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These are the user uploaded subtitles that are being translated: 1 00:00:00,150 --> 00:00:05,430 Today we're going to be talking about how to use you substitution and then integration by parts to evaluate 2 00:00:05,460 --> 00:00:06,310 an integral. 3 00:00:06,390 --> 00:00:12,480 And in this particular case we've been given the definite integral of theta cube times cosign of theta 4 00:00:12,480 --> 00:00:16,910 squared d theta and we'll be evaluating that on the limits of integration. 5 00:00:16,950 --> 00:00:20,450 Square root of pi over to the square root of pi. 6 00:00:20,490 --> 00:00:24,380 I've gone ahead and written the integration by parts formula over here on the right because we're going 7 00:00:24,380 --> 00:00:25,940 to need that in a little bit. 8 00:00:26,160 --> 00:00:32,580 But this problem asks us to make a substitution first and then use integration by parts once we've simplified 9 00:00:32,910 --> 00:00:35,100 the functions inside of our integral. 10 00:00:35,100 --> 00:00:41,430 So we need to identify the substitution that we're going to make in this function and it should be at 11 00:00:41,430 --> 00:00:48,240 least a place to start with that you try to substitute for theta squared because having theta squared 12 00:00:48,600 --> 00:00:53,370 inside of our trigonometric cosigned function here means that we can have a little bit of trouble with 13 00:00:53,370 --> 00:00:53,640 this. 14 00:00:53,640 --> 00:00:57,880 We want to simplify what's inside our trigonometric function as much as we can. 15 00:00:57,900 --> 00:01:02,040 So let's make a substitution for theta squared and see where that gets us. 16 00:01:02,160 --> 00:01:07,560 In this case normally I would use you as the variable for substitution because it's use substitution 17 00:01:07,560 --> 00:01:08,580 most commonly. 18 00:01:08,580 --> 00:01:15,020 But we're going to need to have you available to us later for our integration by parts formula. 19 00:01:15,030 --> 00:01:21,090 So let's actually go ahead and use like X substitution and make a substitution for x in terms of theta 20 00:01:21,420 --> 00:01:24,360 instead of you so that we don't get confused later. 21 00:01:24,360 --> 00:01:30,320 So we'll call X theta squared and that will be the substitution that we're going to make. 22 00:01:30,390 --> 00:01:35,760 Remember that when you're dealing with you substitution you then take the derivative of what you just 23 00:01:35,760 --> 00:01:41,640 identified as X here and we'll call that dx so DX will be 2 theta. 24 00:01:41,820 --> 00:01:44,460 And then of course we add to this D theta. 25 00:01:44,460 --> 00:01:51,180 And now we want to solve for d theta we'll divide both sides by two theta and we'll get the theta is 26 00:01:51,210 --> 00:01:56,270 equal to dx over to theta. 27 00:01:56,640 --> 00:02:03,870 So now we can plug these values back into our integral so we'll end up with the integral with these 28 00:02:03,870 --> 00:02:05,780 limits of integration. 29 00:02:05,940 --> 00:02:12,120 And essentially what we have here is we have theta cubed and then we have cosign. 30 00:02:12,140 --> 00:02:14,390 Remember we substituted X for theta squared. 31 00:02:14,400 --> 00:02:22,340 So now we just have cosign of X and detailer we know is D X divided by two theta. 32 00:02:22,350 --> 00:02:27,100 The first thing we need to realize here is that we can cancel a theta from the numerator and denominator. 33 00:02:27,270 --> 00:02:30,450 So we'll go ahead and cancel out theta here. 34 00:02:30,660 --> 00:02:37,080 This will leave us with just theta squared in the numerator so we can go ahead and call this here theta 35 00:02:37,170 --> 00:02:38,490 squared instead. 