 Welcome to lecture one, probably one out of, I don't know, 60, 61, 62, somewhere in there. Lectures for this course. We're in a studio classroom on NC State's campus. So hopefully we'll try to make it for the people that aren't in the classroom. Well, let me go back. For the people that are in the classroom, as normal as possible, that's probably stating a little too much because we have cameras in each corner and microphones on the desks. But just try to pretend as if they're not there, except that don't cover up the microphones. Things that you would normally ask, as long as they're not like silly, go ahead and ask them. I mean, we want it to be a classroom interactive environment. Just be aware of kind of extra things like getting here late is probably more noticeable in a cable TV class than it is in just a class in Harrelson Hall that has 100 people in it. Same thing with leaving early or any kind of, you know, if you're used to doing this, the whole class, and you can get away with it in a class of, you know, 450. I'm not going to do that the whole time. But if you're used to doing that, this is probably not the class for you. Be aware of Bedhead because your Bedhead is going to be on TV. You might want to have a cap handy in your book bag. No Carolina hats, no Carolina shirts. They're on TV all the time anyway. Every time you turn around, it's Carolina this or that, and we don't want to add to that mess of Carolina being on TV on every station. But I am proud of what we do here at this school. We have two degrees from this school. So I want what we do out there on the airwaves to be a quality product. If I make a mistake, I certainly hope that you're going to catch it and you're going to correct me. It won't be the first and I'm absolutely positive and won't be the last time for that to happen. I'd rather not leave it up here, kind of for all the world to see if it looks like it's wrong. I'm going to get to my attention and you'll fail the class and everybody will be fine. You run the risk of lower grades when you do that. I will tell you, last semester I felt like I had a good group of students, those that persisted in the class. I did learn everybody's name pretty quickly. I've got what, four? Kelly Chandler, Nicole, and Katie. Is that all the repeaters that I have? And I know a couple other people want to come in here. So if you're the kind of person that you skip classes a lot and you're not really attentive and you don't really try real hard, I don't think this is the class for you either because we've got people that would like to kind of interact in this setting and are serious students that aren't able to get in. Did every one of you get kind of an email, kind of that kind of email from me that I was barraged with emails and calls about people that want to get in this class? I have no idea if they knew me as an instructor. But anyway, maybe the time of day or the building or the fact that they get the back of their head on TV, I don't know. But we've got people that are still trying to get in this class. So kind of be honest with yourself if you're one of those that isn't a serious student. I would really appreciate if you'd step aside because there are some that are more serious that want to take your place. My grade distribution for Calc I in here this class last semester, I think this is it. If this isn't it, it's very close to being it. I had seven A's, two B's, four C's, two D's and seven F's. So it's very symmetric. Doesn't that sound symmetric? Seven, two, four, two, seven. Now a couple of the F's were people that quit coming and I really didn't have a choice. They begged me to give them an F and I granted their wish. A couple of them, just people that unfortunately didn't do well. Now I'm kind of reasonably content with that distribution but I think it can be better on the other end, the two D's and seven F's. So let's see what you guys can do to bump some of those numbers up to the more positive end. This is a tough class. 241 is tough not because the content is necessarily tough. It's because it's a little choppy, to be honest with you. We'll do some things for a while, some integration and use of tables. We did some of that. We'll review that. Numerical integration, Simpsons rule, Trapezoidal rule, improper integrals. Then we switch horses to applications of integration, tied together but not, I mean, not really nice flow. And then we'll do some things that are out of the textbook. It's a supplement to the textbook, second-order differential equations. That seems completely different, to tie it together a little bit. If you purchased your book recently, it may or may not have come with a supplement. There is a little kind of brochure that has the same cover put out by the same company, but it's a supplement to this book. Katie, you've got yours. Let me have that show it. If you bought your book at the bookstore new, this should have been with it. If it was not with your book, I'll get a count before we do anything with it, and I'll print you out