 --> 00:11:16,690 Now the last thing we need to do in order to solve for the integral right we're trying to find this 125 00:11:16,780 --> 00:11:18,140 value here right. 126 00:11:18,190 --> 00:11:23,460 The integral of either the 7 x cosign of 2 x x and we have that right here. 127 00:11:23,470 --> 00:11:31,270 So in order to solve for it all we have to do is divide both sides by 53 over 49. 128 00:11:31,300 --> 00:11:35,080 This coefficient goes away and we've solved four integral. 129 00:11:35,080 --> 00:11:42,560 So instead of dividing both sides by 53 over 49 we'll go ahead and multiply both sides. 130 00:11:42,700 --> 00:11:48,470 I 49 over 53 to cancel out what we have here on the left hand side 131 00:11:53,010 --> 00:11:59,670 what we're left with here is 49 over 53 times. 132 00:12:00,490 --> 00:12:02,050 1 7. 133 00:12:02,590 --> 00:12:04,810 And for our second term here 134 00:12:08,060 --> 00:12:16,020 49 over 53 times to over forty nine either the 7 x signed x 135 00:12:18,790 --> 00:12:25,480 our final step is to simplify the seven in the denominator here will cancel and we'll just be left with 136 00:12:25,540 --> 00:12:27,600 seven here in the numerator. 137 00:12:27,820 --> 00:12:40,450 So we'll get seven over fifty three E to the seven x times cosign of 2 x plus here obviously we'll get 138 00:12:40,450 --> 00:12:50,920 the 49ers to cancel and we'll be left with two over the three E to the 7 x times sign of 2 x. 139 00:12:51,280 --> 00:12:54,680 And at this point we can leave our answer this way. 140 00:12:54,730 --> 00:13:02,560 Or you can choose to factor out one over 53 times each to the 7 x so if we factor this out to try to 141 00:13:02,560 --> 00:13:10,180 simplify our answer 153 times each of the 7 x what will be left with in our first term here is just 142 00:13:10,270 --> 00:13:21,580 7 cosign of 2 x plus 2 times sign of 2 x. 143 00:13:21,700 --> 00:13:27,250 And at this point we hadn't included it earlier because we were doing a lot of finagling and moving 144 00:13:27,250 --> 00:13:28,120 things around. 145 00:13:28,210 --> 00:13:33,490 But at this point we have to remember because we did take an integral to go ahead and add plus C to 146 00:13:33,490 --> 00:13:35,910 account for our constant integration. 16346