[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.49,0:00:04.23,Default,,0000,0000,0000,,In the first of the units on\Nalgebraic fractions, we looked
Dialogue: 0,0:00:04.23,0:00:07.97,Default,,0000,0000,0000,,at what happened when we had a\Nproper fraction with linear
Dialogue: 0,0:00:07.97,0:00:11.03,Default,,0000,0000,0000,,factors in the denominator of\Nproper fraction with repeated
Dialogue: 0,0:00:11.03,0:00:13.75,Default,,0000,0000,0000,,linear factors in the\Ndenominator, and what happened
Dialogue: 0,0:00:13.75,0:00:18.17,Default,,0000,0000,0000,,when we had improper fractions.\Nwhat I want to do in this video
Dialogue: 0,0:00:18.17,0:00:21.91,Default,,0000,0000,0000,,is look at what happens when we\Nget an irreducible quadratic
Dialogue: 0,0:00:21.91,0:00:24.63,Default,,0000,0000,0000,,factor when we get an\Nirreducible quadratic factor
Dialogue: 0,0:00:24.63,0:00:28.71,Default,,0000,0000,0000,,will end up with an integral of\Nsomething which looks like this
Dialogue: 0,0:00:28.71,0:00:29.73,Default,,0000,0000,0000,,X plus B.
Dialogue: 0,0:00:30.87,0:00:37.82,Default,,0000,0000,0000,,Over a X squared plus BX\Nplus, see where the A&B are
Dialogue: 0,0:00:37.82,0:00:41.64,Default,,0000,0000,0000,,known constants. And this\Nquadratic in the denominator
Dialogue: 0,0:00:41.64,0:00:44.42,Default,,0000,0000,0000,,cannot be factorized. Now\Nthere's various things that
Dialogue: 0,0:00:44.42,0:00:48.95,Default,,0000,0000,0000,,could happen. It's possible that\Na could turn out to be 0. Now,
Dialogue: 0,0:00:48.95,0:00:53.12,Default,,0000,0000,0000,,if it turns out to be 0, what\Nwould be left with?
Dialogue: 0,0:00:53.79,0:00:55.49,Default,,0000,0000,0000,,Is trying to integrate a
Dialogue: 0,0:00:55.49,0:00:57.53,Default,,0000,0000,0000,,constant. Over this quadratic
Dialogue: 0,0:00:57.53,0:01:04.45,Default,,0000,0000,0000,,factor. So we'll just end up\Nwith a B over AX squared plus BX
Dialogue: 0,0:01:04.45,0:01:09.04,Default,,0000,0000,0000,,plus C. Now the first example\NI'd like to show you is what
Dialogue: 0,0:01:09.04,0:01:12.29,Default,,0000,0000,0000,,we do when we get a situation\Nwhere we've just got a
Dialogue: 0,0:01:12.29,0:01:15.00,Default,,0000,0000,0000,,constant on its own, no ex\Nterms over the irreducible
Dialogue: 0,0:01:15.00,0:01:17.71,Default,,0000,0000,0000,,quadratic factor, so let's\Nhave a look at a specific
Dialogue: 0,0:01:17.71,0:01:17.98,Default,,0000,0000,0000,,example.
Dialogue: 0,0:01:21.24,0:01:24.54,Default,,0000,0000,0000,,Suppose we want to\Nintegrate a constant one.
Dialogue: 0,0:01:25.59,0:01:27.02,Default,,0000,0000,0000,,Over X squared.
Dialogue: 0,0:01:28.22,0:01:32.31,Default,,0000,0000,0000,,Plus X plus one. We want\Nto integrate this with
Dialogue: 0,0:01:32.31,0:01:33.54,Default,,0000,0000,0000,,respect to X.
Dialogue: 0,0:01:35.64,0:01:38.62,Default,,0000,0000,0000,,This denominator will not\Nfactorize if it would factorize,
Dialogue: 0,0:01:38.62,0:01:40.60,Default,,0000,0000,0000,,would be back to expressing it
Dialogue: 0,0:01:40.60,0:01:44.84,Default,,0000,0000,0000,,in partial fractions. The way we\Nproceed is to try to complete
Dialogue: 0,0:01:44.84,0:01:46.24,Default,,0000,0000,0000,,the square in the denominator.
Dialogue: 0,0:01:47.02,0:01:50.44,Default,,0000,0000,0000,,Let me remind you of how we\Ncomplete the square for X
Dialogue: 0,0:01:50.44,0:01:51.86,Default,,0000,0000,0000,,squared plus X plus one.