the material that you need from this supplement for this course. But we do some things in this supplement. Thank you. We also do sequences and series at the end of this course. So it's just a little choppy, to be honest with you. It's really good stuff. I really like it. I hope that I convey that on a daily basis, that I like what I'm doing. I want you to like this stuff. I realize that not all of you are quite as strange as I am, and you're probably not going to like it as much as I do. That'd be wonderful if you did, but I hope that you can at least, by the end of class, tolerate it and tolerate me talking about it. So you have a generic kind of departmental syllabus. I apologize for the fact that some of the gray and shaded areas, they just don't copy all that well, but hopefully the tests are clearly marked, and if you need to know ahead of time, like now, when tests are to schedule your work schedule, you now know when they are. And the exam in this class, which is not on this page, you may want to write it at the end of this page, which is Friday, May 1st from 8 to 11. Friday, May 1st. That's the end of the first week of exams. Friday, May 1st from 8 to 11 in this room. And the exam is cumulative. So hang on to your tests, and those become your best study guide, I think, for your final exam. All right, now what are some things that I'm going to have to say periodically during the semester because of the fact that this is not just a classroom for this semester. I'll be saying things like this. This applies to the people in this classroom, these deadlines, these due dates, but not necessarily to the students that are taking this course this semester on cable TV. So this classroom is broadcast on a, I don't know, several day delay schedule on Raleigh Cable Channel 18. And I've got, I think, seven students taking this course on cable TV. And then after this semester, this broadcast rolls to DVD, and then I'll have students taking this course via a packet of DVDs. They could be in Texas or California or overseas, taking this class from NC State Distance Ed. So we've got you guys, this immediate audience. We've got the cable TV audience. And then we've got the later semesters audience of those that are taking it on DVD. So I'll kind of have to mention periodically which audience that I'm addressing. So I'm going to pass it. So tolerate that. I apologize that that enters into your classroom experience, but it helps people that take this course later, especially to know that, you know, the test isn't going to cover exactly the same material necessarily. So for future semesters, the DVD group, you'll be provided a pacing guide, and it will tell you what test one will cover, when test one will be, because obviously it's going to be different than this group. Does anybody have any questions, issues? Chandler. What are you going to do for a percentage for tests? Yeah, now I've got that almost ready. Okay. The four of you that had me last semester in this setting know that I have an administrative job in the math department too, and that has consumed most of my time up until this morning. So I get to work for a few hours on my own coursework once we get everything else settled over in Harrison Hall. So there's one area that I want to embellish on the kind of the other, my specific syllabus, and it will have those things in it. Let me tell you what I, generically, I think you need to know today. The grading, 60% of your grade, and you don't necessarily have to write this down because it'll be handed to you tomorrow, 60% of your grade will be tests. 10% will be homework and quiz. It is difficult in this setting to do quizzes. I'll be honest with you because, you know, cameras are rolling, you know, people all over Raleigh just be bored to tears that they're sitting there watching you take a quiz. So quizzes are limited in here, but that's 10%. We do WebAssign homework. I'll go ahead and save time there for the applause of WebAssign. You're not going to do that. We know it's not a perfect homework system, but it's the best we can do because the things I love about WebAssign is that it is problems from the book. Sometimes it's the even-numbered problems toward the end of the list of problems, and they tend to get a little more difficult, but it is problems from the book. You get multiple chances at it. If you miss it, you get to try it again and you get immediate feedback. So it's got a lot of things that I think help. If you did homework with pencil and piece of paper and you turned it into me, you'd probably get it back a week later. We're well on our way to something else. So I don't think it's nearly as effective, the kind of the old-fashioned homework. At least we have immediacy that we can deal with. And homework questions that you can ask in class, they can be WebAssign questions. I'd like to get to the point where I could kind of start it or jump-start you on the problem, but occasionally we just... because of how tough the problem is or the fact that everybody across the