 So what we're doing for the next three classes is I'm just reminding you of what we've done for last three months. So to remind you of anything I'll do is I'll give you an outline of the topics we've covered and then I'll turn back to the rest of the topics. So the first thing we did in the class is we talked about integration. And in particular, we focused on techniques of integration, which the institution is that the integral is the integer. Is u v equal to the integral of v v u? It's a matter of the order exactly, but certainly we did partial fractions, so we call this instead of three integrals, we call this integrals of powers, the sine or the cosine. So everyone remember what all these things are? Not necessarily how to do that? No. Not the bottom one. So the bottom one is like how do we do the integral like sine u of x? The whole sine is where x v x, something like that. This one would be an integral involving something like v x or, I don't know, one minus square root of 4, something like that. This one, rather than you just have to do it for the latest thing, this is to let you split up something that's back there. So something like the integral of v x. So those are the various techniques of integration. I will come back and do an example for a little bit. Let me just fill up the board with an outline and then we'll come back to new stuff. I think that's it for techniques of integration. And then there's some other stuff about improper integrals. These are things like the integral from 1 to infinity of 1 over x squared v x or the integral from 0 to 3 of 1 over x minus 1 of v x over x minus 1. Things like that, things where the integral blows up or the bounds of integration are undefined. So we did this sort of thing. It messed up a lot of people who got confused about it. I think it may make a little more sense now that you're a little more familiar with series and so on where you're doing some of these techniques again and again. And then there were also things like differential integration. So this included things like intractability rule. So there's also error estimate with four intervals. How far off is it from the real thing? So that's chapter 5, I think. And I think it's everything in chapter 5 that we covered. Let me just work to make sure I just feel like I didn't get something. Yeah. Okay. And so that's really, yeah, that's the stuff in chapter 5. Then other stuff that we did was applications of this stuff and it all sort of flows together in one. But so beyond just doing integrals, we have things like area between curves and fields. And this includes where you know the cross-section, the surfaces of the revolution, and the washer method of this method. Yeah. Archway. You feel that things were sliding on them. It was not fair today. So we have to be fair to all the publics. Yeah. Yeah, it's like, you know, I remember when I was a kid, I had these toys that I always played with and those other toys that I played with, and sometimes I had to play with them. Yeah. It's the same thing here. Blur, blur, blur. Maybe she should just write with it. Yeah! I know! You know all the answers to the ones that you wrote. Yeah. I think that's it, right? That's it. Okay. And then, of course, we have four or four events. The sequence is just a list of numbers. I know a lot of people get confused between the words, but this is a list of numbers. We want to know what happens to the list of numbers. And the main reason for doing infinite sequences is so that we have, for this class, although sequences come up in all sorts of states and mathematics, the main purpose of infinite sequences is to specialize them to income it. Sounds, we're serious. So this is, I guess, let me just write the notation. So these things are usually written like this and we use a list. And this means... So we spend a lot of time on infinite sounds. A sound that we know how to figure out what they add up to by maybe their special of some sort. So maybe their telescope. And the term cancels out as you go up. We get to geometric series. So that means that it looks like something, and if R is less than one, it has some value than this average. Okay, so we have those. There were some recursive ones, which means that the later terms are defined in terms of the earlier terms. And then we just have... So these are sort of the special ones. And then we have the tests for the arbitrary ones that we can use in the term of these. So we have, say, we have... People test, ratio test, let's look at ratio test. I think that's pretty much it. Is there any... So the main ones that... Well, those are the ones. And so these are to tell us when some series converges, but not necessarily when it converges, too. And then maybe to that, we have starting series, like 3X to the n, then there is a value and a range of X distribution that doesn't converge. Typical methods that you can use for this, if you use the ratio test to write out the interval convergence, and this is starting to get... That was sort of distant past, and now the recent... You get the ratio... You can do the ratio test, you get an interval. You check the end points, you can look for tests. You can see one of these guys, and then there you go. And then a special version of power series is we have Taylor series, form of boring series. So these are ones where this says that a function can be represented by power series near some certain value. So if we take a value, say C, and this tells us that F of X is the value of C plus the derivative on distance from C. So this is just the tangent line, and then the second derivative factorial. This is the Taylor series. The boring series is just Taylor where C is 0. This is from ending to 0 to infinity of the nth derivative. Oh, I forgot the alternating series. The nth derivative is evaluated at some point by my n factorial of X minus the nth derivative. So here I forgot the alternating series. And again with Taylor series because it's just a special kind of power series. This again has an interval of convergence, radius of convergence, that sort of thing. Together with this we also have similar to the other thing where we can say if we go up to five terms and stuff how much are we all? So this is a series that you know or you should know. For example well the geometric series we just copyrighted the cosine the log et cetera. So there's a bunch of these which are ones that we tend to know and then we can either derive the Taylor series directly by just taking derivatives over again. Or if we wanted a Taylor series for something like even the 5x we can plug in here. Or if we wanted a Taylor series for the integral of X sine X then we can multiply sine X by X and then integrate through my term and so on. So to get other guys. In terms of differential equations we have really if you study so there's Euler's method here where