Dialogue: 0,0:01:52.85,0:01:57.10,Default,,0000,0000,0000,,It's a complete the square we\Ntry to write the first 2 terms.
Dialogue: 0,0:01:58.50,0:02:00.34,Default,,0000,0000,0000,,As something squared.
Dialogue: 0,0:02:01.26,0:02:04.70,Default,,0000,0000,0000,,Well, what do we write in\Nthis bracket? We want an X
Dialogue: 0,0:02:04.70,0:02:07.86,Default,,0000,0000,0000,,and clearly when the brackets\Nare all squared out, will get
Dialogue: 0,0:02:07.86,0:02:10.44,Default,,0000,0000,0000,,an X squared which is that\Nterm dealt with.
Dialogue: 0,0:02:11.74,0:02:15.51,Default,,0000,0000,0000,,To get an ex here, we need\Nactually a term 1/2 here because
Dialogue: 0,0:02:15.51,0:02:19.28,Default,,0000,0000,0000,,you imagine when you square the\Nbrackets out you'll get a half X
Dialogue: 0,0:02:19.28,0:02:22.76,Default,,0000,0000,0000,,in another half X, which is the\Nwhole X which is that.
Dialogue: 0,0:02:24.73,0:02:27.29,Default,,0000,0000,0000,,We get something we don't want\Nwhen these brackets are all
Dialogue: 0,0:02:27.29,0:02:28.69,Default,,0000,0000,0000,,squared out, we'll end it with
Dialogue: 0,0:02:28.69,0:02:32.68,Default,,0000,0000,0000,,1/2 squared. Which is 1/4 and we\Ndon't want a quarter, so I'm
Dialogue: 0,0:02:32.68,0:02:34.25,Default,,0000,0000,0000,,going to subtract it again here.
Dialogue: 0,0:02:35.54,0:02:38.99,Default,,0000,0000,0000,,So altogether, all those\Nterms written down there
Dialogue: 0,0:02:38.99,0:02:42.87,Default,,0000,0000,0000,,are equivalent to the\Nfirst 2 terms over here.
Dialogue: 0,0:02:44.50,0:02:47.58,Default,,0000,0000,0000,,And to make these equal, we\Nstill need the plus one.
Dialogue: 0,0:02:50.59,0:02:56.20,Default,,0000,0000,0000,,So tidying this up, we've\Nactually got X plus 1/2 all
Dialogue: 0,0:02:56.20,0:03:02.32,Default,,0000,0000,0000,,squared, and one subtract 1/4 is\N3/4. That is the process of
Dialogue: 0,0:03:02.32,0:03:03.85,Default,,0000,0000,0000,,completing the square.
Dialogue: 0,0:03:12.04,0:03:17.70,Default,,0000,0000,0000,,OK, how will that help us? Well,\Nit means that what we want to do
Dialogue: 0,0:03:17.70,0:03:21.84,Default,,0000,0000,0000,,now is considered instead of the\Nintegral we started with. We
Dialogue: 0,0:03:21.84,0:03:26.37,Default,,0000,0000,0000,,want to consider this integral\None over X plus 1/2 all squared.
Dialogue: 0,0:03:29.32,0:03:33.53,Default,,0000,0000,0000,,Plus 3/4 we want to integrate\Nthat with respect to X.
Dialogue: 0,0:03:34.43,0:03:37.63,Default,,0000,0000,0000,,Now, the way I'm going to\Nproceed is going to make a
Dialogue: 0,0:03:37.63,0:03:40.84,Default,,0000,0000,0000,,substitution in. Here, I'm going\Nto let you be X plus 1/2.
Dialogue: 0,0:03:48.42,0:03:51.88,Default,,0000,0000,0000,,When we do that, are integral\Nwill become the integral of one
Dialogue: 0,0:03:51.88,0:03:55.91,Default,,0000,0000,0000,,over X plus 1/2 will be just\Nyou, so will end up with you
Dialogue: 0,0:03:55.91,0:03:58.63,Default,,0000,0000,0000,,squared. We've got plus 3/4.
Dialogue: 0,0:04:00.21,0:04:02.11,Default,,0000,0000,0000,,We need to take care of the DX.
Dialogue: 0,0:04:03.19,0:04:07.45,Default,,0000,0000,0000,,Now remember that if we want the\Ndifferential du, that's du DX
Dialogue: 0,0:04:07.45,0:04:12.78,Default,,0000,0000,0000,,DX. But in this case du DX is\Njust one. This is just one. So
Dialogue: 0,0:04:12.78,0:04:18.10,Default,,0000,0000,0000,,do you is just DX. So this is\Nnice and simple. The DX we have
Dialogue: 0,0:04:18.10,0:04:19.88,Default,,0000,0000,0000,,here just becomes a du.
