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Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.34,0:00:04.62,Default,,0000,0000,0000,,Sometimes integrals involving\Ntrigonometric functions can be
Dialogue: 0,0:00:04.62,0:00:08.83,Default,,0000,0000,0000,,evaluated by first of all using\Ntrigonometric identities to
Dialogue: 0,0:00:08.83,0:00:13.04,Default,,0000,0000,0000,,rewrite the integrand. That's\Nthe quantity we're trying to
Dialogue: 0,0:00:13.04,0:00:18.19,Default,,0000,0000,0000,,integrate an alternative form,\Nwhich is a bit more amenable to
Dialogue: 0,0:00:18.19,0:00:21.64,Default,,0000,0000,0000,,integration. Sometimes a\Ntrigonometric substitution is
Dialogue: 0,0:00:21.64,0:00:26.18,Default,,0000,0000,0000,,appropriate. Both of these\Ntechniques we look at in this
Dialogue: 0,0:00:26.18,0:00:31.03,Default,,0000,0000,0000,,unit. Before we start I want to\Ngive you a couple of preliminary
Dialogue: 0,0:00:31.03,0:00:35.02,Default,,0000,0000,0000,,results which will be using over\Nand over again and which will be
Dialogue: 0,0:00:35.02,0:00:39.32,Default,,0000,0000,0000,,very important and the first one\Nis I want you to make sure that
Dialogue: 0,0:00:39.32,0:00:43.01,Default,,0000,0000,0000,,you know that the integral of\Nthe cosine of a constant times
Dialogue: 0,0:00:43.01,0:00:45.77,Default,,0000,0000,0000,,X. With respect to X.
Dialogue: 0,0:00:46.29,0:00:49.70,Default,,0000,0000,0000,,Is equal to one over that\Nconstant.
Dialogue: 0,0:00:50.85,0:00:55.23,Default,,0000,0000,0000,,Multiplied by the sign of KX\Nplus a constant of integration
Dialogue: 0,0:00:55.23,0:00:59.61,Default,,0000,0000,0000,,as a very important result. If\Nyou integrate the cosine, you
Dialogue: 0,0:00:59.61,0:01:00.80,Default,,0000,0000,0000,,get a sign.
Dialogue: 0,0:01:01.97,0:01:05.42,Default,,0000,0000,0000,,And if there's a constant in\Nfront of the X that appears down
Dialogue: 0,0:01:05.42,0:01:08.60,Default,,0000,0000,0000,,here will take that as read in\Nall the examples which follow.
Dialogue: 0,0:01:09.36,0:01:15.24,Default,,0000,0000,0000,,Another important results is the\Nintegral of a sign. The integral
Dialogue: 0,0:01:15.24,0:01:22.74,Default,,0000,0000,0000,,of sine KX with respect to X\Nis minus one over K cosine KX.
Dialogue: 0,0:01:23.39,0:01:26.95,Default,,0000,0000,0000,,Plus a constant we're\Nintegrating assign. The result
Dialogue: 0,0:01:26.95,0:01:31.40,Default,,0000,0000,0000,,is minus the cosine and the\Nconstant factor. There appears
Dialogue: 0,0:01:31.40,0:01:34.52,Default,,0000,0000,0000,,out down here as well, so those
Dialogue: 0,0:01:34.52,0:01:37.30,Default,,0000,0000,0000,,two results. Very important.
Dialogue: 0,0:01:37.90,0:01:40.72,Default,,0000,0000,0000,,You should have them at your\Nfingertips and we can call upon
Dialogue: 0,0:01:40.72,0:01:43.30,Default,,0000,0000,0000,,them whenever we want them in\Nthe rest of the video.
Dialogue: 0,0:01:44.09,0:01:47.90,Default,,0000,0000,0000,,We also want to call appan\Ntrigonometric identity's. I'm
Dialogue: 0,0:01:47.90,0:01:52.13,Default,,0000,0000,0000,,going to assume that you've seen\Na lot of trigonometric
Dialogue: 0,0:01:52.13,0:01:55.38,Default,,0000,0000,0000,,identities before. We have a\Ntable of trigonometric
Dialogue: 0,0:01:55.38,0:01:58.65,Default,,0000,0000,0000,,identities here, such as the\Ntable that you might have seen
Dialogue: 0,0:01:58.65,0:02:01.92,Default,,0000,0000,0000,,many times before. If you want\Nthis specific table, you'll find
Dialogue: 0,0:02:01.92,0:02:03.40,Default,,0000,0000,0000,,it in the printed notes
Dialogue: 0,0:02:03.40,0:02:07.02,Default,,0000,0000,0000,,accompanying the video. Why\Nmight we want to use
Dialogue: 0,0:02:07.02,0:02:09.98,Default,,0000,0000,0000,,trigonometric identities?\NWell, for example, we've just
Dialogue: 0,0:02:09.98,0:02:14.11,Default,,0000,0000,0000,,seen that we already know how\Nto integrate the sign of a
Dialogue: 0,0:02:14.11,0:02:17.89,Default,,0000,0000,0000,,quantity and the cosine of the\Nquantity. But suppose we want
Dialogue: 0,0:02:17.89,0:02:21.33,Default,,0000,0000,0000,,to integrate assign multiplied\Nby a cosine or cosine times
Dialogue: 0,0:02:21.33,0:02:24.43,Default,,0000,0000,0000,,cosine or assigned times\Nassign. We don't actually know
Dialogue: 0,0:02:24.43,0:02:27.87,Default,,0000,0000,0000,,how to do those integrals.\NIntegrals at the moment, but
Dialogue: 0,0:02:27.87,0:02:30.62,Default,,0000,0000,0000,,if we use trigonometric\Nidentities, we can rewrite
Dialogue: 0,0:02:30.62,0:02:34.40,Default,,0000,0000,0000,,these in terms of just single\Nsine and cosine terms, which
Dialogue: 0,0:02:34.40,0:02:35.78,Default,,0000,0000,0000,,we can then integrate.
Dialogue: 0,0:02:36.90,0:02:39.66,Default,,0000,0000,0000,,Also, the trigonometric\Nidentities identities allow us
Dialogue: 0,0:02:39.66,0:02:44.01,Default,,0000,0000,0000,,to integrate powers of sines and\Ncosines. You'll see that using
Dialogue: 0,0:02:44.01,0:02:47.96,Default,,0000,0000,0000,,these identity's? We've got\Npowers of cosine powers of sign
Dialogue: 0,0:02:47.96,0:02:53.10,Default,,0000,0000,0000,,and the identity is allow us to\Nwrite into grams in terms of
Dialogue: 0,0:02:53.10,0:02:55.07,Default,,0000,0000,0000,,cosines and sines of double
Dialogue: 0,0:02:55.07,0:02:59.07,Default,,0000,0000,0000,,angles. We know how to integrate\Nthese already using the results.
Dialogue: 0,0:02:59.07,0:03:02.65,Default,,0000,0000,0000,,I've just reminded you of, so\NI'm going to assume that you've
Dialogue: 0,0:03:02.65,0:03:06.22,Default,,0000,0000,0000,,got a table like this at your\Nfingertips, and we can call
Dialogue: 0,0:03:06.22,0:03:08.01,Default,,0000,0000,0000,,appan it whenever we need to.
Dialogue: 0,0:03:08.50,0:03:12.47,Default,,0000,0000,0000,,OK, let's have a look at the\Nfirst example and the example
Dialogue: 0,0:03:12.47,0:03:16.44,Default,,0000,0000,0000,,that I'm going to look at is a\Ndefinite integral. The integral
Dialogue: 0,0:03:16.44,0:03:22.40,Default,,0000,0000,0000,,from X is not to X is π of the\Nsine squared of X DX. So note in
Dialogue: 0,0:03:22.40,0:03:25.71,Default,,0000,0000,0000,,particular, we've gotta power\Nhere. We're looking at the sign
Dialogue: 0,0:03:25.71,0:03:30.35,Default,,0000,0000,0000,,squared of X. What I'm going to\Ndo is go back to the table.
Dialogue: 0,0:03:31.07,0:03:34.93,Default,,0000,0000,0000,,And look for an identity that\Nwill allow us to change the sign
Dialogue: 0,0:03:34.93,0:03:38.50,Default,,0000,0000,0000,,squared X into something else.\NLet me just flip back to the
Dialogue: 0,0:03:38.50,0:03:39.39,Default,,0000,0000,0000,,table of trigonometric
Dialogue: 0,0:03:39.39,0:03:43.96,Default,,0000,0000,0000,,identities. The identity that\NI'm going to use this one, the
Dialogue: 0,0:03:43.96,0:03:45.08,Default,,0000,0000,0000,,cosine of 2A.
Dialogue: 0,0:03:45.65,0:03:48.62,Default,,0000,0000,0000,,Is 1 minus twice\Nsign square day?
Dialogue: 0,0:03:49.65,0:03:52.88,Default,,0000,0000,0000,,If you inspect this carefully,\Nyou'll see that this will enable
Dialogue: 0,0:03:52.88,0:03:56.71,Default,,0000,0000,0000,,us to change a sine squared into\Nthe cosine of a double angle.
Dialogue: 0,0:03:57.69,0:04:00.58,Default,,0000,0000,0000,,Let me write that down again.
Dialogue: 0,0:04:00.58,0:04:07.40,Default,,0000,0000,0000,,Cosine of 2 A\Nis equal to 1
Dialogue: 0,0:04:07.40,0:04:10.80,Default,,0000,0000,0000,,minus twice sign squared
Dialogue: 0,0:04:10.80,0:04:13.99,Default,,0000,0000,0000,,A. First of all, I'm going\Nto rearrange this to get
Dialogue: 0,0:04:13.99,0:04:15.16,Default,,0000,0000,0000,,sine squared on its own.