board had difficulty with it, we'll just do the whole problem in class. All right, 60% test, 10% homework and quiz, 25% is based on the final exam, and possibly more. I'll explain that, 5% of your grade is final exam, and 5% of your grade is maple. So let me address this audience differently from the cable TV bunch, and then future groups, because the nature of our site license with maple, you guys have easy access to maple. So maple's a part of your grade, but to be honest with you, it's a small part. 5%. The cable TV bunch, they are more than likely in the Raleigh area, but not necessarily all of them are on-campus students, so they're not going to get access to maple. So their grade does not include maple. The DVD bunch, their grade does not include maple. So for you guys, 5% of your grade is maple, but the Distance Ed crowd maple is not part of their grade. Absences. This class will meet its 4-hour class, but occasionally meets 5 hours a week, depending on if we're on the syllabus or behind. If we're behind, we'll meet 5 hours that week, 5 times. Test weeks, we'll meet normally 5 times, but if it's not one of those situations, we are where we should be, we don't have a test that week, we'll not meet 5 times that week. So it would be a true 4-hour class, and I'll try to be sensitive. I don't know. What was it, about half and half? Last semester, I think about half of the weeks we met 5 times and the other half we met 4 times. So I try to take attendance. Once I get a seating chart, it's going to help me learn your names. I'll try to take attendance and mark your absences on that. If you come in late, make sure you see me at the end of class and change that absence to a tardy. So if you have 5 absences or fewer in this class, you'll get to replace your worst test with your final exam if it's higher. And that helps some people. I'll be honest with you, that helps some people last semester. The reason I like it, rather than just dropping the test altogether, is that it kind of keeps you working, and if you didn't know it for test 2 and you learn it and you've mastered it for the final exam, guess what? Mission accomplished. You learned the material. So that bad test that you had, if your absences are low, doesn't end up hurting you. If you have a documented disability through DSO here on campus, make sure you give me all the necessary accommodations so I can accommodate you the way you need to be accommodated. Obviously, I do have an academic integrity statement here that you are doing your own work. Those are very ugly situations. I deal with a lot of cheating situations kind of for the whole math department. And, I mean, some of them are just absolutely inane. You know, you have a test A and it's on blue paper. And you have a test B and it's on green paper. And you've got the green test and yet you've got all the answers to the blue test on yours. How did that happen, you know? Believe it or not, we've had a student and we've, in the past, and I've given him that little dilemma, you know, how did that happen? I already knew how it happened. Well, I didn't really know what I was doing and I just memorized a bunch of answers and, you know, he might as well just been going blah, blah, blah, blah, blah, because I didn't buy any of it. I mean, do your own work. We'll find out how your neighbor's doing on his or her paper. So, just do your own work. It is alive and well. Cheating is alive and well on this campus, so, but I'm trying to make sure it doesn't happen in here. We'll take tests in here in the classroom setting with blue books. So, the smaller blue books, which are really this size, not the full page size blue book, or you could get green books. I guess they have the environmentally friendly version now. They're called green books. Much prettier color than this hideous blue. But blue books or green books, turn in six of them to me. Don't put your name on the front. Just put it on a sticky note or something. So, I know you've turned it in and I'll check you off a list. And each time you take a test, I'll bring in 24 blue books, issue a blue book. It'll have a math department stamp on it. That way, I know you're not bringing in a blue book that's got a bunch of formulas or stuff written in the back. And I might have two different versions of tests in here, too. A Form A and a Form B. I didn't do that last semester, but I may have learned my lesson. I do have a home page. I'll now give you that link now, but it has some old tests on it. It may not be exactly the same material that is on your test, but it'll give you an idea of what types of things were important in a prior semester. The web-assigned link is also on there. So, some things that I'll add have to do with the departmental goals and objectives of this class. That's what I don't have the formal version of on here yet. So, when that gets entered in, then I'll give all of you a copy of this. Normally, in a 10-point scale, let's talk about the Bs. 80 to 89, that's the B range. So, 80 and 81 are B minuses. So, the