I said complex numbers where did I put complex numbers. So this includes for approximating differential equations and then there's ways of solving them so really the only ways that we have is if it's separable there are other methods but if the equation is separable we just separate, integrate both sides and solve. We should have been doing a lot of that lately. And then we also have to set for something like y over time y over time y over zero where we find the characteristic polynomial plug in. And then we looked at specific case specific cases of the separable equations exponential variations of that so exponential models which are differential equations like y prime equals constant times y logistic equation which is y prime equals constant times y times one minus P over M which is y over M something like that and then we had some variations like on the shape of the homework or on the problems that I got a lot of emails and the way it was about the last assignment and then we also have we also have we just call it space plane which is a primitive trade which is what we did last time so in terms of the space plane well I said at the end of the last class that you can interpret these things in terms of the space plane by making such a confusion and turning it into a pair of first-order equations and make a picture but I'm not going to do that however you can stuff with the aphids and lay bugs or rabbits and foxes or gingos whatever they were that is scary so I think this is everything does anyone remember anything that I missed so we're good so it's not this, you're fine how many would you like to read this do you want one question it must be a positive number it must be an integer you won't want it I can write a test with one problem and you will be very sad write a test with one problem that is the one so how many would you like then why did you ask this question the test is two and a half hours long how many would you like why does it matter how many questions yeah that's what we're discussing don't ask it because if I tell you okay there's ten problems does that mean anything what does it mean and so how does that affect your life in any way there's ten problems it takes you two and a half hours to do ten problems if I tell you there's a hundred problems so yes so there will be a heavier focus means but if there are ten problems then maybe five of them will be from this or four or five so forty to fifty percent will be on this stuff and fifty to sixty percent will be on this stuff that means like well you have not had any tests on this material you only had homeworks on this material so that means that this is mentor grade but if you view it that way it's not that this is more important although for many okay so let me for those of you that are using math for and a lot of other areas of math this thing which seems completely artificial and weird it comes up a lot this business about differential equations very important but really it's not that important if you're really good at intervals because in applications you should be able to understand what an integral is computers can do them tables can do them or if you can't do them numerically by using say Euler's method or well actually better methods like so that's not so crazy important to be able to do it but still that's part of what we do so obviously you can't do a very simple like x in the x yeah no so well yes and no so in this yes there are equations for telling me what the error bound is I don't remember them why should I expect you to remember them so if I am trying to ask you to say this is a real problem just like on the midterm I will remind you what the relevant formula is you don't have to know how to use that formula on the other hand the error it's just the next term that you did use that's pretty easy to remember the error in Taylor series is essentially the next term it's just not quite at the same place I may or may not choose to remind you of that on these work problems I don't expect you to remember the volume of confidence I'll remind you of the volume of confidence I won't remind you how to use similar triangles I won't remind you how to find the volume on the square hope you could find the volume but I'm going to square zero so that's easy I won't remind you how to find the area of the square but if you need to find the area of an octagon maybe I'll remind you how to do that so the standard things that you should have learned in fifth grade I'm not going to remind you of I'm also not going to remind you that 5 times 7 is 35 but I might remember I might remind you what 3 to the 11 is if you need that okay other formatting type questions you still want to know how many questions so if I tell you 18 I haven't decided because it was written 18 is a possible number it doesn't mean it will be written somewhere in the range of 10, 20 to that one okay other questions no, we're good so we can just like dance a glass for the rest of the time I'm happy with that I'm happy too okay, so I should just start with doing that there's no more so should I start at the beginning and go to the end or is there a specific page okay, so let's start at the beginning and go to the end so trying to do the most simple thing where I have an equation like I mean an integral like an integral of one over log one over so if I have something like this this is an obvious substitution because when you're doing substitutions you have to remember they're going to make you equal something and you shouldn't forget the dU and the biggest so the people who are going to get like 4 on the final are the people who don't remember the dU so by now remember, remind you that when you make a substitution there are two parts yet I keep seeing it even from the good students so we do that and then this becomes very easy so dU is one over X and X so there's dU right in my face and now this is an easy integral this is just the log and then U goes to the log of X so this is the log of the log so this is a very easy substitution problem the other thing that sometimes people screw up when they do substitution is if this had a definite integral like from 3 to 5 then I don't I don't ever have to go back to X here because if this were say an integral from 3 to 5 log X then when X is 3 log 3 when X is 5 we use the log 5 and so then this just becomes the integral from log 3 log 5 of dU that'd get the same answer when you stop okay so substitution this should be easy the only trick in substitution is sometimes being the integral it's not obvious exactly what substitution is today we saw it before there will be at least one substitution that will be sort of frequent more to the point would be something like integration by parts that was like the first real topic that we did in this class so integration by parts we have something like the integral of e to the 