Dialogue: 0,0:04:16.26,0:04:20.73,Default,,0000,0000,0000,,If we add two sine squared data\Nboth sides, then I can get it on
Dialogue: 0,0:04:20.73,0:04:27.06,Default,,0000,0000,0000,,this side. And if I subtract\Ncosine 2A from both sides, are
Dialogue: 0,0:04:27.06,0:04:29.66,Default,,0000,0000,0000,,remove it from the left.
Dialogue: 0,0:04:29.79,0:04:37.19,Default,,0000,0000,0000,,Finally, if I divide both\Nsides by two, I'll be
Dialogue: 0,0:04:37.19,0:04:40.89,Default,,0000,0000,0000,,left with sine squared A.
Dialogue: 0,0:04:40.90,0:04:46.64,Default,,0000,0000,0000,,And this is the result that I\Nwant to use to help me to
Dialogue: 0,0:04:46.64,0:04:51.56,Default,,0000,0000,0000,,evaluate this integral because\Nof what it will allow me to do.
Dialogue: 0,0:04:51.56,0:04:56.89,Default,,0000,0000,0000,,Is it will allow me to change a\Nquantity involving the square of
Dialogue: 0,0:04:56.89,0:05:00.99,Default,,0000,0000,0000,,a trig function into a quantity\Ninvolving double angles. So
Dialogue: 0,0:05:00.99,0:05:03.45,Default,,0000,0000,0000,,let's use it in this case.
Dialogue: 0,0:05:04.41,0:05:10.47,Default,,0000,0000,0000,,The integral will become the\Nintegral from note to pie.
Dialogue: 0,0:05:11.34,0:05:18.45,Default,,0000,0000,0000,,Sine squared X using this\Nformula will be 1 minus cosine
Dialogue: 0,0:05:18.45,0:05:22.27,Default,,0000,0000,0000,,twice X. All divided by
Dialogue: 0,0:05:22.27,0:05:24.71,Default,,0000,0000,0000,,two. Integrated with\Nrespect to X.
Dialogue: 0,0:05:26.12,0:05:31.63,Default,,0000,0000,0000,,I've taken out the fact that 1/2\Nhere and I'm left with the
Dialogue: 0,0:05:31.63,0:05:36.72,Default,,0000,0000,0000,,numerator 1 minus cosine 2X to\Nbe integrated with respect to X.
Dialogue: 0,0:05:37.62,0:05:39.12,Default,,0000,0000,0000,,This is straightforward to
Dialogue: 0,0:05:39.12,0:05:43.88,Default,,0000,0000,0000,,finish off. So definite\Nintegral. So I have square
Dialogue: 0,0:05:43.88,0:05:49.45,Default,,0000,0000,0000,,brackets. The integral of one\Nwith respect to X is simply X.
Dialogue: 0,0:05:50.47,0:05:55.03,Default,,0000,0000,0000,,And the integral of cosine 2 X\Nwe know from our preliminary
Dialogue: 0,0:05:55.03,0:06:00.35,Default,,0000,0000,0000,,work is just going to be sine\N2X divided by two with a minus
Dialogue: 0,0:06:00.35,0:06:03.77,Default,,0000,0000,0000,,sign there and the limits are\Nnot and pie.
Dialogue: 0,0:06:05.77,0:06:10.42,Default,,0000,0000,0000,,We finish this off by first of\Nall, putting the upper limit in,
Dialogue: 0,0:06:10.42,0:06:14.36,Default,,0000,0000,0000,,so we want X replaced by pie\Nhere and pie here.
Dialogue: 0,0:06:15.07,0:06:17.62,Default,,0000,0000,0000,,The sign of 2π is 0.
Dialogue: 0,0:06:18.52,0:06:22.64,Default,,0000,0000,0000,,So when we put the upper limit\Nin will just get.
Dialogue: 0,0:06:22.93,0:06:26.32,Default,,0000,0000,0000,,Pie by substituting for X here.
Dialogue: 0,0:06:27.03,0:06:30.88,Default,,0000,0000,0000,,Let me put the lower limit in.
Dialogue: 0,0:06:30.88,0:06:33.55,Default,,0000,0000,0000,,X being not will be 0 here.
Dialogue: 0,0:06:34.21,0:06:38.83,Default,,0000,0000,0000,,And sign of note here, which is\Nnot so both of those terms will
Dialogue: 0,0:06:38.83,0:06:43.12,Default,,0000,0000,0000,,become zero when we put the\Nlower limit in and so we're just
Dialogue: 0,0:06:43.12,0:06:45.76,Default,,0000,0000,0000,,left with simply 1/2 of Π or π
Dialogue: 0,0:06:45.76,0:06:51.04,Default,,0000,0000,0000,,by 2. And that's our first\Nexample of how we've used a
Dialogue: 0,0:06:51.04,0:06:53.69,Default,,0000,0000,0000,,trigonometric identity to\Nrewrite an integrand involving
Dialogue: 0,0:06:53.69,0:06:58.22,Default,,0000,0000,0000,,powers of a trig function in\Nterms of double angles, which we
Dialogue: 0,0:06:58.22,0:07:00.11,Default,,0000,0000,0000,,already know how to integrate.
Dialogue: 0,0:07:01.36,0:07:09.19,Default,,0000,0000,0000,,Let's have a look at\Nanother example. Suppose we want
Dialogue: 0,0:07:09.19,0:07:17.02,Default,,0000,0000,0000,,to integrate the sign of\Nthree X multiplied by the
Dialogue: 0,0:07:17.02,0:07:20.15,Default,,0000,0000,0000,,cosine of 2 X.
Dialogue: 0,0:07:20.16,0:07:21.25,Default,,0000,0000,0000,,With respect to X.
Dialogue: 0,0:07:21.77,0:07:26.31,Default,,0000,0000,0000,,Now we already know how to\Nintegrate signs. We know how to
Dialogue: 0,0:07:26.31,0:07:30.46,Default,,0000,0000,0000,,integrate cosines, but we have a\Nproblem here because there's a
Dialogue: 0,0:07:30.46,0:07:34.24,Default,,0000,0000,0000,,product. These two terms are\Nmultiplied together and we don't
Dialogue: 0,0:07:34.24,0:07:35.76,Default,,0000,0000,0000,,know how to proceed.
Dialogue: 0,0:07:36.59,0:07:41.28,Default,,0000,0000,0000,,What we do is look in our table\Nof trigonometric identities for
Dialogue: 0,0:07:41.28,0:07:45.19,Default,,0000,0000,0000,,an example where we've gotta\Nsign multiplied by a cosine.
Dialogue: 0,0:07:45.19,0:07:47.54,Default,,0000,0000,0000,,Let's go back to the table.
Dialogue: 0,0:07:47.56,0:07:53.74,Default,,0000,0000,0000,,The first entry in our\Ntable involves assign multiplied
Dialogue: 0,0:07:53.74,0:07:55.80,Default,,0000,0000,0000,,by a cosine.
Dialogue: 0,0:07:56.66,0:08:02.05,Default,,0000,0000,0000,,Let me write this formula down\Nagain. 2 sign a cosine be.
Dialogue: 0,0:08:02.05,0:08:07.69,Default,,0000,0000,0000,,Is\Nequal
Dialogue: 0,0:08:07.69,0:08:14.30,Default,,0000,0000,0000,,to. The\Nsign of the sum of A and be
Dialogue: 0,0:08:14.30,0:08:16.45,Default,,0000,0000,0000,,added to the sign of the
Dialogue: 0,0:08:16.45,0:08:23.94,Default,,0000,0000,0000,,difference A-B. And this is\Nthe identity that I
Dialogue: 0,0:08:23.94,0:08:31.66,Default,,0000,0000,0000,,will use in order\Nto rewrite this integrand
Dialogue: 0,0:08:31.66,0:08:39.38,Default,,0000,0000,0000,,as two separate integrals.\NWe identify the A's
Dialogue: 0,0:08:39.38,0:08:43.28,Default,,0000,0000,0000,,3X. The B is 2 X.
Dialogue: 0,0:08:44.05,0:08:48.71,Default,,0000,0000,0000,,The factor of 2 here isn't a\Nproblem. We can divide
Dialogue: 0,0:08:48.71,0:08:50.41,Default,,0000,0000,0000,,everything through by two.
Dialogue: 0,0:08:50.42,0:08:51.78,Default,,0000,0000,0000,,So we lose it from this side.
Dialogue: 0,0:08:52.77,0:08:57.97,Default,,0000,0000,0000,,So our integral? What will it\Nbecome? Well, the integral of
Dialogue: 0,0:08:57.97,0:09:03.65,Default,,0000,0000,0000,,sign 3X cosine 2X DX will\Nbecome. We want the integral of
Dialogue: 0,0:09:03.65,0:09:06.96,Default,,0000,0000,0000,,the sign of the sum of A&B.
Dialogue: 0,0:09:07.54,0:09:12.24,Default,,0000,0000,0000,,Well, there's some of A&B will\Nbe 3X plus 2X, which is 5X. So
Dialogue: 0,0:09:12.24,0:09:14.26,Default,,0000,0000,0000,,we want the sign of 5X.
Dialogue: 0,0:09:15.52,0:09:19.50,Default,,0000,0000,0000,,Added to the sign of the\Ndifference of amb. Well a
Dialogue: 0,0:09:19.50,0:09:23.85,Default,,0000,0000,0000,,being 3X B being 2X A-B\Nwill be 3X subtract 2 X
Dialogue: 0,0:09:23.85,0:09:28.55,Default,,0000,0000,0000,,which is just One X. So we\Nwant the sign of X all
Dialogue: 0,0:09:28.55,0:09:32.17,Default,,0000,0000,0000,,divided by two and we want\Nto integrate that with
Dialogue: 0,0:09:32.17,0:09:33.26,Default,,0000,0000,0000,,respect to X.
