[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.57,0:00:02.30,Default,,0000,0000,0000,,Sal: Let's see if we can\Ncalculate the definite
Dialogue: 0,0:00:02.30,0:00:07.57,Default,,0000,0000,0000,,integral from zero to\None of x squared times
Dialogue: 0,0:00:07.57,0:00:13.50,Default,,0000,0000,0000,,two to the x to the third power d x.
Dialogue: 0,0:00:13.50,0:00:15.90,Default,,0000,0000,0000,,Like always I encourage\Nyou to pause this video
Dialogue: 0,0:00:15.90,0:00:20.23,Default,,0000,0000,0000,,and see if you can figure\Nthis out on your own.
Dialogue: 0,0:00:20.94,0:00:22.63,Default,,0000,0000,0000,,I'm assuming you've had a go at it.
Dialogue: 0,0:00:22.63,0:00:24.30,Default,,0000,0000,0000,,There's a couple of\Ninteresting things here.
Dialogue: 0,0:00:24.30,0:00:26.10,Default,,0000,0000,0000,,The first thing, at least\Nthat my brain does, it says,
Dialogue: 0,0:00:26.10,0:00:28.37,Default,,0000,0000,0000,,"I'm used to taking derivatives\Nand anti-derivatives
Dialogue: 0,0:00:28.37,0:00:31.70,Default,,0000,0000,0000,,of e to the x, not some\Nother base to the x."
Dialogue: 0,0:00:31.70,0:00:34.70,Default,,0000,0000,0000,,We know that the derivative\Nwith respect to x of e to the x
Dialogue: 0,0:00:34.70,0:00:38.10,Default,,0000,0000,0000,,is e to the x, or we could\Nsay that the anti-derivative
Dialogue: 0,0:00:38.10,0:00:43.70,Default,,0000,0000,0000,,of e to the x is equal\Nto e to the x plus c.
Dialogue: 0,0:00:43.70,0:00:46.90,Default,,0000,0000,0000,,Since I'm dealing with\Nsomething raised to,
Dialogue: 0,0:00:47.93,0:00:49.90,Default,,0000,0000,0000,,this particular situation,\Nsomething raised to
Dialogue: 0,0:00:49.90,0:00:52.50,Default,,0000,0000,0000,,a function of x, it seems\Nlike I might want to put,
Dialogue: 0,0:00:52.50,0:00:56.43,Default,,0000,0000,0000,,I might want to change the base\Nhere, but how do I do that?
Dialogue: 0,0:00:56.43,0:01:00.87,Default,,0000,0000,0000,,The way I would do that is\Nre-express two in terms of e.
Dialogue: 0,0:01:01.10,0:01:03.23,Default,,0000,0000,0000,,What would be two in terms of e?
Dialogue: 0,0:01:03.23,0:01:09.90,Default,,0000,0000,0000,,Two is equal to e, is\Nequal to e raised to the
Dialogue: 0,0:01:09.90,0:01:13.03,Default,,0000,0000,0000,,power that you need to\Nraise e to to get to two.
Dialogue: 0,0:01:13.03,0:01:14.37,Default,,0000,0000,0000,,What's the power that you have
Dialogue: 0,0:01:14.37,0:01:16.50,Default,,0000,0000,0000,,to raise two to to get to two?
Dialogue: 0,0:01:16.50,0:01:18.60,Default,,0000,0000,0000,,Well that's the natural log of two.
Dialogue: 0,0:01:18.63,0:01:21.63,Default,,0000,0000,0000,,Once again the natural\Nlog of two is the exponent
Dialogue: 0,0:01:21.63,0:01:24.10,Default,,0000,0000,0000,,that you have to raise e to to get to two.
Dialogue: 0,0:01:24.10,0:01:27.70,Default,,0000,0000,0000,,If you actually raise e to\Nit you're going to get two.
Dialogue: 0,0:01:27.70,0:01:29.57,Default,,0000,0000,0000,,This is what two is.
Dialogue: 0,0:01:29.57,0:01:31.57,Default,,0000,0000,0000,,Now what is two to the x to the third?