Dialogue: 0,0:04:22.49,0:04:26.38,Default,,0000,0000,0000,,Now this integral is a standard\Nform. There's a standard result
Dialogue: 0,0:04:26.38,0:04:30.63,Default,,0000,0000,0000,,which says that if you want to\Nintegrate one over a squared
Dialogue: 0,0:04:30.63,0:04:32.76,Default,,0000,0000,0000,,plus X squared with respect to
Dialogue: 0,0:04:32.76,0:04:38.42,Default,,0000,0000,0000,,X. That's equal to one over a\Ninverse tangent that's 10 to the
Dialogue: 0,0:04:38.42,0:04:43.50,Default,,0000,0000,0000,,minus one of X over a plus a\Nconstant. Now we will use that
Dialogue: 0,0:04:43.50,0:04:47.49,Default,,0000,0000,0000,,result to write the answer down\Nto this integral, because this
Dialogue: 0,0:04:47.49,0:04:49.67,Default,,0000,0000,0000,,is one of these where a.
Dialogue: 0,0:04:50.75,0:04:52.71,Default,,0000,0000,0000,,Is the square root of 3 over 2?
Dialogue: 0,0:04:53.36,0:04:55.08,Default,,0000,0000,0000,,That's a squared is 3/4.
Dialogue: 0,0:04:57.32,0:05:03.37,Default,,0000,0000,0000,,So A is the square root of 3 /\N2, so we can write down the
Dialogue: 0,0:05:03.37,0:05:08.28,Default,,0000,0000,0000,,answer to this straight away and\Nthis will workout at one over a,
Dialogue: 0,0:05:08.28,0:05:12.82,Default,,0000,0000,0000,,which is one over root 3 over\N210 to the minus one.
Dialogue: 0,0:05:14.39,0:05:18.34,Default,,0000,0000,0000,,Of X over A. In this\Ncase it will be U over a
Dialogue: 0,0:05:18.34,0:05:20.17,Default,,0000,0000,0000,,which is Route 3 over 2.
Dialogue: 0,0:05:21.66,0:05:22.43,Default,,0000,0000,0000,,Plus a constant.
Dialogue: 0,0:05:24.94,0:05:28.74,Default,,0000,0000,0000,,Just to tidy this up a little\Nbit where dividing by a fraction
Dialogue: 0,0:05:28.74,0:05:32.24,Default,,0000,0000,0000,,here. So dividing by Route 3\Nover 2 is like multiplying by
Dialogue: 0,0:05:32.24,0:05:33.41,Default,,0000,0000,0000,,two over Route 3.
Dialogue: 0,0:05:35.30,0:05:37.03,Default,,0000,0000,0000,,We've attempted the minus one.
Dialogue: 0,0:05:37.95,0:05:41.43,Default,,0000,0000,0000,,You we can replace\Nwith X plus 1/2.
Dialogue: 0,0:05:45.04,0:05:49.36,Default,,0000,0000,0000,,And again, dividing by Route 3\Nover 2 is like multiplying by
Dialogue: 0,0:05:49.36,0:05:50.80,Default,,0000,0000,0000,,two over Route 3.
Dialogue: 0,0:05:52.12,0:05:53.58,Default,,0000,0000,0000,,And we have a constant\Nat the end.
Dialogue: 0,0:05:56.56,0:05:59.73,Default,,0000,0000,0000,,And that's the answer. So In\Nother words, to integrate.
Dialogue: 0,0:06:01.00,0:06:03.58,Default,,0000,0000,0000,,A constant over an\Nirreducible quadratic factor.
Dialogue: 0,0:06:03.58,0:06:07.99,Default,,0000,0000,0000,,We can complete the square as\Nwe did here and then use
Dialogue: 0,0:06:07.99,0:06:10.94,Default,,0000,0000,0000,,integration by substitution\Nto finish the problem off.
Dialogue: 0,0:06:12.32,0:06:15.81,Default,,0000,0000,0000,,So that's what happens when\Nwe get a constant over the
Dialogue: 0,0:06:15.81,0:06:16.44,Default,,0000,0000,0000,,quadratic factor.
Dialogue: 0,0:06:20.97,0:06:24.85,Default,,0000,0000,0000,,What else could happen? It may\Nhappen that we get a situation
Dialogue: 0,0:06:24.85,0:06:28.72,Default,,0000,0000,0000,,like this. We end up with a\Nquadratic function at the bottom
Dialogue: 0,0:06:28.72,0:06:30.66,Default,,0000,0000,0000,,and it's derivative at the top.
