1 00:00:00,000 --> 00:00:00,490 2 00:00:00,490 --> 00:00:05,860 In the last video, I showed the Laplace transform of t, or 3 00:00:05,860 --> 00:00:09,410 we could view that as t the first power, is equal to 1/s 4 00:00:09,410 --> 00:00:13,410 squared, if we assume that s is greater than 0. 5 00:00:13,410 --> 00:00:16,100 In this video, we're going to see if we can generalize this 6 00:00:16,100 --> 00:00:20,270 by trying to figure out the Laplace transform of t to the 7 00:00:20,269 --> 00:00:24,660 n, where n is any integer power greater than 0, so n is 8 00:00:24,660 --> 00:00:27,574 any positive integer greater than 0. 9 00:00:27,574 --> 00:00:29,119 So let's try it out. 10 00:00:29,120 --> 00:00:33,060 So we know from our definition of the Laplace transform that 11 00:00:33,060 --> 00:00:38,929 the Laplace transform of t to the n is equal to the integral 12 00:00:38,929 --> 00:00:45,079 from 0 to infinity of our function-- well, let me write 13 00:00:45,079 --> 00:00:48,269 t to the n-- times, and this is just the definition of the 14 00:00:48,270 --> 00:00:54,420 transform, e to the minus st, dt. 15 00:00:54,420 --> 00:00:57,280 And similar to when we figured out this Laplace transform, 16 00:00:57,280 --> 00:00:59,570 your intuition might be that, hey, we should use integration 17 00:00:59,570 --> 00:01:02,240 by parts, and I showed it in the last video. 18 00:01:02,240 --> 00:01:04,769 I always forget it, but I just recorded it, so I do happen to 19 00:01:04,769 --> 00:01:05,859 remember it. 20 00:01:05,859 --> 00:01:08,000 So the integration by parts just tells us that the 21 00:01:08,000 --> 00:01:15,920 integral of uv prime is equal to uv minus the integral of-- 22 00:01:15,920 --> 00:01:21,060 I view this as kind of the swap-- so u prime v. 23 00:01:21,060 --> 00:01:23,174 So this is just our integration by parts formula. 24 00:01:23,174 --> 00:01:26,719 If you ever forget it, you can derive it in about 30 seconds 25 00:01:26,719 --> 00:01:27,379 from the product rule. 26 00:01:27,379 --> 00:01:29,629 And I did it in the last video because I hadn't used it for 27 00:01:29,629 --> 00:01:31,329 awhile, so I had to rederive it. 28 00:01:31,329 --> 00:01:32,459 So let's apply it here. 29 00:01:32,459 --> 00:01:34,849 So what do we want to make our v prime? 30 00:01:34,849 --> 00:01:37,530 It's always good to use the exponential function, because 31 00:01:37,530 --> 00:01:40,140 that's easy to take the antiderivative of. 32 00:01:40,140 --> 00:01:44,540 So this is our v prime, in which case our v is just the 33 00:01:44,540 --> 00:01:46,130 antiderivative of that. 34 00:01:46,129 --> 00:01:51,489 So it's e to the minus st over minus s. 35 00:01:51,489 --> 00:01:54,549 If we take the derivative of this, minus s divided by minus 36 00:01:54,549 --> 00:01:56,659 s cancels out, and you just get that. 37 00:01:56,659 --> 00:02:02,079 And then if we make our u-- let me pick a good color here. 38 00:02:02,079 --> 00:02:05,939 If we make this equal to our u, what's our u prime? 39 00:02:05,939 --> 00:02:12,699 u prime is just going to be n times t to the n minus 1. 40 00:02:12,699 --> 00:02:13,280 Fair enough. 41 00:02:13,280 --> 00:02:16,009 So let's apply the integration by parts. 42 00:02:16,009 --> 00:02:19,329 So this is going to be equal to uv. 43 00:02:19,330 --> 00:02:25,730 u, I'll use this t to the n, so u is t to the n, that's our 44 00:02:25,729 --> 00:02:31,879 u, times v, which is e-- let me write this down-- so it's 45 00:02:31,879 --> 00:02:35,074 minus-- there's a minus sign there, so we put the minus. 