Dialogue: 0,0:01:31.57,0:01:34.50,Default,,0000,0000,0000,,Well if we raise both sides\Nof this to the x to the third
Dialogue: 0,0:01:34.50,0:01:38.17,Default,,0000,0000,0000,,power, we raise both sides\Nto the x to the third power,
Dialogue: 0,0:01:38.17,0:01:40.97,Default,,0000,0000,0000,,two to the x to the third\Nis equal to, if I raise
Dialogue: 0,0:01:40.97,0:01:43.43,Default,,0000,0000,0000,,something to an exponent and\Nthen raise that to an exponent,
Dialogue: 0,0:01:43.43,0:01:49.27,Default,,0000,0000,0000,,it's going to be equal to\Ne to the x to the third,
Dialogue: 0,0:01:49.63,0:01:53.00,Default,,0000,0000,0000,,x to the third, times\Nthe natural log of two,
Dialogue: 0,0:01:53.10,0:01:56.13,Default,,0000,0000,0000,,times the natural log of two.
Dialogue: 0,0:01:56.23,0:01:59.17,Default,,0000,0000,0000,,That already seems pretty interesting.
Dialogue: 0,0:01:59.17,0:02:01.57,Default,,0000,0000,0000,,Let's rewrite this, and\Nactually what I'm going to do,
Dialogue: 0,0:02:01.57,0:02:04.10,Default,,0000,0000,0000,,let's just focus on the\Nindefinite integral first,
Dialogue: 0,0:02:04.10,0:02:05.17,Default,,0000,0000,0000,,see if we can figure that out.
Dialogue: 0,0:02:05.17,0:02:06.50,Default,,0000,0000,0000,,Then we can apply, then we can take,
Dialogue: 0,0:02:06.50,0:02:08.57,Default,,0000,0000,0000,,we can evaluate the definite ones.
Dialogue: 0,0:02:08.57,0:02:11.57,Default,,0000,0000,0000,,Let's just think about\Nthis, let's think about
Dialogue: 0,0:02:11.57,0:02:15.03,Default,,0000,0000,0000,,the indefinite integral of x squared times
Dialogue: 0,0:02:15.03,0:02:18.50,Default,,0000,0000,0000,,two to the x to the third power d x.
Dialogue: 0,0:02:18.50,0:02:20.57,Default,,0000,0000,0000,,I really want to find the\Nanti-derivative of this.
Dialogue: 0,0:02:20.57,0:02:23.17,Default,,0000,0000,0000,,Well this is going to be\Nthe exact same thing as
Dialogue: 0,0:02:23.17,0:02:28.43,Default,,0000,0000,0000,,the integral of, I'll\Nwrite my x squared still,
Dialogue: 0,0:02:28.43,0:02:30.37,Default,,0000,0000,0000,,but instead of two to the x to the third
Dialogue: 0,0:02:30.37,0:02:32.17,Default,,0000,0000,0000,,I'm going to write all of this business.
Dialogue: 0,0:02:32.17,0:02:33.83,Default,,0000,0000,0000,,Let me just copy and paste that.
Dialogue: 0,0:02:33.83,0:02:35.30,Default,,0000,0000,0000,,We already established\Nthat this is the same
Dialogue: 0,0:02:35.30,0:02:38.50,Default,,0000,0000,0000,,thing as two to the x to the third power.
Dialogue: 0,0:02:38.50,0:02:42.87,Default,,0000,0000,0000,,Copy and paste, just like that.
Dialogue: 0,0:02:43.43,0:02:47.80,Default,,0000,0000,0000,,Then let me close it with a d x.
Dialogue: 0,0:02:48.84,0:02:51.77,Default,,0000,0000,0000,,I was able to get it in\Nterms of e as a base.
Dialogue: 0,0:02:51.77,0:02:53.43,Default,,0000,0000,0000,,That makes me a little\Nbit more comfortable
Dialogue: 0,0:02:53.43,0:02:55.37,Default,,0000,0000,0000,,but it still seems pretty complicated.
Dialogue: 0,0:02:55.37,0:02:57.70,Default,,0000,0000,0000,,You might be saying, "Okay, look.
Dialogue: 0,0:02:57.70,0:03:00.97,Default,,0000,0000,0000,,"Maybe u substitution\Ncould be at play here."