Dialogue: 0,0:06:33.57,0:06:36.56,Default,,0000,0000,0000,,If that happens, it's very\Nstraightforward to finish the
Dialogue: 0,0:06:36.56,0:06:40.21,Default,,0000,0000,0000,,integration of because we know\Nfrom a standard result that this
Dialogue: 0,0:06:40.21,0:06:43.53,Default,,0000,0000,0000,,evaluates to the logarithm of\Nthe modulus of the denominator
Dialogue: 0,0:06:43.53,0:06:47.18,Default,,0000,0000,0000,,plus a constant. So, for\Nexample, I'm thinking now of an
Dialogue: 0,0:06:47.18,0:06:48.51,Default,,0000,0000,0000,,example like this one.
Dialogue: 0,0:06:52.96,0:06:56.77,Default,,0000,0000,0000,,Again, irreducible quadratic\Nfactor in the denominator.
Dialogue: 0,0:06:58.43,0:07:01.44,Default,,0000,0000,0000,,Attorney X plus be constant\Ntimes X plus another
Dialogue: 0,0:07:01.44,0:07:04.78,Default,,0000,0000,0000,,constant on the top and if\Nyou inspect this carefully,
Dialogue: 0,0:07:04.78,0:07:08.12,Default,,0000,0000,0000,,if you look at the bottom\Nhere and you differentiate
Dialogue: 0,0:07:08.12,0:07:10.12,Default,,0000,0000,0000,,it, you'll get 2X plus one.
Dialogue: 0,0:07:11.32,0:07:14.52,Default,,0000,0000,0000,,So we've got a situation where\Nwe've got a function at the
Dialogue: 0,0:07:14.52,0:07:17.73,Default,,0000,0000,0000,,bottom and it's derivative at\Nthe top so we can write this
Dialogue: 0,0:07:17.73,0:07:20.66,Default,,0000,0000,0000,,down straight away. The answer\Nis going to be the natural
Dialogue: 0,0:07:20.66,0:07:23.07,Default,,0000,0000,0000,,logarithm of the modulus of\Nwhat's at the bottom.
Dialogue: 0,0:07:27.74,0:07:31.16,Default,,0000,0000,0000,,Let's see and that's finished.\NThat's nice and straightforward.
Dialogue: 0,0:07:31.16,0:07:35.34,Default,,0000,0000,0000,,If you get a situation where\Nyou've got something times X
Dialogue: 0,0:07:35.34,0:07:36.48,Default,,0000,0000,0000,,plus another constant.
Dialogue: 0,0:07:37.07,0:07:39.88,Default,,0000,0000,0000,,And this top line is not the\Nderivative of the bottom
Dialogue: 0,0:07:39.88,0:07:43.44,Default,,0000,0000,0000,,line. Then you gotta do a bit\Nmore work on it as we'll see
Dialogue: 0,0:07:43.44,0:07:44.46,Default,,0000,0000,0000,,in the next example.
Dialogue: 0,0:07:48.83,0:07:53.01,Default,,0000,0000,0000,,Let's have a look at this\Nexample. Suppose we want to
Dialogue: 0,0:07:53.01,0:07:58.33,Default,,0000,0000,0000,,integrate X divided by X squared\Nplus X Plus One, and we want to
Dialogue: 0,0:07:58.33,0:08:00.61,Default,,0000,0000,0000,,integrate it with respect to X.
Dialogue: 0,0:08:01.94,0:08:05.41,Default,,0000,0000,0000,,Still, if we differentiate, the\Nbottom line will get 2X.
Dialogue: 0,0:08:06.04,0:08:08.100,Default,,0000,0000,0000,,Plus One, and that's not\Nwhat we have at the top.
Dialogue: 0,0:08:08.100,0:08:12.23,Default,,0000,0000,0000,,However, what we can do is\Nwe can introduce it to at
Dialogue: 0,0:08:12.23,0:08:15.46,Default,,0000,0000,0000,,the top, so we have two X in\Nthis following way. By
Dialogue: 0,0:08:15.46,0:08:18.14,Default,,0000,0000,0000,,little trick we can put a\Ntwo at the top.
Dialogue: 0,0:08:23.16,0:08:26.05,Default,,0000,0000,0000,,And in order to make this the\Nsame as the integral that we
Dialogue: 0,0:08:26.05,0:08:27.60,Default,,0000,0000,0000,,started with, I'm going to put a
Dialogue: 0,0:08:27.60,0:08:29.66,Default,,0000,0000,0000,,factor of 1/2 outside. Half and
Dialogue: 0,0:08:29.66,0:08:33.76,Default,,0000,0000,0000,,the two canceling. Will will\Nleave the integral that we
Dialogue: 0,0:08:33.76,0:08:34.77,Default,,0000,0000,0000,,started with that.