46 00:02:35,074 --> 00:02:36,509 Let me do it in that color. 47 00:02:36,509 --> 00:02:39,919 48 00:02:39,919 --> 00:02:42,554 I'm just rewriting this. e to the minus st/s. 49 00:02:42,555 --> 00:02:45,930 50 00:02:45,930 --> 00:02:47,724 So that's the uv term right there. 51 00:02:47,724 --> 00:02:50,180 Let me make that clear. 52 00:02:50,180 --> 00:02:51,819 And let me pick a good color here. 53 00:02:51,819 --> 00:02:57,090 So this term right here is this term right here. 54 00:02:57,090 --> 00:03:00,039 And, of course, I'm going to have to evaluate this from 0 55 00:03:00,039 --> 00:03:02,614 to infinity, so let me write that: 0 to infinity. 56 00:03:02,615 --> 00:03:04,420 I could put a little bracket there or something, but you 57 00:03:04,419 --> 00:03:05,989 know we're going to have to evaluate that. 58 00:03:05,990 --> 00:03:07,560 And then from that, we're going to have to 59 00:03:07,560 --> 00:03:09,460 subtract the integral. 60 00:03:09,460 --> 00:03:11,409 And let me not forget our boundaries. 61 00:03:11,409 --> 00:03:20,519 0 to infinity of u prime is n times t to the n minus 1-- 62 00:03:20,520 --> 00:03:26,450 that's our u prime-- times v times minus-- so let me put 63 00:03:26,449 --> 00:03:34,219 this minus out here-- so minus e to the minus st/s. 64 00:03:34,219 --> 00:03:36,810 65 00:03:36,810 --> 00:03:37,469 And then all of that. 66 00:03:37,469 --> 00:03:42,699 Of course, we have our dt, and you have a minus minus. 67 00:03:42,699 --> 00:03:44,539 These things become pluses. 68 00:03:44,539 --> 00:03:47,429 Let's see if we can simplify this a little bit. 69 00:03:47,430 --> 00:03:52,760 So we get our Laplace transform of t to the n is 70 00:03:52,759 --> 00:03:57,099 equal to this evaluated at infinity and evaluated at 0. 71 00:03:57,099 --> 00:04:00,799 So when you've evaluate-- what's the limit of this as t 72 00:04:00,800 --> 00:04:02,430 approaches infinity? 73 00:04:02,430 --> 00:04:04,900 As t approaches infinity, this term, you might say, oh, this 74 00:04:04,900 --> 00:04:05,689 becomes really big. 75 00:04:05,689 --> 00:04:07,609 And I went over this in the last video. 76 00:04:07,610 --> 00:04:09,800 But this term overpowers it, because you're going to have e 77 00:04:09,800 --> 00:04:11,980 to the minus infinity, if we assume that s is 78 00:04:11,979 --> 00:04:13,469 greater than 0. 79 00:04:13,469 --> 00:04:15,960 So if s is greater than 0, this term is going to win out 80 00:04:15,960 --> 00:04:17,970 and go to 0 much faster than this term is 81 00:04:17,970 --> 00:04:18,750 going to go to infinity. 82 00:04:18,750 --> 00:04:22,769 So when you evaluate it at infinity, when you evaluate 83 00:04:22,769 --> 00:04:25,729 this at infinity, you're going to get 0. 84 00:04:25,730 --> 00:04:30,020 And then you're going to subtract this evaluated at 0. 85 00:04:30,019 --> 00:04:35,599 This evaluated at 0, when it's evaluated at 0 is just minus 0 86 00:04:35,600 --> 00:04:40,960 to the n times e to the minus s times 0/s Well, this 87 00:04:40,959 --> 00:04:42,979 becomes 0 as well. 88 00:04:42,980 --> 00:04:46,220 So this whole term evaluated from 0 to infinity is all 0, 89 00:04:46,220 --> 00:04:48,720 which is a nice, convenient thing for us. 90 00:04:48,720 --> 00:04:53,100 And then we're going to have this next term right there. 91 00:04:53,100 --> 00:04:55,470 So let's take out the constant terms. This n 92 00:04:55,470 --> 00:04:57,150 and this s are constant. 