2X times X eX again this could be an indefinite integral or it could be a definite or it could really matter so here when we see when we see an integral that if you take the derivative it's a product maybe it could be a product because maybe it could be a product of eX but when you take the derivative of one piece and it gets better and you can integrate the other piece and it gets no worse then parts is likely so there's this little a lot of people remember this lipid or I think to try and remember which one you do first which parts are good and if you don't see a log then you do what does the I scan for inverse trig and then either algebraic trig and exponential or algebraic exponential trig I don't know what the S is so either one of those things and you should know by now this why there'd be two things where exponential and trig can't have any reward because exponential and trig are really the same thing okay, when we have this we want to pick out one thing to be u and another thing to be dv so in this case when we take the derivative of an exponential it gets no better we also integrate exponential we take the derivative of a of an x it gets better so we should take u e to the 2x dx and so then du dx while I integrate this so this is one half e to the 2x and so then this becomes uv minus the integral of vv u so uv is one half v to the 2x minus the integral of one half v to the 2x vx and so then I just do that integral and I get one half x e to the 2x the body is one corner okay, so integration by parts is very straightforward sometimes you have to manipulate it so something like the integral of e to the x minus vx this is one of the ones does anyone want me to do this example this is one where I integrate by parts once I get something I integrate by parts again and I get this same integral in there and then I solve for the integral so things that tend to be exponentials but string has this property that if you integrate by parts twice you sort of get back where you started and then you can solve you get back where you started plus some other jump and then you can solve for then you started with the other jump does anyone need me to do this example so I'm seeing yes absolutely not so I'll do it for a standard nonsense so here choose one part I don't care which part how about let's let u be e to the x and v be the sign so then v is the cosine and v u is v x v x so then this becomes minus cosine minus e to the x cosine x minus a minus minus the integral of a minus cosine v x right because minus a minus makes it a plus and then do it again so this guy here is in the same form this guy so again I get parts e to the x v v to the cosine x dx integral of the cosine is a sign and v to the v x to the x and v to the z it's really worthwhile to continue to write vx a lot of students leave them off and I think it confused it's worth putting them on okay so do that for this thing that means that this becomes what I have laying around before this plus what I get from this u v is e to the x of sin x minus the integral of v to the u so this part from this example so I have a big equals some stuff minus the thing so that, now we can solve for this is coming to two times the integral of e to the x sin x is this jump e to the x cosine x negative plus e to the x sin x and then just make it a constant so that means that this integral that I want is half of that jump so this is the standard kind of thing you start with something, you push it around and you get back basically what you started with as you may remember on the exam it's a little trickier but if we get two here three there basically this is the model problem any other questions about integrations by parts things like partial fractions, trig substitutions trig integrals, blah blah blah blah are all just variations on integration by parts or substitution in special cases where there are special tricks so the first thing might be say powers of sines and cosines for integrals involving trig functions not inverse trig yet but trig functions so something like integrals cosine x to the I don't know, fourth dx something like this of course I can put two's in 5's and 9's in there so in order to do this kind of thing we use the most commonly used trig identity which is basically which is remember the category of theorem which allows us to turn even numbers of sines even powers of sines and cosines or even powers of cosines so we use this to manipulate it into something where I make this substitution what should I do yeah, split off the sines to x turn them into cosines and then make the substitution so I use the fact here that sines squared x to the one minus the cosine and so then this becomes the integral of sines, probably right in the hand so sines squared one minus cosines squared and I have another sines sitting around the fourth sitting around and I make this substitution u equals cosine so du dx and right here I have du so this becomes then the integral of one minus u squared times u to the fourth minus and then you multiply it out and you go from there so at least all come down to this kind of thing of course there's so if you have one of the powers of sine or cosine being odd then you're in business if both of them are even you have to use a slightly more complicated method different trig identity we use the fact sine squared of x is one half times 2x and cosine squared of x plus cosine 2x so we can use this identity to chop the power in half which then turns us into this situation right so you all do a bunch of rows do you want to do one of those okay good alright so another technique that we have turning an algebraic trig something that if I have an integral that involves something like so so if the integral involves something like one plus x squared then I would make this I would use the fact that the tangent squared is the secant so I would make this a substitution that tangent of x no sorry I would make this a substitution do you want so often when you have something like insure you would make a trig substitution here but it doesn't have to be under the fact it's just that if it is under a radical it makes it better because you can take the right this this is what we're thinking here we're using the fact that one plus tangent squared so if we have a radical then this turns the one plus x squared into the square root of the square we want to turn our x into a tangent so here I would make this a substitution x is the tangent of something there's a constant around we can use a constant so for example if I have 5 here and I would want to transform this into one plus x squared I would here say y cubed over 5 is the square and 1 over square root of 5 square root of 5 if there are specific things that you want me to review more than on the site right now otherwise I'll miss you when I think of it as I go through and I'll waste some