Dialogue: 0,0:08:36.01,0:08:39.59,Default,,0000,0000,0000,,Now. If we differentiate the\Nbottom you see, we get.
Dialogue: 0,0:08:40.14,0:08:44.70,Default,,0000,0000,0000,,2X. Which is what we've got\Nat the top. But we also get a
Dialogue: 0,0:08:44.70,0:08:46.86,Default,,0000,0000,0000,,plus one from differentiating\Nthe extreme and we haven't
Dialogue: 0,0:08:46.86,0:08:49.50,Default,,0000,0000,0000,,got a plus one there, so we\Napply another little trick
Dialogue: 0,0:08:49.50,0:08:50.94,Default,,0000,0000,0000,,now, and we do the following.
Dialogue: 0,0:08:54.46,0:08:56.72,Default,,0000,0000,0000,,We'd like a plus one there.
Dialogue: 0,0:09:00.70,0:09:04.12,Default,,0000,0000,0000,,So that the derivative of\Nthe denominator occurs in
Dialogue: 0,0:09:04.12,0:09:04.88,Default,,0000,0000,0000,,the numerator.
Dialogue: 0,0:09:05.96,0:09:08.91,Default,,0000,0000,0000,,But this is no longer the same\Nas that because I've added a one
Dialogue: 0,0:09:08.91,0:09:10.39,Default,,0000,0000,0000,,here. So I've got to take it
Dialogue: 0,0:09:10.39,0:09:13.44,Default,,0000,0000,0000,,away again. In order that\Nwere still with the same
Dialogue: 0,0:09:13.44,0:09:14.54,Default,,0000,0000,0000,,problem that we started with.
Dialogue: 0,0:09:16.54,0:09:21.26,Default,,0000,0000,0000,,Now what I can do is I can split\Nthis into two integrals. I've
Dialogue: 0,0:09:21.26,0:09:24.63,Default,,0000,0000,0000,,got a half the integral of these\Nfirst 2 terms.
Dialogue: 0,0:09:27.30,0:09:29.95,Default,,0000,0000,0000,,Over X squared plus X plus one.
Dialogue: 0,0:09:31.73,0:09:34.58,Default,,0000,0000,0000,,DX and I've got a half.
Dialogue: 0,0:09:35.82,0:09:40.24,Default,,0000,0000,0000,,The integral of the second term,\Nwhich is minus one over X
Dialogue: 0,0:09:40.24,0:09:45.02,Default,,0000,0000,0000,,squared plus X plus one DX so\Nthat little bit of trickery has
Dialogue: 0,0:09:45.02,0:09:49.07,Default,,0000,0000,0000,,allowed me to split the thing\Ninto two integrals. Now this
Dialogue: 0,0:09:49.07,0:09:52.75,Default,,0000,0000,0000,,first one we've already seen is\Nstraightforward to finish off,
Dialogue: 0,0:09:52.75,0:09:56.43,Default,,0000,0000,0000,,because the numerator now is the\Nderivative of the denominator,
Dialogue: 0,0:09:56.43,0:09:59.00,Default,,0000,0000,0000,,so this is just a half the
Dialogue: 0,0:09:59.00,0:10:02.80,Default,,0000,0000,0000,,natural logarithm. Of the\Nmodulus of X squared plus
Dialogue: 0,0:10:02.80,0:10:03.96,Default,,0000,0000,0000,,X plus one.
Dialogue: 0,0:10:06.62,0:10:10.08,Default,,0000,0000,0000,,And then we've got minus 1/2.\NTake the minus sign out minus
Dialogue: 0,0:10:10.08,0:10:13.53,Default,,0000,0000,0000,,1/2, and this integral integral\Nof one over X squared plus X
Dialogue: 0,0:10:13.53,0:10:17.28,Default,,0000,0000,0000,,Plus one is the one that we did\Nright at the very beginning.
Dialogue: 0,0:10:17.85,0:10:22.34,Default,,0000,0000,0000,,And if we just look back, let's\Nsee the results of finding that
Dialogue: 0,0:10:22.34,0:10:26.48,Default,,0000,0000,0000,,integral. Was this one here, two\Nover Route 3 inverse tan of
Dialogue: 0,0:10:26.48,0:10:28.89,Default,,0000,0000,0000,,twice X plus 1/2 over Route 3?