Dialogue: 0,0:09:34.51,0:09:38.100,Default,,0000,0000,0000,,So what have we done? We've used\Nthe trig identity to change the
Dialogue: 0,0:09:38.100,0:09:43.14,Default,,0000,0000,0000,,product of a signing cosine into\Nthe sum of two separate sign
Dialogue: 0,0:09:43.14,0:09:46.58,Default,,0000,0000,0000,,terms, which we can integrate\Nstraight away. We can integrate
Dialogue: 0,0:09:46.58,0:09:48.66,Default,,0000,0000,0000,,that taking the factor of 1/2
Dialogue: 0,0:09:48.66,0:09:56.31,Default,,0000,0000,0000,,out. The integral of sign 5X\Nwill be minus the cosine of 5X
Dialogue: 0,0:09:56.31,0:09:58.00,Default,,0000,0000,0000,,divided by 5.
Dialogue: 0,0:09:58.74,0:10:03.63,Default,,0000,0000,0000,,And the integral of sine X will\Nbe just minus cosine X, and
Dialogue: 0,0:10:03.63,0:10:05.51,Default,,0000,0000,0000,,they'll be a constant of
Dialogue: 0,0:10:05.51,0:10:10.92,Default,,0000,0000,0000,,integration. And just to tidy it\Nup, at the end we're going to
Dialogue: 0,0:10:10.92,0:10:14.97,Default,,0000,0000,0000,,have minus the half with the\Nfive at the bottom. There will
Dialogue: 0,0:10:14.97,0:10:18.00,Default,,0000,0000,0000,,give you minus cosine 5X all\Ndivided by 10.
Dialogue: 0,0:10:19.06,0:10:23.09,Default,,0000,0000,0000,,And there's a half with this\Nterm here, so it's minus cosine
Dialogue: 0,0:10:23.09,0:10:24.44,Default,,0000,0000,0000,,X divided by two.
Dialogue: 0,0:10:25.22,0:10:27.95,Default,,0000,0000,0000,,Plus a constant of integration.
Dialogue: 0,0:10:28.50,0:10:30.29,Default,,0000,0000,0000,,And that's the solution of this
Dialogue: 0,0:10:30.29,0:10:35.42,Default,,0000,0000,0000,,problem. Let's explore the\Nintegral of products of sines
Dialogue: 0,0:10:35.42,0:10:41.90,Default,,0000,0000,0000,,and cosines a little bit\Nfurther, and what I want to look
Dialogue: 0,0:10:41.90,0:10:48.92,Default,,0000,0000,0000,,at now is integrals of the form\Nthe integral of sign to the
Dialogue: 0,0:10:48.92,0:10:54.32,Default,,0000,0000,0000,,power MX multiplied by cosine to\Nthe power NX DX.
Dialogue: 0,0:10:54.93,0:10:58.23,Default,,0000,0000,0000,,Well, look at a whole family of\Nintegrals like this, but in
Dialogue: 0,0:10:58.23,0:11:01.80,Default,,0000,0000,0000,,particular for the first example\NI'm going to look at the case of
Dialogue: 0,0:11:01.80,0:11:03.73,Default,,0000,0000,0000,,what happens when M is an odd
Dialogue: 0,0:11:03.73,0:11:09.30,Default,,0000,0000,0000,,number. Whenever you have an\Nintegral like this, when M is
Dialogue: 0,0:11:09.30,0:11:14.24,Default,,0000,0000,0000,,odd, the following process will\Nwork. Let's look at a specific
Dialogue: 0,0:11:14.24,0:11:18.28,Default,,0000,0000,0000,,case, supposing I want to\Nintegrate sine cubed X.
Dialogue: 0,0:11:18.82,0:11:23.50,Default,,0000,0000,0000,,Multiplied by cosine\Nsquared XDX.
Dialogue: 0,0:11:24.59,0:11:27.53,Default,,0000,0000,0000,,Notice that M.
Dialogue: 0,0:11:28.22,0:11:30.67,Default,,0000,0000,0000,,Is an odd number and is 3.
Dialogue: 0,0:11:31.56,0:11:35.36,Default,,0000,0000,0000,,There's a little trick here that\Nwe're going to do now, and it's
Dialogue: 0,0:11:35.36,0:11:38.86,Default,,0000,0000,0000,,the sort of trick that comes\Nwith practice and seeing lots of
Dialogue: 0,0:11:38.86,0:11:42.36,Default,,0000,0000,0000,,examples. What we're going to do\Nis we're going to rewrite the
Dialogue: 0,0:11:42.36,0:11:44.12,Default,,0000,0000,0000,,sign cubed X in a slightly
Dialogue: 0,0:11:44.12,0:11:49.28,Default,,0000,0000,0000,,different form. We're going to\Nrecognize that sign cubed can be
Dialogue: 0,0:11:49.28,0:11:53.29,Default,,0000,0000,0000,,written as sine squared X\Nmultiplied by Sign X.
Dialogue: 0,0:11:53.80,0:11:57.57,Default,,0000,0000,0000,,That's a little trick. The sign\Ncubed can be written as sine
Dialogue: 0,0:11:57.57,0:12:01.41,Default,,0000,0000,0000,,squared times sign. So\Nour integral can be
Dialogue: 0,0:12:01.41,0:12:05.20,Default,,0000,0000,0000,,written as sine squared\NX times sign X
Dialogue: 0,0:12:05.20,0:12:08.05,Default,,0000,0000,0000,,multiplied by cosine\Nsquared X DX.
Dialogue: 0,0:12:09.24,0:12:12.67,Default,,0000,0000,0000,,And then I'm going to pick a\Ntrigonometric identity involving
Dialogue: 0,0:12:12.67,0:12:16.44,Default,,0000,0000,0000,,sine squared to write it in\Nterms of cosine squared. Let's
Dialogue: 0,0:12:16.44,0:12:17.47,Default,,0000,0000,0000,,find that identity.
Dialogue: 0,0:12:18.09,0:12:21.12,Default,,0000,0000,0000,,With an identity here, which\Nsays that sine squared of an
Dialogue: 0,0:12:21.12,0:12:22.76,Default,,0000,0000,0000,,angle plus cost squared of an
Dialogue: 0,0:12:22.76,0:12:27.83,Default,,0000,0000,0000,,angle is one. If we rearrange\Nthis, we can write that sine
Dialogue: 0,0:12:27.83,0:12:32.72,Default,,0000,0000,0000,,squared of an angle is 1 minus\Nthe cosine squared of an angle
Dialogue: 0,0:12:32.72,0:12:33.85,Default,,0000,0000,0000,,will use that.
Dialogue: 0,0:12:34.70,0:12:38.60,Default,,0000,0000,0000,,Sine squared of any
Dialogue: 0,0:12:38.60,0:12:44.86,Default,,0000,0000,0000,,angle. Is equal to 1 minus\Nthe cosine squared over any
Dialogue: 0,0:12:44.86,0:12:51.29,Default,,0000,0000,0000,,angle. Will use that in here to\Nchange the sign squared X into
Dialogue: 0,0:12:51.29,0:12:55.74,Default,,0000,0000,0000,,terms involving cosine squared\NX. Let's see what happens. This
Dialogue: 0,0:12:55.74,0:13:00.19,Default,,0000,0000,0000,,integral will become the\Nintegral of or sign squared X.
Dialogue: 0,0:13:00.79,0:13:03.50,Default,,0000,0000,0000,,Will become one minus cosine
Dialogue: 0,0:13:03.50,0:13:09.32,Default,,0000,0000,0000,,squared X. There's still\Nthe terms cynex.
Dialogue: 0,0:13:11.62,0:13:13.66,Default,,0000,0000,0000,,And at the end we still got
Dialogue: 0,0:13:13.66,0:13:17.38,Default,,0000,0000,0000,,cosine squared X. Now this is\Nlooking a bit complicated, but
Dialogue: 0,0:13:17.38,0:13:20.64,Default,,0000,0000,0000,,as we'll see it's all going to\Ncome out in the Wash. Let's
Dialogue: 0,0:13:20.64,0:13:22.15,Default,,0000,0000,0000,,remove the brackets here and see
Dialogue: 0,0:13:22.15,0:13:27.25,Default,,0000,0000,0000,,what we've got. There's a one\Nmultiplied by all this sign X
Dialogue: 0,0:13:27.25,0:13:28.77,Default,,0000,0000,0000,,times cosine squared X.
Dialogue: 0,0:13:29.44,0:13:33.26,Default,,0000,0000,0000,,So that's just sign X\Ntimes cosine squared X
Dialogue: 0,0:13:33.26,0:13:37.09,Default,,0000,0000,0000,,will want to integrate\Nthat with respect to X.
Dialogue: 0,0:13:38.51,0:13:42.44,Default,,0000,0000,0000,,There's also cosine squared X\Nmultiplied by all this.
Dialogue: 0,0:13:42.98,0:13:47.47,Default,,0000,0000,0000,,Now the cosine squared X with\Nthis cosine squared X will give
Dialogue: 0,0:13:47.47,0:13:50.09,Default,,0000,0000,0000,,us a cosine, so the power 4X.
Dialogue: 0,0:13:51.84,0:13:53.92,Default,,0000,0000,0000,,There's also the sign X.
Dialogue: 0,0:13:54.85,0:13:56.80,Default,,0000,0000,0000,,And we want to integrate that.
Dialogue: 0,0:13:57.38,0:14:00.85,Default,,0000,0000,0000,,Also, with respect to X and\Nthere was a minus sign in front,
Dialogue: 0,0:14:00.85,0:14:02.72,Default,,0000,0000,0000,,so that's going to go in there.
