[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.08,0:00:05.27,Default,,0000,0000,0000,,When we're integrating, we need\Nto be able to recognize standard
Dialogue: 0,0:00:05.27,0:00:09.08,Default,,0000,0000,0000,,forms and use them immediately.\NThere are also standard methods
Dialogue: 0,0:00:09.08,0:00:12.89,Default,,0000,0000,0000,,for integration, but for now\Nwe're going to concentrate on
Dialogue: 0,0:00:12.89,0:00:16.32,Default,,0000,0000,0000,,the standard forms of\Nintegration. The first one we're
Dialogue: 0,0:00:16.32,0:00:22.04,Default,,0000,0000,0000,,going to have a look at is the\Nintegral of X to the N with
Dialogue: 0,0:00:22.04,0:00:23.18,Default,,0000,0000,0000,,respect to X.
Dialogue: 0,0:00:24.44,0:00:31.58,Default,,0000,0000,0000,,To do this? We take X to the\Npower N plus one, so we've added
Dialogue: 0,0:00:31.58,0:00:37.13,Default,,0000,0000,0000,,one to the index and then we\Ndivide by the new index.
Dialogue: 0,0:00:37.68,0:00:43.80,Default,,0000,0000,0000,,We must remember to add on a\Nconstant of integration, see.
Dialogue: 0,0:00:45.01,0:00:51.96,Default,,0000,0000,0000,,N can be any number,\Nwhole number or fraction. A
Dialogue: 0,0:00:51.96,0:00:58.41,Default,,0000,0000,0000,,decimal except. It cannot be\Nminus one. If it were minus one,
Dialogue: 0,0:00:58.41,0:01:04.31,Default,,0000,0000,0000,,we have minus one, plus one\Nwould be 0 and we're not allowed
Dialogue: 0,0:01:04.31,0:01:09.76,Default,,0000,0000,0000,,to divide by zero. So in this\Nparticular case an does not
Dialogue: 0,0:01:09.76,0:01:11.12,Default,,0000,0000,0000,,equal minus one.
Dialogue: 0,0:01:11.69,0:01:14.64,Default,,0000,0000,0000,,An extension
Dialogue: 0,0:01:14.64,0:01:22.05,Default,,0000,0000,0000,,to this. Would\Nbe a X plus B where
Dialogue: 0,0:01:22.05,0:01:28.83,Default,,0000,0000,0000,,again A&B are just numbers, just\Nconstants again raised to the
Dialogue: 0,0:01:28.83,0:01:35.31,Default,,0000,0000,0000,,power N. AX plus B\Nis very little different to X,
Dialogue: 0,0:01:35.31,0:01:42.54,Default,,0000,0000,0000,,so again we had one to the index\Nand divide by the new index. But
Dialogue: 0,0:01:42.54,0:01:47.84,Default,,0000,0000,0000,,now because we're multiplying\Nour variable X by a, we've got
Dialogue: 0,0:01:47.84,0:01:53.14,Default,,0000,0000,0000,,to divide by it, because in\Nessence we are reversing the
Dialogue: 0,0:01:53.14,0:01:54.59,Default,,0000,0000,0000,,process of Differentiation.
Dialogue: 0,0:01:55.56,0:02:01.46,Default,,0000,0000,0000,,And again we must have source of\Nconstant of integration, see and
Dialogue: 0,0:02:01.46,0:02:06.38,Default,,0000,0000,0000,,again we cannot have N being\Nequal to negative one.
Dialogue: 0,0:02:07.77,0:02:14.22,Default,,0000,0000,0000,,Next standard form is the\Nintegral of one over X. Saw
Dialogue: 0,0:02:14.22,0:02:20.66,Default,,0000,0000,0000,,this when we did Differentiation\Nas we saw that the derivative
Dialogue: 0,0:02:20.66,0:02:27.69,Default,,0000,0000,0000,,of the log of X was\None over X. So if we
Dialogue: 0,0:02:27.69,0:02:33.55,Default,,0000,0000,0000,,reverse that process the\Nintegral of one over X must
Dialogue: 0,0:02:33.55,0:02:36.48,Default,,0000,0000,0000,,be the log of X.
Dialogue: 0,0:02:36.50,0:02:40.26,Default,,0000,0000,0000,,And we must include the constant\Nof integration, see.