Dialogue: 0,0:03:00.97,0:03:05.03,Default,,0000,0000,0000,,Because I have this crazy\Nexpression, x to the third times
Dialogue: 0,0:03:05.03,0:03:08.00,Default,,0000,0000,0000,,the natural log of two, but\Nwhat's the derivative of that?
Dialogue: 0,0:03:08.03,0:03:09.37,Default,,0000,0000,0000,,Well that's going to be three x
Dialogue: 0,0:03:09.37,0:03:11.17,Default,,0000,0000,0000,,squared times the natural log of two,
Dialogue: 0,0:03:11.17,0:03:13.97,Default,,0000,0000,0000,,or three times the natural\Nlog of two times x squared.
Dialogue: 0,0:03:13.97,0:03:16.17,Default,,0000,0000,0000,,That's just a constant times x squared.
Dialogue: 0,0:03:16.17,0:03:18.77,Default,,0000,0000,0000,,We already have a x squared\Nhere so maybe we can engineer
Dialogue: 0,0:03:18.77,0:03:22.93,Default,,0000,0000,0000,,this a little bit to have\Nthe constant there as well.
Dialogue: 0,0:03:22.97,0:03:24.23,Default,,0000,0000,0000,,Let's think about that.
Dialogue: 0,0:03:24.23,0:03:27.90,Default,,0000,0000,0000,,If we made this, if we defined this as u,
Dialogue: 0,0:03:27.90,0:03:33.23,Default,,0000,0000,0000,,if we said u is equal\Nto x to the third times
Dialogue: 0,0:03:33.23,0:03:36.23,Default,,0000,0000,0000,,the natural log of two,\Nwhat is du going to be?
Dialogue: 0,0:03:36.23,0:03:39.90,Default,,0000,0000,0000,,du is going to be, it's\Ngoing to be, well natural
Dialogue: 0,0:03:39.90,0:03:42.03,Default,,0000,0000,0000,,log of two is just a\Nconstant so it's going to be
Dialogue: 0,0:03:42.03,0:03:45.93,Default,,0000,0000,0000,,three x squared times\Nthe natural log of two.
Dialogue: 0,0:03:46.03,0:03:47.43,Default,,0000,0000,0000,,We could actually just change the order
Dialogue: 0,0:03:47.43,0:03:49.03,Default,,0000,0000,0000,,we're multiplying a little bit.
Dialogue: 0,0:03:49.03,0:03:50.50,Default,,0000,0000,0000,,We could say that this\Nis the same thing as
Dialogue: 0,0:03:50.50,0:03:55.57,Default,,0000,0000,0000,,x squared times three natural log of two,
Dialogue: 0,0:03:55.57,0:03:58.57,Default,,0000,0000,0000,,which is the same thing just\Nusing logarithm properties,
Dialogue: 0,0:03:58.57,0:04:03.77,Default,,0000,0000,0000,,as x squared times the natural\Nlog of two to the third power.
Dialogue: 0,0:04:03.77,0:04:05.10,Default,,0000,0000,0000,,Three natural log of two is the same thing
Dialogue: 0,0:04:05.10,0:04:07.17,Default,,0000,0000,0000,,as the natural log of\Ntwo to the third power.
Dialogue: 0,0:04:07.17,0:04:13.33,Default,,0000,0000,0000,,This is equal to x squared\Ntimes the natural log of eight.
Dialogue: 0,0:04:13.70,0:04:16.43,Default,,0000,0000,0000,,Let's see, if this is u, where is du?
Dialogue: 0,0:04:16.43,0:04:19.47,Default,,0000,0000,0000,,Oh, and of course we can't forget the dx.
Dialogue: 0,0:04:19.97,0:04:25.80,Default,,0000,0000,0000,,This is a dx right over here, dx, dx, dx.
Dialogue: 0,0:04:25.83,0:04:29.77,Default,,0000,0000,0000,,Where is the du? Well we have\Na dx. Let me circle things.
Dialogue: 0,0:04:29.77,0:04:32.77,Default,,0000,0000,0000,,You have a dx here, you have a dx there.