Dialogue: 0,0:14:03.35,0:14:05.70,Default,,0000,0000,0000,,So we've expanded the\Nbrackets here and written.
Dialogue: 0,0:14:05.70,0:14:07.17,Default,,0000,0000,0000,,This is 2 separate integrals.
Dialogue: 0,0:14:08.41,0:14:13.59,Default,,0000,0000,0000,,Now, each of these integrals can\Nbe evaluated by making a
Dialogue: 0,0:14:13.59,0:14:18.30,Default,,0000,0000,0000,,substitution. If we make a\Nsubstitution and let you equals
Dialogue: 0,0:14:18.30,0:14:24.17,Default,,0000,0000,0000,,cosine X. The differential du\Nis du DX.
Dialogue: 0,0:14:24.97,0:14:30.88,Default,,0000,0000,0000,,DX Do you DX if we\Ndifferentiate cosine, X will get
Dialogue: 0,0:14:30.88,0:14:32.56,Default,,0000,0000,0000,,minus the sign X.
Dialogue: 0,0:14:33.11,0:14:36.58,Default,,0000,0000,0000,,So we've got du is minus sign X
Dialogue: 0,0:14:36.58,0:14:42.56,Default,,0000,0000,0000,,DX. Now look at what we've got\Nwhen we make this substitution.
Dialogue: 0,0:14:42.56,0:14:48.02,Default,,0000,0000,0000,,The cosine squared X will become\Nsimply you squared and sign X DX
Dialogue: 0,0:14:48.02,0:14:53.06,Default,,0000,0000,0000,,altogether can be written as a\Nminus du, so this will become.
Dialogue: 0,0:14:53.83,0:14:55.32,Default,,0000,0000,0000,,Minus the integral.
Dialogue: 0,0:14:56.01,0:14:57.42,Default,,0000,0000,0000,,Of you squared.
Dialogue: 0,0:14:58.01,0:14:58.94,Default,,0000,0000,0000,,Do you?
Dialogue: 0,0:15:01.25,0:15:06.40,Default,,0000,0000,0000,,What about this term? We've got\Ncosine to the power four cosine
Dialogue: 0,0:15:06.40,0:15:09.83,Default,,0000,0000,0000,,to the power 4X will be you to
Dialogue: 0,0:15:09.83,0:15:15.24,Default,,0000,0000,0000,,the powerful. And sign X\NDX sign X DX is minus DU.
Dialogue: 0,0:15:15.24,0:15:18.42,Default,,0000,0000,0000,,There's another minus\Nsign here, so overall
Dialogue: 0,0:15:18.42,0:15:22.52,Default,,0000,0000,0000,,will have plus the\Nintegral of you to the
Dialogue: 0,0:15:22.52,0:15:23.88,Default,,0000,0000,0000,,four, do you?
Dialogue: 0,0:15:25.45,0:15:29.64,Default,,0000,0000,0000,,Now these are very very simple\Nintegrals to finish the integral
Dialogue: 0,0:15:29.64,0:15:32.31,Default,,0000,0000,0000,,of you squared is you cubed over
Dialogue: 0,0:15:32.31,0:15:38.63,Default,,0000,0000,0000,,3? The integral of you to the\Nfour is due to the five over 5
Dialogue: 0,0:15:38.63,0:15:40.46,Default,,0000,0000,0000,,plus a constant of integration.
Dialogue: 0,0:15:42.47,0:15:48.01,Default,,0000,0000,0000,,All we need to do to finish off\Nis return to our original
Dialogue: 0,0:15:48.01,0:15:53.12,Default,,0000,0000,0000,,variables. Remember, you was\Ncosine of X, so we finish off by
Dialogue: 0,0:15:53.12,0:15:54.40,Default,,0000,0000,0000,,writing minus 1/3.
Dialogue: 0,0:15:54.97,0:15:59.30,Default,,0000,0000,0000,,You being cosine X means that\Nwe've got cosine cubed X.
Dialogue: 0,0:16:00.67,0:16:07.18,Default,,0000,0000,0000,,Plus 1/5. You to\Nthe five will be Co sign
Dialogue: 0,0:16:07.18,0:16:09.10,Default,,0000,0000,0000,,to the power 5X.
Dialogue: 0,0:16:10.26,0:16:11.76,Default,,0000,0000,0000,,Plus a constant of integration.
Dialogue: 0,0:16:12.57,0:16:17.19,Default,,0000,0000,0000,,And that's the solution to the\Nproblem that we started with.
Dialogue: 0,0:16:18.22,0:16:24.18,Default,,0000,0000,0000,,Let's stick with the same sort\Nof family of integrals, so we're
Dialogue: 0,0:16:24.18,0:16:30.15,Default,,0000,0000,0000,,still sticking with the integral\Nof sign to the power MX cosine
Dialogue: 0,0:16:30.15,0:16:32.63,Default,,0000,0000,0000,,to the power NX DX.
Dialogue: 0,0:16:33.21,0:16:37.65,Default,,0000,0000,0000,,And now I'm going to have a look\Nat what happens in the case when
Dialogue: 0,0:16:37.65,0:16:39.13,Default,,0000,0000,0000,,M is an even number.
Dialogue: 0,0:16:39.65,0:16:42.56,Default,,0000,0000,0000,,And N is an odd number.
Dialogue: 0,0:16:44.48,0:16:47.25,Default,,0000,0000,0000,,This method will always work\Nwhen M is even. An is odd.
Dialogue: 0,0:16:47.79,0:16:52.27,Default,,0000,0000,0000,,Let's look at a specific case.\NSuppose we want to integrate the
Dialogue: 0,0:16:52.27,0:16:54.13,Default,,0000,0000,0000,,sign to the power 4X.
Dialogue: 0,0:16:55.19,0:16:57.75,Default,,0000,0000,0000,,Cosine cubed X.
Dialogue: 0,0:16:58.35,0:16:59.29,Default,,0000,0000,0000,,DX
Dialogue: 0,0:17:01.84,0:17:07.12,Default,,0000,0000,0000,,Notice that M the power of sign\Nis now even em is full.
Dialogue: 0,0:17:08.43,0:17:12.69,Default,,0000,0000,0000,,And N which is the power of\Ncosine, is odd an IS3.
Dialogue: 0,0:17:13.43,0:17:17.34,Default,,0000,0000,0000,,What I'm going to do is I'm\Ngoing to use the identity that
Dialogue: 0,0:17:17.34,0:17:21.26,Default,,0000,0000,0000,,cosine squared of an angle is 1\Nminus sign squared of an angle
Dialogue: 0,0:17:21.26,0:17:25.17,Default,,0000,0000,0000,,and you'll be able to lift that\Ndirectly from the table we had
Dialogue: 0,0:17:25.17,0:17:28.18,Default,,0000,0000,0000,,at the beginning, which stated\Nthe very important and well
Dialogue: 0,0:17:28.18,0:17:31.19,Default,,0000,0000,0000,,known results that cosine\Nsquared of an angle plus the
Dialogue: 0,0:17:31.19,0:17:34.20,Default,,0000,0000,0000,,sine squared of an angle is\Nalways equal to 1.
Dialogue: 0,0:17:34.89,0:17:40.11,Default,,0000,0000,0000,,What I'm going to do is I'm\Ngoing to use this to rewrite the
Dialogue: 0,0:17:40.11,0:17:42.85,Default,,0000,0000,0000,,cosine term. In here, in terms
Dialogue: 0,0:17:42.85,0:17:47.29,Default,,0000,0000,0000,,of signs. First of all, I'm\Ngoing to apply the little trick
Dialogue: 0,0:17:47.29,0:17:53.28,Default,,0000,0000,0000,,we had before. And split the\Ncosine turn up like this cosine
Dialogue: 0,0:17:53.28,0:17:56.31,Default,,0000,0000,0000,,cubed. I'm going to write this
Dialogue: 0,0:17:56.31,0:17:59.28,Default,,0000,0000,0000,,cosine squared. Multiplied by
Dialogue: 0,0:17:59.28,0:18:05.40,Default,,0000,0000,0000,,cosine. So I've changed the\Ncosine cubed to these two terms
Dialogue: 0,0:18:05.40,0:18:12.22,Default,,0000,0000,0000,,here. Now I can use\Nthe identity to change cosine
Dialogue: 0,0:18:12.22,0:18:15.37,Default,,0000,0000,0000,,squared X into terms involving
Dialogue: 0,0:18:15.37,0:18:20.56,Default,,0000,0000,0000,,sine squared. So the integral\Nwill become the integral of
Dialogue: 0,0:18:20.56,0:18:23.25,Default,,0000,0000,0000,,sign. To the power 4X.
Dialogue: 0,0:18:24.06,0:18:30.00,Default,,0000,0000,0000,,Cosine squared X. We can write\Nas one minus sign, squared X.
Dialogue: 0,0:18:31.46,0:18:35.94,Default,,0000,0000,0000,,And there's still this term\Ncosine X here as well.
Dialogue: 0,0:18:37.73,0:18:41.30,Default,,0000,0000,0000,,And all that has to be\Nintegrated with respect to X.
Dialogue: 0,0:18:44.24,0:18:48.98,Default,,0000,0000,0000,,Let me remove the brackets here.\NWhen we remove the brackets,
Dialogue: 0,0:18:48.98,0:18:54.58,Default,,0000,0000,0000,,there will be signed to the 4th\NX Times one all multiplied by
Dialogue: 0,0:18:54.58,0:19:01.68,Default,,0000,0000,0000,,cosine X. That'll be signed\Nto the 4th X
Dialogue: 0,0:19:01.68,0:19:05.18,Default,,0000,0000,0000,,multiplied by sign squared
Dialogue: 0,0:19:05.18,0:19:10.68,Default,,0000,0000,0000,,X. Which is signed to the 6X\Nor multiplied by cosine X.