36 00:02:38,700 --> 00:02:40,340 But remember that data squared. 37 00:02:40,380 --> 00:02:42,570 We set equal to x. 38 00:02:42,600 --> 00:02:48,450 So instead of theta squared here we will actually have is just x. 39 00:02:48,480 --> 00:02:54,210 And keep in mind that we have this this one half here the two in the denominator that we can move outside 40 00:02:54,330 --> 00:02:55,730 of the integral. 41 00:02:55,740 --> 00:03:01,620 So what we end up with before we start our integration by parts as we move the one half out in front 42 00:03:01,620 --> 00:03:02,520 of the integral. 43 00:03:02,640 --> 00:03:12,500 So we have one half our limits of integration and then what we're left with is just x cosign of x x 44 00:03:13,020 --> 00:03:19,110 and now we're in a great position to start integration by parts because we have two components inside 45 00:03:19,110 --> 00:03:20,200 of our integral. 46 00:03:20,250 --> 00:03:25,900 We have X and we have cosign of X so we need to identify you and DVH. 47 00:03:26,220 --> 00:03:32,430 Well in this case it's really obvious what you should pick for you because remember we want to pick 48 00:03:32,430 --> 00:03:39,690 something for you that will become simpler when we take its derivative d u and the derivative of X is 49 00:03:39,690 --> 00:03:49,620 just one which is much simpler than X so if we set you equal to x that means that divi has to be everything 50 00:03:49,620 --> 00:03:50,900 else inside of our integral. 51 00:03:50,970 --> 00:03:58,990 So divi automatically becomes everything else which is cosigned XTi X. So cosign x x. 52 00:03:59,010 --> 00:04:02,150 Now we take the derivative of you to get d u. 53 00:04:02,160 --> 00:04:05,650 So do you as equal to the derivative of X is just one. 54 00:04:05,850 --> 00:04:12,220 So we would have one DX which is just dx and then we take the integral of divi to get V. 55 00:04:12,270 --> 00:04:17,510 So we get a V equals the integral of cosign is sine. 56 00:04:17,610 --> 00:04:23,670 So we get sign of X and now we have all four of our components we can plug these into our integration 57 00:04:23,670 --> 00:04:24,740 by parts formula. 58 00:04:24,900 --> 00:04:27,540 So remember our integration by parts formula. 59 00:04:27,540 --> 00:04:35,670 Essentially we're going to be replacing this integral here this entire piece with what we plug in to 60 00:04:35,670 --> 00:04:38,680 the right hand side of our integration by parts formula over here. 61 00:04:38,880 --> 00:04:42,010 So we still need to include this one half that's out in front. 62 00:04:42,270 --> 00:04:48,030 So we'll get one half and then we'll multiply this by the right hand sidebar formula which is you Times 63 00:04:48,030 --> 00:04:51,160 V minus the integral of V times d u. 64 00:04:51,360 --> 00:04:52,710 And we have those components here. 65 00:04:52,710 --> 00:04:57,190 So we just plug in u and v US X and V is sign of x. 66 00:04:57,240 --> 00:05:05,930 So you Times V is just x sign of X and then we subtract the integral and again we're just following 67 00:05:05,930 --> 00:05:11,070 here our integration by parts formula V and D U. 68 00:05:11,090 --> 00:05:14,820 So we have V as sign of X and D u as DIAK. 69 00:05:14,840 --> 00:05:22,760 So we're just left with sign of x dx and remember that because we're dealing with a definite integral 70 00:05:22,760 --> 00:05:27,980 here we have these limits of integration these limits of integration apply to everything inside these 71 00:05:28,280 --> 00:05:30,130 parentheses here these big brackets. 