Dialogue: 0,0:04:32.77,0:04:36.10,Default,,0000,0000,0000,,You have an x squared here,\Nyou have an x squared here.
Dialogue: 0,0:04:36.10,0:04:38.23,Default,,0000,0000,0000,,So really all we need is,
Dialogue: 0,0:04:38.23,0:04:40.90,Default,,0000,0000,0000,,all we need here is the\Nnatural log of eight.
Dialogue: 0,0:04:40.90,0:04:43.83,Default,,0000,0000,0000,,Ideally we would have the\Nnatural log of eight right over
Dialogue: 0,0:04:43.83,0:04:46.97,Default,,0000,0000,0000,,here, and we could put it\Nthere as long as we also,
Dialogue: 0,0:04:46.97,0:04:49.23,Default,,0000,0000,0000,,we could multiply by\Nthe natural log of eight
Dialogue: 0,0:04:49.23,0:04:53.33,Default,,0000,0000,0000,,as long as we also divide\Nby a natural log of eight.
Dialogue: 0,0:04:53.37,0:04:56.30,Default,,0000,0000,0000,,We can do it like right over here,
Dialogue: 0,0:04:56.30,0:04:58.83,Default,,0000,0000,0000,,we could divide by natural log of eight.
Dialogue: 0,0:04:58.83,0:05:01.37,Default,,0000,0000,0000,,But we know that the\Nanti-derivative of some constant
Dialogue: 0,0:05:01.37,0:05:04.03,Default,,0000,0000,0000,,times a function is the\Nsame thing as a constant
Dialogue: 0,0:05:04.03,0:05:06.30,Default,,0000,0000,0000,,times the anti-derivative\Nof that function.
Dialogue: 0,0:05:06.30,0:05:08.43,Default,,0000,0000,0000,,We could just take that on the outside.
Dialogue: 0,0:05:08.43,0:05:12.33,Default,,0000,0000,0000,,It's one over the natural log of eight.
Dialogue: 0,0:05:12.57,0:05:15.43,Default,,0000,0000,0000,,Let's write this in terms of u and du.
Dialogue: 0,0:05:15.43,0:05:18.57,Default,,0000,0000,0000,,This simplifies to one over the natural
Dialogue: 0,0:05:18.57,0:05:23.35,Default,,0000,0000,0000,,log of eight times the anti-derivative of
Dialogue: 0,0:05:24.45,0:05:31.77,Default,,0000,0000,0000,,e to the u, e to the u, that's the u, du.
Dialogue: 0,0:05:31.77,0:05:36.51,Default,,0000,0000,0000,,This times this times that is du, du.
Dialogue: 0,0:05:36.51,0:05:38.70,Default,,0000,0000,0000,,And this is straightforward,
Dialogue: 0,0:05:38.70,0:05:40.90,Default,,0000,0000,0000,,we know what this is going to be.
Dialogue: 0,0:05:40.90,0:05:43.10,Default,,0000,0000,0000,,This is going to be equal\Nto, let me just write
Dialogue: 0,0:05:43.10,0:05:45.60,Default,,0000,0000,0000,,the one over natural\Nlog of eight out here,
Dialogue: 0,0:05:45.63,0:05:53.88,Default,,0000,0000,0000,,one over natural log of\Neight times e to the u,
Dialogue: 0,0:05:55.77,0:05:57.70,Default,,0000,0000,0000,,and of course if we're\Nthinking in terms of just
Dialogue: 0,0:05:57.70,0:06:00.43,Default,,0000,0000,0000,,anti-derivative there would\Nbe some constant out there.
Dialogue: 0,0:06:00.43,0:06:02.90,Default,,0000,0000,0000,,Then we would just\Nreverse the substitution.
Dialogue: 0,0:06:02.90,0:06:04.50,Default,,0000,0000,0000,,We already know what u is.
Dialogue: 0,0:06:04.50,0:06:07.30,Default,,0000,0000,0000,,This is going to be equal\Nto, the anti-derivative of
Dialogue: 0,0:06:07.30,0:06:11.57,Default,,0000,0000,0000,,this expression is one over\Nthe natural log of eight
Dialogue: 0,0:06:11.57,0:06:15.37,Default,,0000,0000,0000,,times e to the, instead\Nof u, we know that u is
Dialogue: 0,0:06:15.37,0:06:19.10,Default,,0000,0000,0000,,x to the third times\Nthe natural log of two.