Dialogue: 0,0:19:12.13,0:19:19.02,Default,,0000,0000,0000,,And there's a minus sign in the\Nmiddle, and we want to integrate
Dialogue: 0,0:19:19.02,0:19:23.15,Default,,0000,0000,0000,,all that. With\Nrespect to X.
Dialogue: 0,0:19:25.34,0:19:29.10,Default,,0000,0000,0000,,Again, a simple substitution\Nwill allow us to finish this
Dialogue: 0,0:19:29.10,0:19:30.98,Default,,0000,0000,0000,,off. If we let you.
Dialogue: 0,0:19:31.51,0:19:33.25,Default,,0000,0000,0000,,Be sign X.
Dialogue: 0,0:19:34.35,0:19:35.97,Default,,0000,0000,0000,,So do you.
Dialogue: 0,0:19:36.48,0:19:39.14,Default,,0000,0000,0000,,Is cosine X DX.
Dialogue: 0,0:19:39.79,0:19:43.56,Default,,0000,0000,0000,,This will become immediately the\Nintegral of well signed to the
Dialogue: 0,0:19:43.56,0:19:48.02,Default,,0000,0000,0000,,4th X sign to the 4th X will be\Nyou to the four.
Dialogue: 0,0:19:48.78,0:19:54.13,Default,,0000,0000,0000,,The cosine X times the DX cosine\NX DX becomes du.
Dialogue: 0,0:19:55.72,0:20:02.52,Default,,0000,0000,0000,,Subtract. Sign\Nto the six, X will become you to
Dialogue: 0,0:20:02.52,0:20:09.39,Default,,0000,0000,0000,,the six. And the cosine\NX DX is du.
Dialogue: 0,0:20:09.47,0:20:13.20,Default,,0000,0000,0000,,So what we've achieved are two\Nvery simple integrals that we
Dialogue: 0,0:20:13.20,0:20:14.89,Default,,0000,0000,0000,,can complete to finish the
Dialogue: 0,0:20:14.89,0:20:20.75,Default,,0000,0000,0000,,problem. The integral of you to\Nthe four is due to the five over
Dialogue: 0,0:20:20.75,0:20:25.60,Default,,0000,0000,0000,,5. The integral of you to the\Nsix is due to the 7 over 7.
Dialogue: 0,0:20:26.46,0:20:27.55,Default,,0000,0000,0000,,Plus a constant.
Dialogue: 0,0:20:29.01,0:20:33.69,Default,,0000,0000,0000,,And then just to finish off, we\Nreturn to the original variables
Dialogue: 0,0:20:33.69,0:20:37.98,Default,,0000,0000,0000,,and replace EU with sign X,\Nwhich will give us 1/5.
Dialogue: 0,0:20:38.51,0:20:42.36,Default,,0000,0000,0000,,Sign next to the five or sign to\Nthe power 5X.
Dialogue: 0,0:20:44.16,0:20:45.09,Default,,0000,0000,0000,,Minus.
Dialogue: 0,0:20:46.11,0:20:52.44,Default,,0000,0000,0000,,One 7th. You to the\NSeven will be signed to the 7X.
Dialogue: 0,0:20:53.05,0:20:56.27,Default,,0000,0000,0000,,Plus a constant of integration.
Dialogue: 0,0:20:58.30,0:21:01.71,Default,,0000,0000,0000,,So that's how we deal with\Nintegrals of this family. In the
Dialogue: 0,0:21:01.71,0:21:05.97,Default,,0000,0000,0000,,case when M is an even number\Nand when N is an odd number. Now
Dialogue: 0,0:21:05.97,0:21:09.66,Default,,0000,0000,0000,,in the case when both M&N are\Neven, you should try using the
Dialogue: 0,0:21:09.66,0:21:13.07,Default,,0000,0000,0000,,double angle formulas, and I'm\Nnot going to do an example of
Dialogue: 0,0:21:13.07,0:21:16.76,Default,,0000,0000,0000,,that because there isn't time in\Nthis video to do that. But there
Dialogue: 0,0:21:16.76,0:21:19.60,Default,,0000,0000,0000,,are examples in the exercises\Naccompanying the video and you
Dialogue: 0,0:21:19.60,0:21:21.02,Default,,0000,0000,0000,,should try those for yourself.
Dialogue: 0,0:21:21.73,0:21:28.61,Default,,0000,0000,0000,,I'm not going to look\Nat some integrals for which
Dialogue: 0,0:21:28.61,0:21:31.36,Default,,0000,0000,0000,,a trigonometric substitution is
Dialogue: 0,0:21:31.36,0:21:36.79,Default,,0000,0000,0000,,appropriate. Suppose we want to\Nevaluate this integral.
Dialogue: 0,0:21:36.79,0:21:43.10,Default,,0000,0000,0000,,The integral of\N1 / 1
Dialogue: 0,0:21:43.10,0:21:46.26,Default,,0000,0000,0000,,plus X squared.
Dialogue: 0,0:21:47.03,0:21:48.18,Default,,0000,0000,0000,,With respect to X.
Dialogue: 0,0:21:49.71,0:21:53.10,Default,,0000,0000,0000,,Now the trigonometric\Nsubstitution that I want to use
Dialogue: 0,0:21:53.10,0:21:59.14,Default,,0000,0000,0000,,is this one. I want to let X be\Nthe tangent of a new variable, X
Dialogue: 0,0:21:59.14,0:22:00.27,Default,,0000,0000,0000,,equals 10 theater.
Dialogue: 0,0:22:00.92,0:22:04.12,Default,,0000,0000,0000,,While I picked this particular\Nsubstitution well, all will
Dialogue: 0,0:22:04.12,0:22:09.08,Default,,0000,0000,0000,,become clear in time, but I want\Nto just look ahead a little bit
Dialogue: 0,0:22:09.08,0:22:11.22,Default,,0000,0000,0000,,by letting X equal 10 theater.
Dialogue: 0,0:22:11.75,0:22:14.84,Default,,0000,0000,0000,,What will have at the\Ndenominator down here is
Dialogue: 0,0:22:14.84,0:22:16.55,Default,,0000,0000,0000,,1 + 10 squared theater.
Dialogue: 0,0:22:17.57,0:22:22.95,Default,,0000,0000,0000,,One plus X squared will become\N1 + 10 squared and we have an
Dialogue: 0,0:22:22.95,0:22:27.17,Default,,0000,0000,0000,,identity already which says\Nthat 1 + 10 squared of an
Dialogue: 0,0:22:27.17,0:22:31.39,Default,,0000,0000,0000,,angle is equal to the sequence\Nsquared of the angle. That's
Dialogue: 0,0:22:31.39,0:22:36.00,Default,,0000,0000,0000,,an identity that we had on the\Ntable right at the beginning,
Dialogue: 0,0:22:36.00,0:22:40.61,Default,,0000,0000,0000,,so the idea is that by making\Nthis substitution, 1 + 10
Dialogue: 0,0:22:40.61,0:22:44.45,Default,,0000,0000,0000,,squared can be replaced by a\Nsingle term sequence squared,
Dialogue: 0,0:22:44.45,0:22:47.52,Default,,0000,0000,0000,,as we'll see, so let's\Nprogress with that
Dialogue: 0,0:22:47.52,0:22:47.91,Default,,0000,0000,0000,,substitution.
Dialogue: 0,0:22:49.39,0:22:54.78,Default,,0000,0000,0000,,If we let X be tongue theater,\Nthe integrals going to become 1
Dialogue: 0,0:22:54.78,0:22:59.35,Default,,0000,0000,0000,,/ 1 plus X squared will become 1\N+ 10 squared.
Dialogue: 0,0:23:00.48,0:23:04.89,Default,,0000,0000,0000,,Theater. And we have to take\Ncare of the DX in an appropriate
Dialogue: 0,0:23:04.89,0:23:11.74,Default,,0000,0000,0000,,way. Now remember that DX is\Ngoing to be given by the XD
Dialogue: 0,0:23:11.74,0:23:14.23,Default,,0000,0000,0000,,theater multiplied by D theater.
Dialogue: 0,0:23:14.37,0:23:18.06,Default,,0000,0000,0000,,DXD theater we want to\Ndifferentiate X is 10 theater
Dialogue: 0,0:23:18.06,0:23:19.54,Default,,0000,0000,0000,,with respect to theater.
Dialogue: 0,0:23:20.45,0:23:24.82,Default,,0000,0000,0000,,Now the derivative of tongue\Ntheater is the secant squared,
Dialogue: 0,0:23:24.82,0:23:27.88,Default,,0000,0000,0000,,so we get secret squared Theta D
Dialogue: 0,0:23:27.88,0:23:32.94,Default,,0000,0000,0000,,theater. So this will allow us\Nto change the DX in here.
Dialogue: 0,0:23:33.60,0:23:40.49,Default,,0000,0000,0000,,Two, secant squared, Theta D\NTheta over on the right.
Dialogue: 0,0:23:40.49,0:23:44.55,Default,,0000,0000,0000,,At this stage I'm going to use\Nthe trigonometric identity,
Dialogue: 0,0:23:44.55,0:23:50.23,Default,,0000,0000,0000,,which says that 1 + 10 squared\Nof an angle is equal to the
Dialogue: 0,0:23:50.23,0:23:54.70,Default,,0000,0000,0000,,sequence squared of the angle.\NSo In other words, all this
Dialogue: 0,0:23:54.70,0:23:58.76,Default,,0000,0000,0000,,quantity down here is just the\Nsequence squared of Theta.