93 00:04:57,149 --> 00:04:59,334 They are constant with respect to t. 94 00:04:59,334 --> 00:05:06,989 So you have plus n/s times the integral from 0 to infinity of 95 00:05:06,990 --> 00:05:16,740 t to the n minus 1 times e to the minus st, dt. 96 00:05:16,740 --> 00:05:20,670 Now, this should look reasonably familiar to you. 97 00:05:20,670 --> 00:05:23,170 98 00:05:23,170 --> 00:05:26,009 What's the definition of the Laplace transform? 99 00:05:26,009 --> 00:05:31,289 The Laplace transform of any function is equal to the 100 00:05:31,290 --> 00:05:36,750 integral from 0 to infinity of that function times e to the 101 00:05:36,750 --> 00:05:39,029 minus st, dt. 102 00:05:39,029 --> 00:05:41,389 Well, when we have an e to the minus st, dt, we're taking the 103 00:05:41,389 --> 00:05:44,500 integral from 0 to infinity, so this whole integral is 104 00:05:44,500 --> 00:05:48,310 equal to the Laplace transform of this, of t 105 00:05:48,310 --> 00:05:50,930 to the n minus 1. 106 00:05:50,930 --> 00:05:54,530 So just that easily, because this term went to 0, we've 107 00:05:54,529 --> 00:05:56,129 simplified things. 108 00:05:56,129 --> 00:06:04,110 We get the Laplace transform of t to the n is equal to-- 109 00:06:04,110 --> 00:06:08,830 this is all 0-- it's equal to n/s-- that's right there-- 110 00:06:08,829 --> 00:06:11,639 times this integral right here, which we just figured 111 00:06:11,639 --> 00:06:17,000 out was the Laplace transform of t to the n minus 1. 112 00:06:17,000 --> 00:06:19,069 Well, this is a nice, neat simplification. 113 00:06:19,069 --> 00:06:22,089 We can now figure out the Laplace transform of a higher 114 00:06:22,089 --> 00:06:24,644 power in terms of the one power lower that, but it still 115 00:06:24,644 --> 00:06:25,974 doesn't give me a generalized formula. 116 00:06:25,975 --> 00:06:31,060 So let's see if we can use this with this information to 117 00:06:31,060 --> 00:06:32,480 get a generalized formula. 118 00:06:32,480 --> 00:06:36,790 So the Laplace transform of just t-- so let me write that 119 00:06:36,790 --> 00:06:39,030 down; I wrote that at the beginning of the problem. 120 00:06:39,029 --> 00:06:42,269 We get the Laplace transform, I could write this as t to the 121 00:06:42,269 --> 00:06:47,509 1, which is just t, is equal to 1/s squared, where s is 122 00:06:47,509 --> 00:06:52,050 greater than 0. 123 00:06:52,050 --> 00:06:54,420 Now, what happens if we take the Laplace 124 00:06:54,420 --> 00:06:59,480 transform of t squared? 125 00:06:59,480 --> 00:07:01,160 Well, we can just use this formula up here. 126 00:07:01,160 --> 00:07:05,460 The Laplace transform of t squared is equal to 2/s times 127 00:07:05,459 --> 00:07:10,759 the Laplace transform of t, of just t to the 1, right? 128 00:07:10,759 --> 00:07:11,579 2 minus 1. 129 00:07:11,579 --> 00:07:16,550 So times the Laplace transform of t to the 1. 130 00:07:16,550 --> 00:07:18,520 Well, t, we know what that is. 131 00:07:18,519 --> 00:07:24,509 This is equal to 2/s times this, times 1/s squared, which 132 00:07:24,509 --> 00:07:27,519 is equal to 2/s to the third. 133 00:07:27,519 --> 00:07:28,310 Interesting. 134 00:07:28,310 --> 00:07:29,819 Let's see if we can do another one. 135 00:07:29,819 --> 00:07:33,949 What is-- I'll do it in the dark blue-- the Laplace 136 00:07:33,949 --> 00:07:37,659 transform of t to the third? 137 00:07:37,660 --> 00:07:39,370 Well, we just use this formula up here. 138 00:07:39,370 --> 00:07:40,199 It's n/s. 139 00:07:40,199 --> 00:07:41,879 In this case, n is 3. 