Dialogue: 0,0:02:40.99,0:02:47.97,Default,,0000,0000,0000,,If we have one over\Na X plus BTX.
Dialogue: 0,0:02:48.50,0:02:51.56,Default,,0000,0000,0000,,And again, A&B are constants.
Dialogue: 0,0:02:52.60,0:03:00.43,Default,,0000,0000,0000,,Then what we must have is one\Nover A the log of AX plus
Dialogue: 0,0:03:00.43,0:03:05.46,Default,,0000,0000,0000,,B. Again, we must include a\Nconstant of integration.
Dialogue: 0,0:03:06.10,0:03:13.12,Default,,0000,0000,0000,,Let's just have a look at\Nan example of this. The integral
Dialogue: 0,0:03:13.12,0:03:16.63,Default,,0000,0000,0000,,of one over 2 - 3
Dialogue: 0,0:03:16.63,0:03:23.29,Default,,0000,0000,0000,,XDX. Here we can see that\Nthe A is negative three, so
Dialogue: 0,0:03:23.29,0:03:29.67,Default,,0000,0000,0000,,that's going to be minus one\Nover three log of the modulus of
Dialogue: 0,0:03:29.67,0:03:35.08,Default,,0000,0000,0000,,2 - 3 X plus. Again, the\Nconstant of integration, see.
Dialogue: 0,0:03:35.96,0:03:43.85,Default,,0000,0000,0000,,Next standard form the integral\Nof E to the X
Dialogue: 0,0:03:43.85,0:03:47.01,Default,,0000,0000,0000,,with respect to X.
Dialogue: 0,0:03:48.20,0:03:53.67,Default,,0000,0000,0000,,We know that E to the X is its\Nown derivative, so when we
Dialogue: 0,0:03:53.67,0:03:58.76,Default,,0000,0000,0000,,integrate it, we must get\Nexactly the same form E to the X
Dialogue: 0,0:03:58.76,0:04:06.25,Default,,0000,0000,0000,,again. What if we've\Ngot the to the MXDX?
Dialogue: 0,0:04:07.54,0:04:12.48,Default,,0000,0000,0000,,Again. Thinking as integration\Nas reversing differentiation. We
Dialogue: 0,0:04:12.48,0:04:18.06,Default,,0000,0000,0000,,need one over ME to the MX\Nbecause if we were to
Dialogue: 0,0:04:18.06,0:04:24.58,Default,,0000,0000,0000,,differentiate E to the MX, we'd\Nget me to the MX and we want
Dialogue: 0,0:04:24.58,0:04:27.83,Default,,0000,0000,0000,,the ends to cancel out and again
Dialogue: 0,0:04:27.83,0:04:34.66,Default,,0000,0000,0000,,plus C. Let's just have a\Nlook at an example of that. The
Dialogue: 0,0:04:34.66,0:04:37.91,Default,,0000,0000,0000,,integral of E to the four XDX.
Dialogue: 0,0:04:38.64,0:04:46.57,Default,,0000,0000,0000,,And that will be 1/4 E\Nto the 4X plus a constant
Dialogue: 0,0:04:46.57,0:04:48.56,Default,,0000,0000,0000,,of integration, see.
Dialogue: 0,0:04:50.68,0:04:56.59,Default,,0000,0000,0000,,Will move on now and look at the\Ntrig functions and look at the
Dialogue: 0,0:04:56.59,0:04:57.85,Default,,0000,0000,0000,,standard integrals associated
Dialogue: 0,0:04:57.85,0:05:04.66,Default,,0000,0000,0000,,with those. So the integral\Nof Cos X DX.
Dialogue: 0,0:05:06.29,0:05:12.34,Default,,0000,0000,0000,,The derivative of sine X is\Ncause eggs, so if we reverse
Dialogue: 0,0:05:12.34,0:05:18.39,Default,,0000,0000,0000,,that, the integral of cars must\Nbe signed X again plus a
Dialogue: 0,0:05:18.39,0:05:19.90,Default,,0000,0000,0000,,constant of integration.