Dialogue: 0,0:23:58.78,0:24:04.72,Default,,0000,0000,0000,,And this is very nice now\Nbecause this term here will
Dialogue: 0,0:24:04.72,0:24:10.66,Default,,0000,0000,0000,,cancel out with this term down\Nin the denominator down there,
Dialogue: 0,0:24:10.66,0:24:17.14,Default,,0000,0000,0000,,and we're left purely with the\Nintegral of one with respect to
Dialogue: 0,0:24:17.14,0:24:19.84,Default,,0000,0000,0000,,theater. Very simple to finish.
Dialogue: 0,0:24:20.52,0:24:24.70,Default,,0000,0000,0000,,The integral of one with respect\Nto theater is just theater.
Dialogue: 0,0:24:24.71,0:24:26.39,Default,,0000,0000,0000,,Plus a constant of integration.
Dialogue: 0,0:24:28.05,0:24:32.91,Default,,0000,0000,0000,,We want to return to our\Noriginal variables and if X was
Dialogue: 0,0:24:32.91,0:24:37.77,Default,,0000,0000,0000,,10 theater than theater is the\Nangle whose tangent, his ex. So
Dialogue: 0,0:24:37.77,0:24:40.60,Default,,0000,0000,0000,,theater is 10 to the minus one
Dialogue: 0,0:24:40.60,0:24:43.75,Default,,0000,0000,0000,,of X. Plus a constant.
Dialogue: 0,0:24:46.01,0:24:47.77,Default,,0000,0000,0000,,And that's the problem finished.
Dialogue: 0,0:24:48.29,0:24:50.96,Default,,0000,0000,0000,,This is a very important\Nstandard result that the
Dialogue: 0,0:24:50.96,0:24:54.82,Default,,0000,0000,0000,,integral of one over 1 plus\NX squared DX is equal to the
Dialogue: 0,0:24:54.82,0:24:58.39,Default,,0000,0000,0000,,inverse tan 10 to the minus\None of X plus a constant.
Dialogue: 0,0:24:58.39,0:25:01.36,Default,,0000,0000,0000,,That's a result that you'll\Nsee in all the standard
Dialogue: 0,0:25:01.36,0:25:04.03,Default,,0000,0000,0000,,tables of integrals, and\Nit's a result that you'll
Dialogue: 0,0:25:04.03,0:25:07.00,Default,,0000,0000,0000,,need to call appan very\Nfrequently, and if you can't
Dialogue: 0,0:25:07.00,0:25:09.97,Default,,0000,0000,0000,,remember it, then at least\Nyou'll need to know that
Dialogue: 0,0:25:09.97,0:25:13.54,Default,,0000,0000,0000,,there is such a formula that\Nexists and you want to be
Dialogue: 0,0:25:13.54,0:25:15.02,Default,,0000,0000,0000,,able to look it up.
Dialogue: 0,0:25:16.72,0:25:20.49,Default,,0000,0000,0000,,I want to generalize this a\Nlittle bit to look at the case
Dialogue: 0,0:25:20.49,0:25:24.84,Default,,0000,0000,0000,,when we deal with not just a one\Nhere, but a more general case of
Dialogue: 0,0:25:24.84,0:25:28.32,Default,,0000,0000,0000,,an arbitrary constant in there.\NSo let's look at what happens if
Dialogue: 0,0:25:28.32,0:25:30.06,Default,,0000,0000,0000,,we have a situation like this.
Dialogue: 0,0:25:30.90,0:25:36.90,Default,,0000,0000,0000,,Suppose we want to integrate one\Nover a squared plus X squared
Dialogue: 0,0:25:36.90,0:25:38.90,Default,,0000,0000,0000,,with respect to X.
Dialogue: 0,0:25:39.48,0:25:42.79,Default,,0000,0000,0000,,Where a is a
Dialogue: 0,0:25:42.79,0:25:49.86,Default,,0000,0000,0000,,constant. This time I'm going to\Nmake this substitution let X be
Dialogue: 0,0:25:49.86,0:25:55.54,Default,,0000,0000,0000,,a town theater, and we'll see\Nwhy we've made that substitution
Dialogue: 0,0:25:55.54,0:25:58.13,Default,,0000,0000,0000,,in just a little while.
Dialogue: 0,0:25:58.81,0:26:04.41,Default,,0000,0000,0000,,With this substitution, X is\Na Tan Theta. The differential
Dialogue: 0,0:26:04.41,0:26:08.89,Default,,0000,0000,0000,,DX becomes a secant squared\NTheta D Theta.
Dialogue: 0,0:26:11.69,0:26:14.48,Default,,0000,0000,0000,,Let's put all this into this
Dialogue: 0,0:26:14.48,0:26:19.70,Default,,0000,0000,0000,,integral here. Will have the\Nintegral of one over a squared.
Dialogue: 0,0:26:20.98,0:26:26.98,Default,,0000,0000,0000,,Plus And X squared\Nwill become a squared 10.
Dialogue: 0,0:26:26.98,0:26:28.45,Default,,0000,0000,0000,,Squared feet are.
Dialogue: 0,0:26:29.46,0:26:31.80,Default,,0000,0000,0000,,The
Dialogue: 0,0:26:31.80,0:26:39.43,Default,,0000,0000,0000,,DX Will\Nbecome a sex squared Theta D
Dialogue: 0,0:26:39.43,0:26:47.26,Default,,0000,0000,0000,,Theta. Now what I can do\Nnow is I can take out a common
Dialogue: 0,0:26:47.26,0:26:50.21,Default,,0000,0000,0000,,factor of A squared from the
Dialogue: 0,0:26:50.21,0:26:57.31,Default,,0000,0000,0000,,denominator. Taking an A squared\Nout from this term will leave me
Dialogue: 0,0:26:57.31,0:27:03.80,Default,,0000,0000,0000,,one taking a squared out from\Nthis term will leave me tan
Dialogue: 0,0:27:03.80,0:27:09.37,Default,,0000,0000,0000,,squared theater. And it's still\Non the top. I've got a sex
Dialogue: 0,0:27:09.37,0:27:10.95,Default,,0000,0000,0000,,squared Theta D Theta.
Dialogue: 0,0:27:13.36,0:27:20.50,Default,,0000,0000,0000,,We have the trig identity that 1\N+ 10 squared of any angle is sex
Dialogue: 0,0:27:20.50,0:27:22.40,Default,,0000,0000,0000,,squared of the angle.
Dialogue: 0,0:27:22.66,0:27:28.86,Default,,0000,0000,0000,,So I can use that identity in\Nhere to write the denominator as
Dialogue: 0,0:27:28.86,0:27:34.58,Default,,0000,0000,0000,,one over a squared and the 1 +\N10 squared becomes simply
Dialogue: 0,0:27:34.58,0:27:36.02,Default,,0000,0000,0000,,sequence squared theater.
Dialogue: 0,0:27:36.63,0:27:41.44,Default,,0000,0000,0000,,We still gotten a secant squared\Ntheater in the numerator, and a
Dialogue: 0,0:27:41.44,0:27:45.45,Default,,0000,0000,0000,,lot of this is going to simplify\Nand cancel now.
Dialogue: 0,0:27:46.20,0:27:47.65,Default,,0000,0000,0000,,The secant squared will go the
Dialogue: 0,0:27:47.65,0:27:52.18,Default,,0000,0000,0000,,top and the bottom. The one of\Nthese at the bottom will go with
Dialogue: 0,0:27:52.18,0:27:56.03,Default,,0000,0000,0000,,the others at the top, and we're\Nleft with the integral of one
Dialogue: 0,0:27:56.03,0:27:57.80,Default,,0000,0000,0000,,over A with respect to theater.
Dialogue: 0,0:28:00.17,0:28:02.89,Default,,0000,0000,0000,,Again, this is straightforward\Nto finish. The integral of one
Dialogue: 0,0:28:02.89,0:28:06.43,Default,,0000,0000,0000,,over a one over as a constant\Nwith respect to Theta is just
Dialogue: 0,0:28:06.43,0:28:08.33,Default,,0000,0000,0000,,going to give me one over a.
Dialogue: 0,0:28:08.87,0:28:11.78,Default,,0000,0000,0000,,Theater. Plus the constant of
Dialogue: 0,0:28:11.78,0:28:16.85,Default,,0000,0000,0000,,integration. To return to the\Noriginal variables, we've got to
Dialogue: 0,0:28:16.85,0:28:21.73,Default,,0000,0000,0000,,go back to our original\Nsubstitution. If X is a tan
Dialogue: 0,0:28:21.73,0:28:27.06,Default,,0000,0000,0000,,Theta, then we can write that X\Nover A is 10 theater.
Dialogue: 0,0:28:27.09,0:28:30.66,Default,,0000,0000,0000,,And In other words, that\Ntheater is the angle whose
Dialogue: 0,0:28:30.66,0:28:35.30,Default,,0000,0000,0000,,tangent is 10 to the minus\None of all this X over a.
Dialogue: 0,0:28:36.59,0:28:41.24,Default,,0000,0000,0000,,That will enable me to write our\Nfinal results as one over a town
Dialogue: 0,0:28:41.24,0:28:42.57,Default,,0000,0000,0000,,to the minus one.
Dialogue: 0,0:28:43.25,0:28:45.94,Default,,0000,0000,0000,,X over a.
Dialogue: 0,0:28:46.06,0:28:47.62,Default,,0000,0000,0000,,Plus a constant of integration.
