[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.51,0:00:02.80,Default,,0000,0000,0000,,We already know.
Dialogue: 0,0:00:03.96,0:00:08.42,Default,,0000,0000,0000,,When we differentiate\Nlog X.
Dialogue: 0,0:00:09.43,0:00:10.77,Default,,0000,0000,0000,,We end up with.
Dialogue: 0,0:00:13.87,0:00:15.94,Default,,0000,0000,0000,,Is one over X?
Dialogue: 0,0:00:17.23,0:00:24.74,Default,,0000,0000,0000,,We also know that if we've\Ngot Y equals log of a
Dialogue: 0,0:00:24.74,0:00:26.62,Default,,0000,0000,0000,,function of X.
Dialogue: 0,0:00:27.42,0:00:29.06,Default,,0000,0000,0000,,And we differentiate it.
Dialogue: 0,0:00:32.17,0:00:37.66,Default,,0000,0000,0000,,Then what we end up with is the\Nderivative of that function over
Dialogue: 0,0:00:37.66,0:00:39.34,Default,,0000,0000,0000,,the function of X.
Dialogue: 0,0:00:40.52,0:00:44.56,Default,,0000,0000,0000,,Now the point about integrating\Nis if we can recognize something
Dialogue: 0,0:00:44.56,0:00:48.23,Default,,0000,0000,0000,,that's a differential, then we\Ncan simply reverse the process.
Dialogue: 0,0:00:48.23,0:00:52.63,Default,,0000,0000,0000,,So what we're going to be\Nlooking for or looking at in
Dialogue: 0,0:00:52.63,0:00:56.30,Default,,0000,0000,0000,,this case, is functions that\Nlook like this that require
Dialogue: 0,0:00:56.30,0:00:59.97,Default,,0000,0000,0000,,integration, so we can go back\Nfrom there to there.
Dialogue: 0,0:01:01.91,0:01:05.58,Default,,0000,0000,0000,,So let's see if we can just\Nwrite that little bit down
Dialogue: 0,0:01:05.58,0:01:08.34,Default,,0000,0000,0000,,again and then have a look\Nat some examples.
Dialogue: 0,0:01:09.90,0:01:16.86,Default,,0000,0000,0000,,So no, that is why is\Nthe log of a function of
Dialogue: 0,0:01:16.86,0:01:23.24,Default,,0000,0000,0000,,X then divide by The X\Nis the derivative of that
Dialogue: 0,0:01:23.24,0:01:26.14,Default,,0000,0000,0000,,function divided by the\Nfunction.
Dialogue: 0,0:01:27.87,0:01:31.40,Default,,0000,0000,0000,,So therefore, if\Nwe can recognize.
Dialogue: 0,0:01:33.92,0:01:35.25,Default,,0000,0000,0000,,That form.
Dialogue: 0,0:01:38.11,0:01:43.03,Default,,0000,0000,0000,,And we want to integrate it.\NThen we can claim straight away
Dialogue: 0,0:01:43.03,0:01:48.77,Default,,0000,0000,0000,,that this is the log of the\Nfunction of X plus. Of course a
Dialogue: 0,0:01:48.77,0:01:52.46,Default,,0000,0000,0000,,constant of integration because\Nthere are no limits here.
Dialogue: 0,0:01:53.42,0:01:57.19,Default,,0000,0000,0000,,So we're going to be looking for\Nthis. We're going to be looking
Dialogue: 0,0:01:57.19,0:02:00.38,Default,,0000,0000,0000,,at what we've been given to\Nintegrate and can we spot?
Dialogue: 0,0:02:01.46,0:02:05.56,Default,,0000,0000,0000,,A derivative. Over the\Nfunction, or something
Dialogue: 0,0:02:05.56,0:02:06.97,Default,,0000,0000,0000,,approaching a derivative.
Dialogue: 0,0:02:08.05,0:02:11.62,Default,,0000,0000,0000,,So now we've got the result.\NLet's look at some examples.
Dialogue: 0,0:02:12.98,0:02:19.12,Default,,0000,0000,0000,,So we take the integral\Nof tan XDX.
Dialogue: 0,0:02:19.98,0:02:24.90,Default,,0000,0000,0000,,Now it doesn't look much like\None of the examples. We've just
Dialogue: 0,0:02:24.90,0:02:30.23,Default,,0000,0000,0000,,been talking about, but we know\Nthat we can redefine Tan X sign
Dialogue: 0,0:02:30.23,0:02:31.87,Default,,0000,0000,0000,,X over cause X.
Dialogue: 0,0:02:33.22,0:02:38.19,Default,,0000,0000,0000,,And now when we look at the\Nderivative of cars is minus sign
Dialogue: 0,0:02:38.19,0:02:43.01,Default,,0000,0000,0000,,so. The numerator is very nearly\Nthe derivative of the
Dialogue: 0,0:02:43.01,0:02:48.13,Default,,0000,0000,0000,,denominator, so let's make it\Nso. Let's put in minus sign X.