Dialogue: 0,0:05:20.96,0:05:24.94,Default,,0000,0000,0000,,If we integrate cause
Dialogue: 0,0:05:24.94,0:05:30.71,Default,,0000,0000,0000,,NX DX. Well, we\Nknow what happens when we
Dialogue: 0,0:05:30.71,0:05:35.19,Default,,0000,0000,0000,,differentiate cynex. We get\NEnsign Annex. So when we do the
Dialogue: 0,0:05:35.19,0:05:40.48,Default,,0000,0000,0000,,integration, we're going to have\Nthe sign an X. But we've got to
Dialogue: 0,0:05:40.48,0:05:45.36,Default,,0000,0000,0000,,have one over N sign NX and\Nagain plus the constant of
Dialogue: 0,0:05:45.36,0:05:51.89,Default,,0000,0000,0000,,integration, see. What about\Ndoing sign integral of
Dialogue: 0,0:05:51.89,0:05:58.62,Default,,0000,0000,0000,,sine X? Again, we know\Nthat the derivative of causes
Dialogue: 0,0:05:58.62,0:06:03.67,Default,,0000,0000,0000,,minus sign. So if we integrate\Nsign reversing the
Dialogue: 0,0:06:03.67,0:06:09.86,Default,,0000,0000,0000,,Differentiation, it's got to be\Nminus Cos X Plus C. And
Dialogue: 0,0:06:09.86,0:06:17.16,Default,,0000,0000,0000,,similarly if we have sign NX,\NDX, then it's got to be minus
Dialogue: 0,0:06:17.16,0:06:21.10,Default,,0000,0000,0000,,one over N cause NX plus C.
Dialogue: 0,0:06:22.49,0:06:24.60,Default,,0000,0000,0000,,What
Dialogue: 0,0:06:24.60,0:06:28.83,Default,,0000,0000,0000,,about?\N10
Dialogue: 0,0:06:30.13,0:06:33.88,Default,,0000,0000,0000,,Integral of Tan
Dialogue: 0,0:06:33.88,0:06:36.87,Default,,0000,0000,0000,,X. The X.
Dialogue: 0,0:06:38.20,0:06:41.78,Default,,0000,0000,0000,,We know that the integral of
Dialogue: 0,0:06:41.78,0:06:48.34,Default,,0000,0000,0000,,Tan. We can change to be\Nthe integral of sign over cause
Dialogue: 0,0:06:48.34,0:06:55.91,Default,,0000,0000,0000,,the X. Now when we look\Nwe can see that the top is the
Dialogue: 0,0:06:55.91,0:07:00.65,Default,,0000,0000,0000,,derivative of the bottom within\Na minus sign, 'cause the
Dialogue: 0,0:07:00.65,0:07:06.34,Default,,0000,0000,0000,,derivative of causes minus sign,\Nand so we have minus the natural
Dialogue: 0,0:07:06.34,0:07:08.71,Default,,0000,0000,0000,,log of cause of X.
Dialogue: 0,0:07:09.27,0:07:15.98,Default,,0000,0000,0000,,When my subtracting a log where\Ndividing by what's inside and if
Dialogue: 0,0:07:15.98,0:07:23.24,Default,,0000,0000,0000,,we divide by cosine then that's\Nthe same as SEK and so we
Dialogue: 0,0:07:23.24,0:07:31.07,Default,,0000,0000,0000,,can show that this is the log\Nof set of X and again we
Dialogue: 0,0:07:31.07,0:07:36.66,Default,,0000,0000,0000,,must have plus a constant of\Nintegration because these are
Dialogue: 0,0:07:36.66,0:07:42.18,Default,,0000,0000,0000,,indefinite integrals. There's\Nanother one that we can
Dialogue: 0,0:07:42.18,0:07:45.81,Default,,0000,0000,0000,,take and that is sex
Dialogue: 0,0:07:45.81,0:07:51.48,Default,,0000,0000,0000,,squared XDX. When we did, the\Ndifferentiation of Tangent, we
Dialogue: 0,0:07:51.48,0:07:55.66,Default,,0000,0000,0000,,saw that when you\Ndifferentiate it tan you gots
Dialogue: 0,0:07:55.66,0:08:01.23,Default,,0000,0000,0000,,X squared and so if we reverse\Nthat, the integral of sex
Dialogue: 0,0:08:01.23,0:08:06.33,Default,,0000,0000,0000,,squared must be Tan X plus a\Nconstant of integration, see.