Dialogue: 0,0:10:29.89,0:10:34.58,Default,,0000,0000,0000,,So we've got two over Route\N3 inverse tangent.
Dialogue: 0,0:10:35.75,0:10:37.45,Default,,0000,0000,0000,,Twice X plus 1/2.
Dialogue: 0,0:10:38.54,0:10:39.73,Default,,0000,0000,0000,,Over Route 3.
Dialogue: 0,0:10:41.58,0:10:43.01,Default,,0000,0000,0000,,Plus a constant of integration.
Dialogue: 0,0:10:44.54,0:10:49.34,Default,,0000,0000,0000,,I can just tidy this up so it's\Nnice and neat to finish it off a
Dialogue: 0,0:10:49.34,0:10:52.34,Default,,0000,0000,0000,,half the logarithm of X squared\Nplus X plus one.
Dialogue: 0,0:10:54.02,0:10:59.41,Default,,0000,0000,0000,,The Twos will counsel here, so\NI'm left with minus one over
Dialogue: 0,0:10:59.41,0:11:01.20,Default,,0000,0000,0000,,Route 3 inverse tangent.
Dialogue: 0,0:11:01.92,0:11:05.20,Default,,0000,0000,0000,,And it might be nice just to\Nmultiply these brackets out to
Dialogue: 0,0:11:05.20,0:11:08.47,Default,,0000,0000,0000,,finish it off, so I'll have two\NX and 2 * 1/2.
Dialogue: 0,0:11:09.11,0:11:09.82,Default,,0000,0000,0000,,Is one.
Dialogue: 0,0:11:10.89,0:11:12.18,Default,,0000,0000,0000,,All over Route 3.
Dialogue: 0,0:11:13.80,0:11:15.37,Default,,0000,0000,0000,,Plus a constant of integration.
Dialogue: 0,0:11:17.67,0:11:19.05,Default,,0000,0000,0000,,And that's the problem solved.
Dialogue: 0,0:11:26.10,0:11:30.28,Default,,0000,0000,0000,,Let's have a look at one\Nfinal example where we can
Dialogue: 0,0:11:30.28,0:11:33.70,Default,,0000,0000,0000,,draw some of these threads\Ntogether. Supposing we want
Dialogue: 0,0:11:33.70,0:11:37.50,Default,,0000,0000,0000,,to integrate 1 divided by XX\Nsquared plus one DX.
Dialogue: 0,0:11:39.40,0:11:42.70,Default,,0000,0000,0000,,What if we got? In this case,\Nit's a proper fraction.
Dialogue: 0,0:11:44.03,0:11:45.48,Default,,0000,0000,0000,,And we've got a linear factor
Dialogue: 0,0:11:45.48,0:11:47.88,Default,,0000,0000,0000,,here. And a quadratic\Nfactor here.
Dialogue: 0,0:11:48.90,0:11:52.38,Default,,0000,0000,0000,,You can try, but you'll find\Nthat this quadratic factor will
Dialogue: 0,0:11:52.38,0:11:54.27,Default,,0000,0000,0000,,not factorize, so this is an
Dialogue: 0,0:11:54.27,0:11:57.75,Default,,0000,0000,0000,,irreducible quadratic factor.\NSo what we're going to do is
Dialogue: 0,0:11:57.75,0:11:59.93,Default,,0000,0000,0000,,we're going to, first of all,\Nexpress the integrand.
Dialogue: 0,0:12:02.58,0:12:06.05,Default,,0000,0000,0000,,As the sum of its partial\Nfractions and the appropriate
Dialogue: 0,0:12:06.05,0:12:09.87,Default,,0000,0000,0000,,form of partial fractions are\Ngoing to be a constant over
Dialogue: 0,0:12:09.87,0:12:10.91,Default,,0000,0000,0000,,the linear factor.
Dialogue: 0,0:12:12.87,0:12:15.59,Default,,0000,0000,0000,,And then we'll need BX plus C.
Dialogue: 0,0:12:16.85,0:12:20.38,Default,,0000,0000,0000,,Over the irreducible quadratic\Nfactor X squared plus one.
Dialogue: 0,0:12:22.45,0:12:28.26,Default,,0000,0000,0000,,We now have to find abian. See,\Nwe do that in the usual way by
Dialogue: 0,0:12:28.26,0:12:31.74,Default,,0000,0000,0000,,adding these together, the\Ncommon denominator will be XX
Dialogue: 0,0:12:31.74,0:12:32.90,Default,,0000,0000,0000,,squared plus one.