Dialogue: 0,0:02:51.72,0:02:54.97,Default,,0000,0000,0000,,Now having putting the minus\Nsign, we've achieved what we
Dialogue: 0,0:02:54.97,0:02:57.90,Default,,0000,0000,0000,,want. The numerator is the\Nderivative of the denominator,
Dialogue: 0,0:02:57.90,0:03:00.17,Default,,0000,0000,0000,,the top is the derivative of the
Dialogue: 0,0:03:00.17,0:03:05.30,Default,,0000,0000,0000,,bottom. But we need to put in\Nthat balancing minus sign so
Dialogue: 0,0:03:05.30,0:03:08.74,Default,,0000,0000,0000,,that we can retain the\Nequality of these two
Dialogue: 0,0:03:08.74,0:03:12.18,Default,,0000,0000,0000,,expressions. Having done\Nthat, we can now write this
Dialogue: 0,0:03:12.18,0:03:12.56,Default,,0000,0000,0000,,down.
Dialogue: 0,0:03:13.86,0:03:20.68,Default,,0000,0000,0000,,Minus and it's that minus sign.\NThe log of caused X plus a
Dialogue: 0,0:03:20.68,0:03:22.78,Default,,0000,0000,0000,,constant of integration, see.
Dialogue: 0,0:03:24.51,0:03:29.20,Default,,0000,0000,0000,,We're subtracting a log, which\Nmeans we're dividing by what's
Dialogue: 0,0:03:29.20,0:03:30.61,Default,,0000,0000,0000,,within the log.
Dialogue: 0,0:03:31.75,0:03:36.11,Default,,0000,0000,0000,,Function, so we're dividing\Nby cause what we do know is
Dialogue: 0,0:03:36.11,0:03:39.67,Default,,0000,0000,0000,,with. With dividing by\Ncars, then that's one over
Dialogue: 0,0:03:39.67,0:03:44.82,Default,,0000,0000,0000,,cars and that sank. So this\Nis log of sex X Plus C.
Dialogue: 0,0:03:50.52,0:03:54.54,Default,,0000,0000,0000,,Now let's go on again and have a\Nlook at another example.
Dialogue: 0,0:03:56.46,0:04:01.59,Default,,0000,0000,0000,,Integral of X\Nover one plus
Dialogue: 0,0:04:01.59,0:04:04.16,Default,,0000,0000,0000,,X squared DX.
Dialogue: 0,0:04:06.25,0:04:10.92,Default,,0000,0000,0000,,Look at the bottom and\Ndifferentiate it. Its derivative
Dialogue: 0,0:04:10.92,0:04:17.67,Default,,0000,0000,0000,,is 2X only got X on top,\Nthat's no problem. Let's make it
Dialogue: 0,0:04:17.67,0:04:21.30,Default,,0000,0000,0000,,2X on top by multiplying by two.
Dialogue: 0,0:04:22.82,0:04:25.98,Default,,0000,0000,0000,,If we've multiplied by two,\Nwe've got to divide by two,
Dialogue: 0,0:04:25.98,0:04:29.13,Default,,0000,0000,0000,,and that means we want a\Nhalf of that result there.
Dialogue: 0,0:04:29.13,0:04:32.58,Default,,0000,0000,0000,,So now this is balanced out\Nand it's the same as that.
Dialogue: 0,0:04:34.15,0:04:38.45,Default,,0000,0000,0000,,What we've got on the top\Nnow is very definitely the
Dialogue: 0,0:04:38.45,0:04:41.97,Default,,0000,0000,0000,,derivative of what's on\Nthe bottom, so again, we
Dialogue: 0,0:04:41.97,0:04:47.05,Default,,0000,0000,0000,,can have a half the log of\None plus X squared plus C.
Dialogue: 0,0:04:48.36,0:04:53.81,Default,,0000,0000,0000,,We can even have this with\Nlook like very complicated
Dialogue: 0,0:04:53.81,0:04:59.80,Default,,0000,0000,0000,,functions, so let's have a\Nlook at one over X Times
Dialogue: 0,0:04:59.80,0:05:02.53,Default,,0000,0000,0000,,the natural log of X.
Dialogue: 0,0:05:07.78,0:05:12.92,Default,,0000,0000,0000,,Doesn't look like what we've got\Ndoes it? But let's remember that
Dialogue: 0,0:05:12.92,0:05:19.34,Default,,0000,0000,0000,,the derivative of log X is one\Nover X. So if I write this a
Dialogue: 0,0:05:19.34,0:05:24.04,Default,,0000,0000,0000,,little bit differently, one over\NX divided by log X DX.
Dialogue: 0,0:05:25.59,0:05:30.74,Default,,0000,0000,0000,,Then we can see that what's on\Ntop is indeed the derivative of
Dialogue: 0,0:05:30.74,0:05:35.49,Default,,0000,0000,0000,,what's on the bottom, and so,\Nagain, this is the log of.