Dialogue: 0,0:08:08.25,0:08:15.07,Default,,0000,0000,0000,,If we've got sex\Nsquared NXDX, and again,
Dialogue: 0,0:08:15.07,0:08:21.90,Default,,0000,0000,0000,,this must be one\Nover N 10 of
Dialogue: 0,0:08:21.90,0:08:24.46,Default,,0000,0000,0000,,NX plus C.
Dialogue: 0,0:08:26.36,0:08:29.97,Default,,0000,0000,0000,,Now we're going to look at some\Nrather more complicated ones,
Dialogue: 0,0:08:29.97,0:08:32.59,Default,,0000,0000,0000,,but they are worthwhile\Nremembering because they can
Dialogue: 0,0:08:32.59,0:08:35.22,Default,,0000,0000,0000,,save an awful lot of work if you
Dialogue: 0,0:08:35.22,0:08:39.93,Default,,0000,0000,0000,,can identify them. These are\Nconnected with the inverse trig
Dialogue: 0,0:08:39.93,0:08:46.59,Default,,0000,0000,0000,,functions. So we have the\Nintegral of one over the square
Dialogue: 0,0:08:46.59,0:08:52.12,Default,,0000,0000,0000,,root of 1 minus X squared with\Nrespect to X.
Dialogue: 0,0:08:53.91,0:08:58.01,Default,,0000,0000,0000,,With the differentiation of the\Ninverse trig function, we solve
Dialogue: 0,0:08:58.01,0:09:03.34,Default,,0000,0000,0000,,that when we differentiate it\Nsigned to the minus one of X, we
Dialogue: 0,0:09:03.34,0:09:09.49,Default,,0000,0000,0000,,got this and so if we reverse\Nthat sign to the minus one of X
Dialogue: 0,0:09:09.49,0:09:13.18,Default,,0000,0000,0000,,must be the answer to the\Nintegral because integration
Dialogue: 0,0:09:13.18,0:09:17.23,Default,,0000,0000,0000,,reverses differentiation.\NSimilarly, we have something
Dialogue: 0,0:09:17.23,0:09:24.49,Default,,0000,0000,0000,,that looks very much the same. A\Nsquared minus X squared. The X
Dialogue: 0,0:09:24.49,0:09:27.85,Default,,0000,0000,0000,,instead of 1 minus X squared.
Dialogue: 0,0:09:28.43,0:09:32.46,Default,,0000,0000,0000,,Then the integral is\Nsigned to the minus
Dialogue: 0,0:09:32.46,0:09:35.49,Default,,0000,0000,0000,,One X over a plus C.
Dialogue: 0,0:09:36.90,0:09:40.46,Default,,0000,0000,0000,,Let's have a look at some\Nexample using this result
Dialogue: 0,0:09:40.46,0:09:44.73,Default,,0000,0000,0000,,because sometimes we have to\Nwork a little bit hard at this
Dialogue: 0,0:09:44.73,0:09:50.31,Default,,0000,0000,0000,,side. Let's take a\Nstraightforward one, square root
Dialogue: 0,0:09:50.31,0:09:53.53,Default,,0000,0000,0000,,of 4 minus X squared.
Dialogue: 0,0:09:53.58,0:09:58.75,Default,,0000,0000,0000,,Now we can identify the four\Nwith the A squared, so it's
Dialogue: 0,0:09:58.75,0:10:04.79,Default,,0000,0000,0000,,fairly clear that a must be\Nequal to two, so we end up with
Dialogue: 0,0:10:04.79,0:10:10.82,Default,,0000,0000,0000,,sign to the minus one of X over\N2 plus see the constant of
Dialogue: 0,0:10:10.82,0:10:14.25,Default,,0000,0000,0000,,integration. What if it doesn't\Nquite look like that? What if
Dialogue: 0,0:10:14.25,0:10:17.16,Default,,0000,0000,0000,,we've got something here in\Nfront of the X squared?
Dialogue: 0,0:10:18.88,0:10:25.74,Default,,0000,0000,0000,,So we take the integral\Nof one over the square
Dialogue: 0,0:10:25.74,0:10:31.23,Default,,0000,0000,0000,,root of 4 - 9\NX squared DX.
Dialogue: 0,0:10:31.81,0:10:37.03,Default,,0000,0000,0000,,Here we can identify the A as\Nbeing too, but we've got 9 here.