Dialogue: 0,0:12:35.20,0:12:39.21,Default,,0000,0000,0000,,Will need to multiply top and\Nbottom here by X squared plus
Dialogue: 0,0:12:39.21,0:12:42.88,Default,,0000,0000,0000,,one to achieve the correct\Ndenominator so we'll have an AX
Dialogue: 0,0:12:42.88,0:12:43.88,Default,,0000,0000,0000,,squared plus one.
Dialogue: 0,0:12:45.53,0:12:51.00,Default,,0000,0000,0000,,And we need to multiply top and\Nbottom here by X to achieve that
Dialogue: 0,0:12:51.00,0:12:54.91,Default,,0000,0000,0000,,denominator. So we'll have VX\Nplus C4 multiplied by X.
Dialogue: 0,0:12:57.33,0:12:59.16,Default,,0000,0000,0000,,This quantity is equal to that
Dialogue: 0,0:12:59.16,0:13:03.14,Default,,0000,0000,0000,,quantity. The denominators are\Nalready the same, so we can
Dialogue: 0,0:13:03.14,0:13:07.30,Default,,0000,0000,0000,,equate the numerators. If we\Njust look at the numerators
Dialogue: 0,0:13:07.30,0:13:11.25,Default,,0000,0000,0000,,will have one is equal to a X\Nsquared plus one.
Dialogue: 0,0:13:13.65,0:13:15.52,Default,,0000,0000,0000,,Plus BX Plus C.
Dialogue: 0,0:13:16.82,0:13:18.41,Default,,0000,0000,0000,,Multiplied by X.
Dialogue: 0,0:13:20.40,0:13:23.78,Default,,0000,0000,0000,,What's a sensible value to\Nsubstitute for X so we can
Dialogue: 0,0:13:23.78,0:13:27.15,Default,,0000,0000,0000,,find abian? See while a\Nsensible value is clearly X is
Dialogue: 0,0:13:27.15,0:13:30.84,Default,,0000,0000,0000,,zero, whi is that sensible?\NWell, if X is zero, both of
Dialogue: 0,0:13:30.84,0:13:32.99,Default,,0000,0000,0000,,these terms at the end will\Ndisappear.
Dialogue: 0,0:13:34.52,0:13:38.30,Default,,0000,0000,0000,,So X being zero will have one is\Nequal to A.
Dialogue: 0,0:13:39.08,0:13:41.08,Default,,0000,0000,0000,,0 squared is 0 + 1.
Dialogue: 0,0:13:42.08,0:13:45.03,Default,,0000,0000,0000,,Is still one, so we'll have one\Na. So a is one.
Dialogue: 0,0:13:46.83,0:13:48.09,Default,,0000,0000,0000,,That's our value for a.
Dialogue: 0,0:13:49.41,0:13:54.17,Default,,0000,0000,0000,,What can we do to find B and see\Nwhat I'm going to do now is I'm
Dialogue: 0,0:13:54.17,0:13:56.69,Default,,0000,0000,0000,,going to equate some\Ncoefficients and let's start by
Dialogue: 0,0:13:56.69,0:13:59.49,Default,,0000,0000,0000,,looking at the coefficients of X\Nsquared on both side.
Dialogue: 0,0:14:01.56,0:14:04.73,Default,,0000,0000,0000,,On the left hand side there\Nare no X squared's.
Dialogue: 0,0:14:06.35,0:14:09.63,Default,,0000,0000,0000,,What about on the right\Nhand side? There's clearly
Dialogue: 0,0:14:09.63,0:14:10.72,Default,,0000,0000,0000,,AX squared here.
Dialogue: 0,0:14:12.99,0:14:15.34,Default,,0000,0000,0000,,And when we multiply the\Nbrackets out here, that
Dialogue: 0,0:14:15.34,0:14:16.38,Default,,0000,0000,0000,,would be X squared.
Dialogue: 0,0:14:19.96,0:14:24.20,Default,,0000,0000,0000,,There are no more X squares, so\NA plus B must be zero. That
Dialogue: 0,0:14:24.20,0:14:28.44,Default,,0000,0000,0000,,means that B must be the minus\Nnegative of a must be the minor
Dialogue: 0,0:14:28.44,0:14:32.08,Default,,0000,0000,0000,,say, but a is already one, so be\Nmust be minus one.
Dialogue: 0,0:14:34.15,0:14:38.52,Default,,0000,0000,0000,,We still need to find C and\Nwill do that by equating
Dialogue: 0,0:14:38.52,0:14:39.61,Default,,0000,0000,0000,,coefficients of X.