Dialogue: 0,0:05:37.00,0:05:43.00,Default,,0000,0000,0000,,Log of X plus a constant of\Nintegration. See, so even in
Dialogue: 0,0:05:43.00,0:05:48.50,Default,,0000,0000,0000,,something like that we can find\Nwhat it is we're actually
Dialogue: 0,0:05:48.50,0:05:51.50,Default,,0000,0000,0000,,looking for. Let's take one more
Dialogue: 0,0:05:51.50,0:05:57.32,Default,,0000,0000,0000,,example. A little bit contrived,\Nbut it does show you how you
Dialogue: 0,0:05:57.32,0:06:00.16,Default,,0000,0000,0000,,need to work and look to see.
Dialogue: 0,0:06:01.72,0:06:03.51,Default,,0000,0000,0000,,If what you've got on the.
Dialogue: 0,0:06:04.22,0:06:08.68,Default,,0000,0000,0000,,Numerator is in fact the\Nderivative of the denominator.
Dialogue: 0,0:06:14.27,0:06:15.93,Default,,0000,0000,0000,,So let's have a look at this.
Dialogue: 0,0:06:18.34,0:06:23.16,Default,,0000,0000,0000,,Looks quite fearsome as it's\Nwritten, but let's just think
Dialogue: 0,0:06:23.16,0:06:29.43,Default,,0000,0000,0000,,about what we would get if we\Ndifferentiate it X Sign X. I'll
Dialogue: 0,0:06:29.43,0:06:36.17,Default,,0000,0000,0000,,just do that over here. Let's\Nsay Y equals X sign X. Now this
Dialogue: 0,0:06:36.17,0:06:43.89,Default,,0000,0000,0000,,is a product, it's a U times by\NAV, so we know that if Y equals
Dialogue: 0,0:06:43.89,0:06:46.30,Default,,0000,0000,0000,,UV when we do the
Dialogue: 0,0:06:46.30,0:06:53.59,Default,,0000,0000,0000,,differentiation. Why by DX\Nis UDV by the X
Dialogue: 0,0:06:53.59,0:06:56.79,Default,,0000,0000,0000,,plus VDU by DX?
Dialogue: 0,0:06:57.72,0:07:04.91,Default,,0000,0000,0000,,So in this case you is\NX&V is synex, so that's X
Dialogue: 0,0:07:04.91,0:07:09.10,Default,,0000,0000,0000,,cause X Plus V which is\Nsynex.
Dialogue: 0,0:07:11.35,0:07:16.76,Default,,0000,0000,0000,,Times by du by DX, but you was X\Nor do you buy X is just one?
Dialogue: 0,0:07:17.37,0:07:20.20,Default,,0000,0000,0000,,So if we look what we can see
Dialogue: 0,0:07:20.20,0:07:23.22,Default,,0000,0000,0000,,here. Is that the numerator?
Dialogue: 0,0:07:23.91,0:07:29.56,Default,,0000,0000,0000,,Is X cause X sign X, which is\Nthe derivative of the
Dialogue: 0,0:07:29.56,0:07:34.27,Default,,0000,0000,0000,,denominator X sign X, and so\Nagain complicated though it
Dialogue: 0,0:07:34.27,0:07:39.45,Default,,0000,0000,0000,,looks we've been able to spot\Nthat the numerator is again
Dialogue: 0,0:07:39.45,0:07:44.16,Default,,0000,0000,0000,,the derivative of the\Ndenominator, and so we can say
Dialogue: 0,0:07:44.16,0:07:49.82,Default,,0000,0000,0000,,straight away that the result\Nof this integral is the log of
Dialogue: 0,0:07:49.82,0:07:52.17,Default,,0000,0000,0000,,the denominator X Sign X.
Dialogue: 0,0:07:54.38,0:07:58.22,Default,,0000,0000,0000,,Sometimes you have to look very\Nclosely and let's just remember
Dialogue: 0,0:07:58.22,0:08:02.41,Default,,0000,0000,0000,,if we just look back at this\None. But sometimes you might
Dialogue: 0,0:08:02.41,0:08:06.94,Default,,0000,0000,0000,,have to balance the function in\Norder to be able to make it
Dialogue: 0,0:08:06.94,0:08:11.13,Default,,0000,0000,0000,,look like you want it to look.\NBut quite often it's fairly
Dialogue: 0,0:08:11.13,0:08:15.67,Default,,0000,0000,0000,,clear that that's what you have\Nto do. So do remember, this is
Dialogue: 0,0:08:15.67,0:08:19.16,Default,,0000,0000,0000,,a very typical standard form of\Nintegration of very important
Dialogue: 0,0:08:19.16,0:08:22.100,Default,,0000,0000,0000,,one, and one that occurs a\Ngreat deal when looking at
Dialogue: 0,0:08:22.100,0:08:23.70,Default,,0000,0000,0000,,differential equations.