Dialogue: 0,0:10:37.03,0:10:42.25,Default,,0000,0000,0000,,We really do need that just to\Nbe an X squared, so we're going
Dialogue: 0,0:10:42.25,0:10:47.10,Default,,0000,0000,0000,,to do is take the nine out\Nthrough the square root. And why
Dialogue: 0,0:10:47.10,0:10:51.58,Default,,0000,0000,0000,,is it comes through the square\Nroot, we will have to square
Dialogue: 0,0:10:51.58,0:10:56.43,Default,,0000,0000,0000,,rooted, which is going to leave\Nit as a three, and so we're
Dialogue: 0,0:10:56.43,0:10:58.67,Default,,0000,0000,0000,,going to come to this the
Dialogue: 0,0:10:58.67,0:11:02.50,Default,,0000,0000,0000,,integral of. One over 3.
Dialogue: 0,0:11:03.25,0:11:07.14,Default,,0000,0000,0000,,One over the square root of.
Dialogue: 0,0:11:07.79,0:11:14.49,Default,,0000,0000,0000,,For over 9 minus\NX squared with respect
Dialogue: 0,0:11:14.49,0:11:17.01,Default,,0000,0000,0000,,to X now.
Dialogue: 0,0:11:17.80,0:11:22.31,Default,,0000,0000,0000,,This does look like one we've\Ndone before. We can keep the one
Dialogue: 0,0:11:22.31,0:11:23.70,Default,,0000,0000,0000,,over three. That's fine.
Dialogue: 0,0:11:24.29,0:11:27.66,Default,,0000,0000,0000,,And now we need sign to the
Dialogue: 0,0:11:27.66,0:11:34.82,Default,,0000,0000,0000,,minus one. Of X over a?\NWhat is a this time? Will a
Dialogue: 0,0:11:34.82,0:11:41.42,Default,,0000,0000,0000,,IS2 over three because a squared\Nis 4 over 9, so that's X
Dialogue: 0,0:11:41.42,0:11:44.47,Default,,0000,0000,0000,,over 2 over 3 plus C.
Dialogue: 0,0:11:45.18,0:11:50.63,Default,,0000,0000,0000,,Dividing by a fraction, we know\Nhow to do that. We convert the
Dialogue: 0,0:11:50.63,0:11:55.24,Default,,0000,0000,0000,,fraction and multiply, which\Ngives us sign to the minus one
Dialogue: 0,0:11:55.24,0:11:58.17,Default,,0000,0000,0000,,of three X over 2 plus C.
Dialogue: 0,0:11:59.74,0:12:04.58,Default,,0000,0000,0000,,So we see we can do these quite\Neasily by trying to get that
Dialogue: 0,0:12:04.58,0:12:08.74,Default,,0000,0000,0000,,coefficient of X to be one and\Nnot the number that's actually
Dialogue: 0,0:12:08.74,0:12:15.68,Default,,0000,0000,0000,,there. The other hyperbolic\Nfunction that we met was 10 to
Dialogue: 0,0:12:15.68,0:12:17.37,Default,,0000,0000,0000,,the minus one.
Dialogue: 0,0:12:17.44,0:12:22.52,Default,,0000,0000,0000,,And the derivative of 10 to\Nthe minus one was this
Dialogue: 0,0:12:22.52,0:12:28.07,Default,,0000,0000,0000,,function one over 1 plus X\Nsquared. So if we want to
Dialogue: 0,0:12:28.07,0:12:33.61,Default,,0000,0000,0000,,integrate one over 1 plus X\Nsquared, the answer must be 10
Dialogue: 0,0:12:33.61,0:12:37.31,Default,,0000,0000,0000,,to the minus one of X Plus C.
Dialogue: 0,0:12:38.55,0:12:44.16,Default,,0000,0000,0000,,In the same way as we had an A\Nsquared in there, let's put one
Dialogue: 0,0:12:44.16,0:12:46.03,Default,,0000,0000,0000,,in. Now in this one.
Dialogue: 0,0:12:46.61,0:12:49.38,Default,,0000,0000,0000,,The X.
Dialogue: 0,0:12:50.48,0:12:57.54,Default,,0000,0000,0000,,We get this one standard result,\None over a 10 to the
Dialogue: 0,0:12:57.54,0:13:04.00,Default,,0000,0000,0000,,minus One X over a plus\Na constant of integration, see.