Dialogue: 0,0:28:49.03,0:28:52.54,Default,,0000,0000,0000,,And this is another very\Nimportant standard result that
Dialogue: 0,0:28:52.54,0:28:56.83,Default,,0000,0000,0000,,the integral of one over a\Nsquared plus X squared with
Dialogue: 0,0:28:56.83,0:29:03.85,Default,,0000,0000,0000,,respect to X is one over a 10 to\Nthe minus one of X over a plus a
Dialogue: 0,0:29:03.85,0:29:07.75,Default,,0000,0000,0000,,constant, and as before, that's\Na standard result that you'll
Dialogue: 0,0:29:07.75,0:29:12.43,Default,,0000,0000,0000,,see frequently in all the tables\Nof integrals, and you'll need to
Dialogue: 0,0:29:12.43,0:29:16.72,Default,,0000,0000,0000,,call a pawn that in lots of\Nsituations when you're required
Dialogue: 0,0:29:16.72,0:29:17.89,Default,,0000,0000,0000,,to do integration.
Dialogue: 0,0:29:17.94,0:29:23.94,Default,,0000,0000,0000,,OK, so now we've got the\Nstandard result that the
Dialogue: 0,0:29:23.94,0:29:31.14,Default,,0000,0000,0000,,integral of one over a squared\Nplus X squared DX is equal
Dialogue: 0,0:29:31.14,0:29:38.34,Default,,0000,0000,0000,,to one over a town to\Nthe minus one of X of
Dialogue: 0,0:29:38.34,0:29:40.40,Default,,0000,0000,0000,,A. As a constant of integration.
Dialogue: 0,0:29:41.04,0:29:46.41,Default,,0000,0000,0000,,Let's see how we might use\Nthis formula in a slightly
Dialogue: 0,0:29:46.41,0:29:52.26,Default,,0000,0000,0000,,different case. Suppose we\Nhave the integral of 1 / 4 +
Dialogue: 0,0:29:52.26,0:29:54.22,Default,,0000,0000,0000,,9 X squared DX.
Dialogue: 0,0:29:55.36,0:29:58.52,Default,,0000,0000,0000,,Now this looks very similar to\Nthe standard formula we have
Dialogue: 0,0:29:58.52,0:30:00.77,Default,,0000,0000,0000,,here. Except there's a slight
Dialogue: 0,0:30:00.77,0:30:04.94,Default,,0000,0000,0000,,problem. And the problem is that\Ninstead of One X squared, which
Dialogue: 0,0:30:04.94,0:30:08.07,Default,,0000,0000,0000,,we have in the standard result,\NI've got nine X squared.
Dialogue: 0,0:30:08.85,0:30:11.83,Default,,0000,0000,0000,,What I'm going to do is I'm\Ngoing to divide everything at
Dialogue: 0,0:30:11.83,0:30:15.55,Default,,0000,0000,0000,,the bottom by 9, take a factor\Nof nine out so that we end up
Dialogue: 0,0:30:15.55,0:30:19.27,Default,,0000,0000,0000,,with just a One X squared here.\NSo what I'm going to do is I'm
Dialogue: 0,0:30:19.27,0:30:20.51,Default,,0000,0000,0000,,going to write the denominator
Dialogue: 0,0:30:20.51,0:30:25.49,Default,,0000,0000,0000,,like this. So I've taken a\Nfactor of nine out. You'll see
Dialogue: 0,0:30:25.49,0:30:30.05,Default,,0000,0000,0000,,if we multiply the brackets\Nagain here, there's 9 * 4 over
Dialogue: 0,0:30:30.05,0:30:35.37,Default,,0000,0000,0000,,9, which is just four and the\Nnine times the X squared, so I
Dialogue: 0,0:30:35.37,0:30:39.55,Default,,0000,0000,0000,,haven't changed anything. I've\Njust taken a factor of nine out
Dialogue: 0,0:30:39.55,0:30:45.25,Default,,0000,0000,0000,,the point of doing that is that\Nnow I have a single. I have a
Dialogue: 0,0:30:45.25,0:30:49.81,Default,,0000,0000,0000,,One X squared here, which will\Nmatch the formula I have there.
Dialogue: 0,0:30:50.45,0:30:53.54,Default,,0000,0000,0000,,If I take the 9 outside the
Dialogue: 0,0:30:53.54,0:30:59.31,Default,,0000,0000,0000,,integral. I'm left with 1 /, 4\Nninths plus X squared integrated
Dialogue: 0,0:30:59.31,0:31:05.55,Default,,0000,0000,0000,,with respect to X and I hope you\Ncan see that this is exactly one
Dialogue: 0,0:31:05.55,0:31:10.95,Default,,0000,0000,0000,,of the standard forms. Now when\Nwe let A squared B4 over nine
Dialogue: 0,0:31:10.95,0:31:16.78,Default,,0000,0000,0000,,with a squared is 4 over 9. We\Nhave the standard form. If A
Dialogue: 0,0:31:16.78,0:31:23.02,Default,,0000,0000,0000,,squared is 4 over 9A will be 2\Nover 3 and we can complete this
Dialogue: 0,0:31:23.02,0:31:27.53,Default,,0000,0000,0000,,integration. Using the standard\Nresult that one over 9 stays
Dialogue: 0,0:31:27.53,0:31:29.76,Default,,0000,0000,0000,,there, we want one over A.
Dialogue: 0,0:31:30.54,0:31:34.74,Default,,0000,0000,0000,,Or A is 2/3. So\Nwe want 1 / 2/3.
Dialogue: 0,0:31:35.81,0:31:37.65,Default,,0000,0000,0000,,10 to the minus one.
Dialogue: 0,0:31:38.39,0:31:40.24,Default,,0000,0000,0000,,Of X over a.
Dialogue: 0,0:31:40.88,0:31:44.80,Default,,0000,0000,0000,,X divided by a is X divided by
Dialogue: 0,0:31:44.80,0:31:48.24,Default,,0000,0000,0000,,2/3. Plus a constant of
Dialogue: 0,0:31:48.24,0:31:53.45,Default,,0000,0000,0000,,integration. Just to tide to\Nthese fractions up, three will
Dialogue: 0,0:31:53.45,0:31:58.04,Default,,0000,0000,0000,,divide into 9 three times, so\Nwe'll have 326 in the
Dialogue: 0,0:31:58.04,0:32:03.71,Default,,0000,0000,0000,,denominator. 10 to the minus one\Nand dividing by 2/3 is like
Dialogue: 0,0:32:03.71,0:32:09.56,Default,,0000,0000,0000,,multiplying by three over 2, so\NI'll have 10 to the minus one of
Dialogue: 0,0:32:09.56,0:32:12.49,Default,,0000,0000,0000,,three X over 2 plus the constant
Dialogue: 0,0:32:12.49,0:32:16.55,Default,,0000,0000,0000,,of integration. So the point\Nhere is you might have to do a
Dialogue: 0,0:32:16.55,0:32:19.40,Default,,0000,0000,0000,,bit of work on the integrand\Nin order to be able to write
Dialogue: 0,0:32:19.40,0:32:21.59,Default,,0000,0000,0000,,it in the form of one of the\Nstandard results.
Dialogue: 0,0:32:22.87,0:32:28.87,Default,,0000,0000,0000,,OK, let's have a look at another\Ncase where another integral to
Dialogue: 0,0:32:28.87,0:32:32.87,Default,,0000,0000,0000,,look at where a trigonometric\Nsubstitution is appropriate.
Dialogue: 0,0:32:32.87,0:32:38.87,Default,,0000,0000,0000,,Suppose we want to find the\Nintegral of one over the square
Dialogue: 0,0:32:38.87,0:32:41.87,Default,,0000,0000,0000,,root of A squared minus X
Dialogue: 0,0:32:41.87,0:32:47.16,Default,,0000,0000,0000,,squared DX. Again,\NA is a constant.
Dialogue: 0,0:32:49.71,0:32:55.61,Default,,0000,0000,0000,,The substitution that I'm\Ngoing to make is this one.
Dialogue: 0,0:32:55.61,0:33:00.92,Default,,0000,0000,0000,,I'm going to write X equals\Na sign theater.
Dialogue: 0,0:33:02.13,0:33:08.55,Default,,0000,0000,0000,,If I do that, what will happen\Nto my integral, let's see.
Dialogue: 0,0:33:09.08,0:33:11.16,Default,,0000,0000,0000,,And have the integral of one
Dialogue: 0,0:33:11.16,0:33:17.98,Default,,0000,0000,0000,,over. The square root. The A\Nsquared will stay the same, but
Dialogue: 0,0:33:17.98,0:33:22.80,Default,,0000,0000,0000,,the X squared will become a\Nsquared sine squared.
Dialogue: 0,0:33:22.80,0:33:25.97,Default,,0000,0000,0000,,I squared sine squared Theta.
Dialogue: 0,0:33:26.75,0:33:30.79,Default,,0000,0000,0000,,Now the reason I've done that\Nis because in a minute I'm
Dialogue: 0,0:33:30.79,0:33:35.18,Default,,0000,0000,0000,,going to take out a factor of a\Nsquared, which will leave me
Dialogue: 0,0:33:35.18,0:33:39.56,Default,,0000,0000,0000,,one 1 minus sign squared, and I\Ndo have an identity involving 1
Dialogue: 0,0:33:39.56,0:33:43.26,Default,,0000,0000,0000,,minus sign squared as we'll\Nsee, but just before we do
Dialogue: 0,0:33:43.26,0:33:47.31,Default,,0000,0000,0000,,that, let's substitute for the\Ndifferential as well. If X is a
Dialogue: 0,0:33:47.31,0:33:51.01,Default,,0000,0000,0000,,sign theater, then DX will be a\Ncosine, Theta, D, Theta.
Dialogue: 0,0:33:52.35,0:33:59.03,Default,,0000,0000,0000,,So we have a cosine Theta D\NTheta for the differential DX.
Dialogue: 0,0:34:01.40,0:34:07.39,Default,,0000,0000,0000,,Let me take out the factor of a\Nsquared in the denominator.