140 00:07:41,879 --> 00:07:47,629 So it's 3/s times the Laplace transform of t to the n minus 141 00:07:47,629 --> 00:07:49,569 1, so t squared. 142 00:07:49,569 --> 00:07:52,120 143 00:07:52,120 --> 00:07:54,769 We know what the Laplace transform of this one was. 144 00:07:54,769 --> 00:07:56,399 This is just this right there. 145 00:07:56,399 --> 00:08:01,979 So it's equal to 3/s times this thing. 146 00:08:01,980 --> 00:08:03,610 And I'm going to actually write it this way, because I 147 00:08:03,610 --> 00:08:04,389 think it's interesting. 148 00:08:04,389 --> 00:08:05,560 So I'll write the numerator. 149 00:08:05,560 --> 00:08:13,310 Times 2 times 1/s over s squared, which is-- we could 150 00:08:13,310 --> 00:08:18,019 write it as 3 factorial over-- what is this? s 151 00:08:18,019 --> 00:08:19,269 to the fourth power. 152 00:08:19,269 --> 00:08:22,339 153 00:08:22,339 --> 00:08:23,339 Let's do another one. 154 00:08:23,339 --> 00:08:26,939 And I think you already are getting the idea of 155 00:08:26,939 --> 00:08:28,469 what's going on. 156 00:08:28,470 --> 00:08:33,750 The Laplace transform of t to the fourth power is what? 157 00:08:33,750 --> 00:08:40,269 It's equal to 4/s times the Laplace transform of t the 158 00:08:40,269 --> 00:08:41,710 third power. 159 00:08:41,710 --> 00:08:43,168 And that's just 4/s times this. 160 00:08:43,168 --> 00:08:50,149 So it's 4/s times 3 factorial over s to the fourth. 161 00:08:50,149 --> 00:08:54,789 So now 4 times 3 factorial, that's just 4 factorial over s 162 00:08:54,789 --> 00:08:56,120 to the fifth. 163 00:08:56,120 --> 00:08:59,490 And so you can just get the general principle. 164 00:08:59,490 --> 00:09:01,100 And we can prove this by induction. 165 00:09:01,100 --> 00:09:03,235 It's almost trivial based on what we've already done. 166 00:09:03,235 --> 00:09:14,710 That the Laplace transform of t to the n is equal to n 167 00:09:14,710 --> 00:09:21,740 factorial over s to the n plus 1. 168 00:09:21,740 --> 00:09:25,250 We proved it directly for this base case right here. 169 00:09:25,250 --> 00:09:29,289 This is 1 factorial over s to the 1 plus 1. 170 00:09:29,289 --> 00:09:33,610 And then if we know it's true for this, we know it's going 171 00:09:33,610 --> 00:09:34,940 to be true for the next increment. 172 00:09:34,940 --> 00:09:37,850 So induction proof is almost obvious, but you can even see 173 00:09:37,850 --> 00:09:38,970 it based on this. 174 00:09:38,970 --> 00:09:41,330 If you have to figure out the Laplace transform of t to the 175 00:09:41,330 --> 00:09:43,500 tenth, you could just keep doing this over and over 176 00:09:43,500 --> 00:09:46,000 again, but I think you see the pattern pretty clearly. 177 00:09:46,000 --> 00:09:48,940 So anyway, I thought that was a neat problem in and of 178 00:09:48,940 --> 00:09:52,730 itself, outside of the fact, it'll be useful when we figure 179 00:09:52,730 --> 00:09:55,620 out inverse and Laplace transforms. But this is a 180 00:09:55,620 --> 00:09:56,639 pretty neat result. 181 00:09:56,639 --> 00:10:00,730 The Laplace transform of t to the n, where n is some integer 182 00:10:00,730 --> 00:10:04,480 greater than 0 is equal to n factorial over s to the n plus 183 00:10:04,480 --> 00:10:06,690 1, where s is also greater than 0. 184 00:10:06,690 --> 00:10:09,290 That was an assumption we had to make early on when we took 185 00:10:09,289 --> 00:10:12,250 our limits as t approaches infinity. 186 00:10:12,250 --> 00:10:15,389 Anyway, hopefully you found that useful. 187 00:10:15,389 --> 00:10:15,865