Dialogue: 0,0:13:05.60,0:13:12.28,Default,,0000,0000,0000,,Let's take an example\None over 9 plus X
Dialogue: 0,0:13:12.28,0:13:13.76,Default,,0000,0000,0000,,squared DX.
Dialogue: 0,0:13:15.05,0:13:21.92,Default,,0000,0000,0000,,Nine and a square to the same.\NSo a must be equal to three, so
Dialogue: 0,0:13:21.92,0:13:28.79,Default,,0000,0000,0000,,that gives us one over 310 to\Nthe minus One X over 3 plus C.
Dialogue: 0,0:13:29.68,0:13:34.04,Default,,0000,0000,0000,,Same questions we had before.\NWhat if it's not?
Dialogue: 0,0:13:34.76,0:13:39.25,Default,,0000,0000,0000,,A1 there what if there's a\Nnumber in there? Well, again,
Dialogue: 0,0:13:39.25,0:13:43.74,Default,,0000,0000,0000,,let's have a look. Let's take\Nthe integral of one over.
Dialogue: 0,0:13:44.25,0:13:50.70,Default,,0000,0000,0000,,25 + 16\NX squared DX.
Dialogue: 0,0:13:52.08,0:13:53.23,Default,,0000,0000,0000,,What do we do?
Dialogue: 0,0:13:54.68,0:14:02.52,Default,,0000,0000,0000,,We try to get a one there,\Nwhich means we take the 16 out
Dialogue: 0,0:14:02.52,0:14:09.80,Default,,0000,0000,0000,,so we'll have one over 16\Nintegral of one over 25 over 16
Dialogue: 0,0:14:09.80,0:14:12.04,Default,,0000,0000,0000,,plus X squared DX.
Dialogue: 0,0:14:13.01,0:14:16.16,Default,,0000,0000,0000,,The one over 16 can stay\Nas it is.
Dialogue: 0,0:14:17.24,0:14:25.13,Default,,0000,0000,0000,,This is now a squared, so a must\Nbe 5 over 4. So we want one
Dialogue: 0,0:14:25.13,0:14:28.58,Default,,0000,0000,0000,,over a one over 5 over 4.
Dialogue: 0,0:14:29.85,0:14:37.39,Default,,0000,0000,0000,,10 to the minus one of\NX over a so X over
Dialogue: 0,0:14:37.39,0:14:41.15,Default,,0000,0000,0000,,5 over 4 plus the constant
Dialogue: 0,0:14:41.15,0:14:48.42,Default,,0000,0000,0000,,of integration. Now we need to\Ntidy that off a little bit. One
Dialogue: 0,0:14:48.42,0:14:56.17,Default,,0000,0000,0000,,over 16 * 1 over 5 over 4\Nis 4 over 510 to the minus one
Dialogue: 0,0:14:56.17,0:15:01.98,Default,,0000,0000,0000,,of four X over 5 plus the\Nconstant of integration. See and
Dialogue: 0,0:15:01.98,0:15:07.78,Default,,0000,0000,0000,,we can make a simplification\NHere by dividing by 4, giving us
Dialogue: 0,0:15:07.78,0:15:13.11,Default,,0000,0000,0000,,120th, four and four cancel and\N4 * 5 is 20.
Dialogue: 0,0:15:13.11,0:15:21.10,Default,,0000,0000,0000,,10 to the minus one\N4X over 5 plus ever
Dialogue: 0,0:15:21.10,0:15:24.30,Default,,0000,0000,0000,,present constant of integration,
Dialogue: 0,0:15:24.30,0:15:29.11,Default,,0000,0000,0000,,see. Now those are all the\Nstandard forms. Once you've seen
Dialogue: 0,0:15:29.11,0:15:32.46,Default,,0000,0000,0000,,them and use them there. Well\Nworth learning learning because
Dialogue: 0,0:15:32.46,0:15:36.48,Default,,0000,0000,0000,,you can recognize them and you\Ncan use them straight away and
Dialogue: 0,0:15:36.48,0:15:40.50,Default,,0000,0000,0000,,you don't have to worry about\Nhow to do certain forms of
Dialogue: 0,0:15:40.50,0:15:43.85,Default,,0000,0000,0000,,Integration 'cause they just\Nthere, they just part of what
Dialogue: 0,0:15:43.85,0:15:47.54,Default,,0000,0000,0000,,you do all the time, but\Nlearning them is very important.