72 00:05:30,170 --> 00:05:34,020 So we have to remember both this integral here and the sine x. 73 00:05:34,070 --> 00:05:42,850 So we have to remember to evaluate at the square of Pi over to 2 the square root of pi. 74 00:05:42,860 --> 00:05:48,050 Now thanks to integration by parts we have an integral that's really manageable we just have the integral 75 00:05:48,080 --> 00:05:49,230 of sign of X. 76 00:05:49,250 --> 00:05:54,540 Well we know that the integral of sign of x is negative cosign of x. 77 00:05:54,560 --> 00:06:03,560 So what we end up with here is 1 1/2 times x sign of X and because we're going to get minus and then 78 00:06:03,650 --> 00:06:12,530 negative cosign of x that will be plus cosign of X and that is our integral evaluated. 79 00:06:12,830 --> 00:06:20,090 And of course we have our limits of integration here that we'll have to evaluate this function at. 80 00:06:20,300 --> 00:06:25,700 But before we do that now that we've finished taking the integral of everything we have no integrals 81 00:06:25,700 --> 00:06:29,930 left we need to go ahead and back substitute for X. 82 00:06:29,930 --> 00:06:33,890 Remember we said that X was equal to theta squared. 83 00:06:33,890 --> 00:06:39,020 Well now that all our integrals are gone it's time to put X back in terms of theta. 84 00:06:39,020 --> 00:06:53,020 So what we'll have is one halftimes theta squared times sign of theta squared plus cosign of theta squared. 85 00:06:53,270 --> 00:06:59,360 And now we want to evaluate on these limits of integration and where we plugging these limits of integration 86 00:06:59,450 --> 00:07:01,720 in for theta. 87 00:07:01,730 --> 00:07:07,190 Remember when you're evaluating at limits of integration like this you always plug in this top number 88 00:07:07,190 --> 00:07:11,570 first and then subtract whatever you get when you plug in the bottom number. 89 00:07:11,570 --> 00:07:14,620 So we'll plug in the top number which is square of Pi. 90 00:07:14,620 --> 00:07:20,280 First and what we'll get is the square root of pi squared. 91 00:07:20,590 --> 00:07:23,750 Well a square root squared takes away that square root. 92 00:07:23,750 --> 00:07:26,670 So the square root of pi squared is just pi. 93 00:07:26,860 --> 00:07:31,480 So we end up with pi times sign of data squared. 94 00:07:31,480 --> 00:07:35,070 And again the squared of Pi squared is just pi. 95 00:07:35,080 --> 00:07:41,170 So we end up with sign of pi plus cosign obviously of Pi again. 96 00:07:41,380 --> 00:07:45,520 And then we'll subtract whatever we get when we plug in the lower limit of integration. 97 00:07:45,520 --> 00:07:50,990 So we need a parentheses here so that this negative sign applies to everything we've got in here. 98 00:07:51,280 --> 00:07:58,450 So we'll get the square root of pi over two squared which will just give us of course PI over 2 and 99 00:07:58,450 --> 00:08:07,620 then we'll have sign of Pi over to plus cosign of Pi over to. 100 00:08:07,620 --> 00:08:14,030 So now we just evaluate We have 1 1/2 times sign of Pi right here. 101 00:08:14,040 --> 00:08:16,040 Sign of Pi is zero. 102 00:08:16,080 --> 00:08:19,620 So zero times pi will obviously just give us still 0. 103 00:08:19,620 --> 00:08:23,890 This is going to go away completely cosign of Pi is negative 1. 104 00:08:23,900 --> 00:08:31,620 So we've got a negative one there than we have here minus sign of Pi over 2 is 1 so 1 times pi over 105 00:08:31,620 --> 00:08:34,200 2 is Pi over 2. 106 00:08:34,500 --> 00:08:38,260 And then cosign of Pi over two is zero. 107 00:08:38,310 --> 00:08:40,410 So we just have a minus zero. 108 00:08:40,410 --> 00:08:42,130 We don't need to account for that. 109 00:08:42,300 --> 00:08:45,140 So we just have negative 1 minus PI over 2. 110 00:08:45,330 --> 00:08:54,080 Now if we distribute the 1 half we'll get negative 1 half minus PI over 4 and that's it. 111 00:08:54,090 --> 00:08:55,210 That's our final answer. 12171