Dialogue: 0,0:14:41.74,0:14:43.48,Default,,0000,0000,0000,,There are no ex terms on\Nthe left.
Dialogue: 0,0:14:45.24,0:14:46.78,Default,,0000,0000,0000,,There are no ex terms in here.
Dialogue: 0,0:14:48.46,0:14:53.25,Default,,0000,0000,0000,,There's an X squared term there,\Nand the only ex term is CX, so
Dialogue: 0,0:14:53.25,0:14:54.62,Default,,0000,0000,0000,,see must be 0.
Dialogue: 0,0:14:56.31,0:14:59.95,Default,,0000,0000,0000,,So when we express this in its\Npartial fractions, will end up
Dialogue: 0,0:14:59.95,0:15:01.16,Default,,0000,0000,0000,,with a being one.
Dialogue: 0,0:15:02.72,0:15:06.70,Default,,0000,0000,0000,,Be being minus one\Nand see being 0.
Dialogue: 0,0:15:09.02,0:15:14.99,Default,,0000,0000,0000,,So we'll be left with trying to\Nintegrate one over X minus X
Dialogue: 0,0:15:14.99,0:15:17.28,Default,,0000,0000,0000,,over X squared plus one.
Dialogue: 0,0:15:18.99,0:15:19.64,Default,,0000,0000,0000,,DX
Dialogue: 0,0:15:21.61,0:15:25.29,Default,,0000,0000,0000,,so we've used partial fractions\Nto split this up into two terms,
Dialogue: 0,0:15:25.29,0:15:28.98,Default,,0000,0000,0000,,and all we have to do now is\Ncompletely integration. Let me
Dialogue: 0,0:15:28.98,0:15:30.21,Default,,0000,0000,0000,,write that down again.
Dialogue: 0,0:15:31.75,0:15:38.39,Default,,0000,0000,0000,,We want to integrate one over X\Nminus X over X squared plus one
Dialogue: 0,0:15:38.39,0:15:43.60,Default,,0000,0000,0000,,and all that wants to be\Nintegrated with respect to X.
Dialogue: 0,0:15:46.66,0:15:48.43,Default,,0000,0000,0000,,First term straightforward.
Dialogue: 0,0:15:49.26,0:15:52.39,Default,,0000,0000,0000,,The integral of one\Nover X is the
Dialogue: 0,0:15:52.39,0:15:54.73,Default,,0000,0000,0000,,logarithm of the\Nmodulus of X.
Dialogue: 0,0:15:56.57,0:15:58.32,Default,,0000,0000,0000,,To integrate the second term.
Dialogue: 0,0:15:59.04,0:16:03.51,Default,,0000,0000,0000,,We notice that the numerator is\Nalmost the derivative of the
Dialogue: 0,0:16:03.51,0:16:05.94,Default,,0000,0000,0000,,denominator. If we\Ndifferentiate, the denominator
Dialogue: 0,0:16:05.94,0:16:07.16,Default,,0000,0000,0000,,will get 2X.
Dialogue: 0,0:16:08.11,0:16:11.23,Default,,0000,0000,0000,,There is really only want 1X\Nnow. We fiddle that by putting
Dialogue: 0,0:16:11.23,0:16:13.31,Default,,0000,0000,0000,,it to at the top and a half
Dialogue: 0,0:16:13.31,0:16:17.60,Default,,0000,0000,0000,,outside like that. So this\Nintegral is going to workout
Dialogue: 0,0:16:17.60,0:16:21.54,Default,,0000,0000,0000,,to be minus 1/2 the\Nlogarithm of the modulus of
Dialogue: 0,0:16:21.54,0:16:23.11,Default,,0000,0000,0000,,X squared plus one.
Dialogue: 0,0:16:24.71,0:16:27.36,Default,,0000,0000,0000,,And there's a constant of\Nintegration at the end.
Dialogue: 0,0:16:29.40,0:16:32.61,Default,,0000,0000,0000,,And I'll leave the answer like\Nthat if you wanted to do. We
Dialogue: 0,0:16:32.61,0:16:34.59,Default,,0000,0000,0000,,could combine these using the\Nlaws of logarithms.
Dialogue: 0,0:16:35.67,0:16:37.86,Default,,0000,0000,0000,,And that's integration of\Nalgebraic fractions. You
Dialogue: 0,0:16:37.86,0:16:41.30,Default,,0000,0000,0000,,need a lot of practice at\Nthat, and there are more
Dialogue: 0,0:16:41.30,0:16:43.18,Default,,0000,0000,0000,,practice exercises in the\Naccompanying text.