Dialogue: 0,0:34:08.04,0:34:13.84,Default,,0000,0000,0000,,Taking a squad from this\Nfirst term will leave me one
Dialogue: 0,0:34:13.84,0:34:20.16,Default,,0000,0000,0000,,and a squared from the second\Nterm will leave me one minus
Dialogue: 0,0:34:20.16,0:34:21.74,Default,,0000,0000,0000,,sign squared Theta.
Dialogue: 0,0:34:22.81,0:34:27.09,Default,,0000,0000,0000,,I have still gotten a costly to\Nthe theater at the top.
Dialogue: 0,0:34:28.49,0:34:31.93,Default,,0000,0000,0000,,Now let me remind you there's a\Ntrig identity which says that
Dialogue: 0,0:34:31.93,0:34:35.38,Default,,0000,0000,0000,,the cosine squared of an angle\Nplus the sine squared of an
Dialogue: 0,0:34:35.38,0:34:36.53,Default,,0000,0000,0000,,angle is always one.
Dialogue: 0,0:34:37.16,0:34:40.76,Default,,0000,0000,0000,,So if we have one minus the sine\Nsquared of an angle, we can
Dialogue: 0,0:34:40.76,0:34:42.04,Default,,0000,0000,0000,,replace it with cosine squared.
Dialogue: 0,0:34:42.67,0:34:49.63,Default,,0000,0000,0000,,So 1 minus sign squared Theta\Nwe can replace with simply
Dialogue: 0,0:34:49.63,0:34:51.53,Default,,0000,0000,0000,,cosine squared Theta.
Dialogue: 0,0:34:51.54,0:34:54.41,Default,,0000,0000,0000,,Is the A squared out the\Nfrontier and we want the square
Dialogue: 0,0:34:54.41,0:34:55.60,Default,,0000,0000,0000,,root of the whole lot.
Dialogue: 0,0:34:56.28,0:35:03.35,Default,,0000,0000,0000,,Now this is very simple. We want\Nthe square root of A squared
Dialogue: 0,0:35:03.35,0:35:08.79,Default,,0000,0000,0000,,cosine squared Theta. We square\Nroot. These squared terms will
Dialogue: 0,0:35:08.79,0:35:10.97,Default,,0000,0000,0000,,be just left with.
Dialogue: 0,0:35:10.98,0:35:12.60,Default,,0000,0000,0000,,A cosine Theta.
Dialogue: 0,0:35:13.14,0:35:18.62,Default,,0000,0000,0000,,In the denominator and within a\Ncosine Theta in the numerator.
Dialogue: 0,0:35:19.77,0:35:21.96,Default,,0000,0000,0000,,And these were clearly\Ncancel out.
Dialogue: 0,0:35:23.08,0:35:27.70,Default,,0000,0000,0000,,And we're left with the integral\Nof one with respect to theater,
Dialogue: 0,0:35:27.70,0:35:31.16,Default,,0000,0000,0000,,which is just theater plus a\Nconstant of integration.
Dialogue: 0,0:35:33.94,0:35:39.23,Default,,0000,0000,0000,,Just to return to the original\Nvariables, given that X was a
Dialogue: 0,0:35:39.23,0:35:43.64,Default,,0000,0000,0000,,sign theater, then clearly X\Nover A is sign theater.
Dialogue: 0,0:35:44.40,0:35:50.05,Default,,0000,0000,0000,,So theater is the angle who sign\Nis or sign to the minus one of X
Dialogue: 0,0:35:50.05,0:35:54.99,Default,,0000,0000,0000,,over a, so replacing the theater\Nwith sign to the minus one of X
Dialogue: 0,0:35:54.99,0:35:57.11,Default,,0000,0000,0000,,over a will get this result.
Dialogue: 0,0:35:57.65,0:36:02.40,Default,,0000,0000,0000,,And this is a very important\Nstandard result that if you want
Dialogue: 0,0:36:02.40,0:36:07.55,Default,,0000,0000,0000,,to integrate 1 divided by the\Nsquare root of A squared minus X
Dialogue: 0,0:36:07.55,0:36:12.30,Default,,0000,0000,0000,,squared, the result is the\Ninverse sine or the sign to the
Dialogue: 0,0:36:12.30,0:36:14.68,Default,,0000,0000,0000,,minus one of X over a.
Dialogue: 0,0:36:15.25,0:36:16.67,Default,,0000,0000,0000,,Plus a constant of integration.
Dialogue: 0,0:36:18.01,0:36:25.26,Default,,0000,0000,0000,,Will have a look one final\Nexample which is a variant on
Dialogue: 0,0:36:25.26,0:36:31.90,Default,,0000,0000,0000,,the previous example. Suppose we\Nwant to integrate 1 divided by
Dialogue: 0,0:36:31.90,0:36:37.94,Default,,0000,0000,0000,,the square root of 4 -\N9 X squared DX.
Dialogue: 0,0:36:38.57,0:36:42.90,Default,,0000,0000,0000,,Now that's very similar to the\None we just looked at. Remember
Dialogue: 0,0:36:42.90,0:36:47.60,Default,,0000,0000,0000,,that we had the results that the\Nintegral of one over the square
Dialogue: 0,0:36:47.60,0:36:49.76,Default,,0000,0000,0000,,root of A squared minus X
Dialogue: 0,0:36:49.76,0:36:55.38,Default,,0000,0000,0000,,squared DX. Was the inverse sine\Nof X over a plus a constant?
Dialogue: 0,0:36:55.38,0:36:59.29,Default,,0000,0000,0000,,That's keep that in mind. That's\Nthe standard result we've
Dialogue: 0,0:36:59.29,0:37:03.86,Default,,0000,0000,0000,,already proved. We're almost\Nthere. In this case. The problem
Dialogue: 0,0:37:03.86,0:37:08.12,Default,,0000,0000,0000,,is that instead of a single X\Nsquared, we've got nine X
Dialogue: 0,0:37:08.12,0:37:12.19,Default,,0000,0000,0000,,squared. So like we did in the\Nother example, I'm going to take
Dialogue: 0,0:37:12.19,0:37:15.78,Default,,0000,0000,0000,,the factor of nine out to leave\Nus just a single X squared in
Dialogue: 0,0:37:15.78,0:37:17.57,Default,,0000,0000,0000,,there, and I do that like this.
Dialogue: 0,0:37:18.84,0:37:26.06,Default,,0000,0000,0000,,Taking a nine out from\Nthese terms here, I'll have
Dialogue: 0,0:37:26.06,0:37:29.67,Default,,0000,0000,0000,,four ninths minus X squared.
Dialogue: 0,0:37:30.29,0:37:33.75,Default,,0000,0000,0000,,Again, the nine times the four\Nninths leaves the four which we
Dialogue: 0,0:37:33.75,0:37:37.20,Default,,0000,0000,0000,,had originally, and then we've\Ngot the nine X squared, which we
Dialogue: 0,0:37:37.20,0:37:41.91,Default,,0000,0000,0000,,have there. The whole point of\Ndoing that is that then I'm
Dialogue: 0,0:37:41.91,0:37:46.19,Default,,0000,0000,0000,,going to extract the Route 9,\Nwhich is 3 and bring it right
Dialogue: 0,0:37:46.19,0:37:52.29,Default,,0000,0000,0000,,outside. And inside under the\Nintegral sign, I'll be left with
Dialogue: 0,0:37:52.29,0:37:58.31,Default,,0000,0000,0000,,one over the square root of 4\Nninths minus X squared DX.
Dialogue: 0,0:38:00.34,0:38:05.46,Default,,0000,0000,0000,,Now in this form, I hope you can\Nspot that we can use the
Dialogue: 0,0:38:05.46,0:38:08.76,Default,,0000,0000,0000,,standard result immediately with\Nthe standard results, with a
Dialogue: 0,0:38:08.76,0:38:12.05,Default,,0000,0000,0000,,being with a squared being equal\Nto four ninths.
Dialogue: 0,0:38:12.79,0:38:16.70,Default,,0000,0000,0000,,In other words, a being equal to
Dialogue: 0,0:38:16.70,0:38:21.14,Default,,0000,0000,0000,,2/3. Putting all that together\Nwill have a third. That's the
Dialogue: 0,0:38:21.14,0:38:24.63,Default,,0000,0000,0000,,third and the integral will\Nbecome the inverse sine.
Dialogue: 0,0:38:25.94,0:38:29.67,Default,,0000,0000,0000,,X. Divided by AA
Dialogue: 0,0:38:29.67,0:38:35.17,Default,,0000,0000,0000,,was 2/3. Plus a\Nconstant of integration.
Dialogue: 0,0:38:37.14,0:38:43.01,Default,,0000,0000,0000,,And just to tidy that up will be\Nleft with the third inverse sine
Dialogue: 0,0:38:43.01,0:38:48.45,Default,,0000,0000,0000,,dividing by 2/3 is the same as\Nmultiplying by three over 2, so
Dialogue: 0,0:38:48.45,0:38:52.64,Default,,0000,0000,0000,,will have 3X over 2 plus a\Nconstant of integration.
Dialogue: 0,0:38:52.67,0:38:56.06,Default,,0000,0000,0000,,And that's our final\Nresult. So we've seen a lot
Dialogue: 0,0:38:56.06,0:38:58.09,Default,,0000,0000,0000,,of examples that have\Nintegration using
Dialogue: 0,0:38:58.09,0:39:00.13,Default,,0000,0000,0000,,trigonometric identities\Nand integration using trig
Dialogue: 0,0:39:00.13,0:39:03.18,Default,,0000,0000,0000,,substitutions. You need a\Nlot of practice, and there
Dialogue: 0,0:39:03.18,0:39:06.23,Default,,0000,0000,0000,,are a lot of exercises in\Nthe accompanying text.