Dialogue: 0,0:06:19.10,0:06:21.87,Default,,0000,0000,0000,,And of course we could put a plus c there.
Dialogue: 0,0:06:22.10,0:06:24.23,Default,,0000,0000,0000,,Now, going back to the original problem.
Dialogue: 0,0:06:24.23,0:06:26.57,Default,,0000,0000,0000,,We just need to evaluate\Nthe anti-derivative
Dialogue: 0,0:06:26.57,0:06:29.50,Default,,0000,0000,0000,,of this at each of these points.
Dialogue: 0,0:06:29.50,0:06:30.97,Default,,0000,0000,0000,,Let's rewrite that.
Dialogue: 0,0:06:30.97,0:06:36.07,Default,,0000,0000,0000,,Given what we just figured out,\Nlet me copy and paste that.
Dialogue: 0,0:06:36.30,0:06:39.12,Default,,0000,0000,0000,,This is just going to be equal to,
Dialogue: 0,0:06:40.18,0:06:43.97,Default,,0000,0000,0000,,it's going to be equal to\Nthe anti-derivative evaluated
Dialogue: 0,0:06:43.97,0:06:47.10,Default,,0000,0000,0000,,at one minus the anti-derivative\Nevaluated at zero.
Dialogue: 0,0:06:47.10,0:06:48.03,Default,,0000,0000,0000,,We don't have to worry about the
Dialogue: 0,0:06:48.03,0:06:50.03,Default,,0000,0000,0000,,constants because those will cancel out.
Dialogue: 0,0:06:50.03,0:06:53.57,Default,,0000,0000,0000,,So we are going to get,\Nwe are going to get one--
Dialogue: 0,0:06:53.57,0:06:55.97,Default,,0000,0000,0000,,Let me evaluate it first at one.
Dialogue: 0,0:06:56.60,0:06:59.50,Default,,0000,0000,0000,,You're going to get one\Nover the natural log of
Dialogue: 0,0:06:59.50,0:07:05.43,Default,,0000,0000,0000,,eight times e to the\None to the third power,
Dialogue: 0,0:07:05.43,0:07:08.10,Default,,0000,0000,0000,,which is just one, times\Nthe natural log of two,
Dialogue: 0,0:07:08.10,0:07:10.83,Default,,0000,0000,0000,,natural log of two,\Nthat's evaluated at one.
Dialogue: 0,0:07:10.83,0:07:15.07,Default,,0000,0000,0000,,Then we're going to have\Nminus it evaluated it at zero.
Dialogue: 0,0:07:15.10,0:07:18.03,Default,,0000,0000,0000,,It's going to be one\Nover the natural log of
Dialogue: 0,0:07:18.03,0:07:21.63,Default,,0000,0000,0000,,eight times e to the, well when x is zero
Dialogue: 0,0:07:21.63,0:07:23.77,Default,,0000,0000,0000,,this whole thing is going to be zero.
Dialogue: 0,0:07:23.77,0:07:29.10,Default,,0000,0000,0000,,Well e to the zero is just\None, and e to the natural
Dialogue: 0,0:07:29.10,0:07:32.23,Default,,0000,0000,0000,,log of two, well that's\Njust going to be two,
Dialogue: 0,0:07:32.23,0:07:33.97,Default,,0000,0000,0000,,we already established that early on,
Dialogue: 0,0:07:33.97,0:07:35.77,Default,,0000,0000,0000,,this is just going to be equal to two.
Dialogue: 0,0:07:35.77,0:07:39.37,Default,,0000,0000,0000,,We are left with two over the\Nnatural log of eight minus
Dialogue: 0,0:07:39.37,0:07:42.50,Default,,0000,0000,0000,,one over the natural log of\Neight, which is just going
Dialogue: 0,0:07:42.50,0:07:47.53,Default,,0000,0000,0000,,to be equal to one over\Nthe natural log of eight.
Dialogue: 0,0:07:47.90,0:07:52.27,Default,,0000,0000,0000,,And we are, and we are done.