bottom two numbers are B minuses or the minuses in general. The top two are the pluses. So, 88 and 89 would be B plus. Everything in the middle would just be a B. So, we do use the plus-minus system. Any questions that any of you have before we actually do what we're supposed to be doing anyway? Which is mathematics. Okay. There is some built-in overlap. In 141 and 241. From experience, we know that the end of 141 can sometimes be a little bit of a blur, and things don't get covered either with the depth that they should be covered or possibly not covered at all. And we do have people that have kind of come from, hopefully, not just 141s on this campus, but community college equivalents of 141 that maybe this is your first semester here and you just took AP Calculus, got credit for Calc 1. Now you're starting Calc 2. A lot of different backgrounds. So, I think it's probably wise to start off overlapping. So, the two sections that we overlap that were actually part of 141 are 5.7, which actually has three things in it. It has trig integrals. They actually have trigonometry in them in the integrand, and we'll make some substitutions to hopefully make them integrable. Decomposition into partial fractions. Once we get it decomposed into partial fractions, then hopefully each piece is quite a bit easier than the original fraction that we had. And trig substitution. So, we'll review pretty briefly, but we'll review those three things in 5.7. By the way, this doesn't have any trig in it in the original problem, but we introduce trig to... the main purpose, I guess, is to get a square root so that it simplifies, and we get rid of the square root. We're dealing with trig, and then we substitute back into the original. Now, that looks a whole lot easier than it actually ends up being. Table of integrals is you've got 20 integration formulas in the front of your book or the back of your book on some of these card stock pages, and you think, well, I don't know how to do this problem. I look it up in the table of integrals, and I jot down, you know, A is 4 and B is 3. It's not quite that simple. You have to make your problem actually fit into the formula that you see in the table of integrals, and sometimes that becomes the work. That is the preparation so that you can actually use the formula that's in the back of the book. So we'll spend some time reviewing how to use the table of integrals. How would a question be asked about the table of integrals? Typically what I do is I give a subset of the table of integrals, and then I give a problem for which normally you have to use the table of integrals, or you have a choice of 10 or 12 formulas to try to match up your problem with one of those. So I won't give you the whole table because I'd be giving you some stuff that I actually want you to know. In fact, I know you already know. So that's the goal of the first week is to review. If it takes us a little longer, that's okay. But I want to make sure we're kind of all on the same page on these things. So let's move on to new material, which was not part of Math 141. All right. So let's do some trigonometric integrals. So this is in 5.7. If you have your notes from last semester in 141, you might want to go back to them. Re-read it in your textbook. This is a well-written textbook. It's not some what I would call kind of a typical old math book that just kind of gives the formulas and gives an example and then moves on. It does give some pretty good pros in there about how to do these problems. And I think a lot of times it tells you not only kind of how to go about it, but why you're doing it this way and what are some common pitfalls in doing the problem that way. So make sure you read the book. I think it's valuable. I think you probably paid $140 for it, $120. Something like that. Otherwise, we'll get your money for it. So one-third of 5.7 would be trig integrals. One of the things you want to look for would be odd exponents. So we're going to have, let's say, some sines and cosines in the integrand. So let's say we have sine cubed. Let me just kind of get an idea here. I mean, is this something that looks pretty familiar? Is this something that was addressed in your 141? I hope. Yes? Okay. So if you've got an odd power, then one of the techniques that will work is to take one of those aside. So if we have sine cubed, it leaves a sine squared. So then we've taken one of those and kind of written it out to the side. What purpose can I like for you guys since you've had this most of you and it indicated that you felt reasonably comfortable with this, and I know you were doing a lot of leisure reading over the holidays about calculus and some of the procedures, some of the processes. Right, Katie? Leisure reading, yeah. Why would we want to send that single sign out there by itself? What's the purpose? Okay, this stuff out here is going to be derivative of u. So if we want sine of x dx to be du, what is the u thing going to have to be such that sine x is derivative of u? Cosine. Cosine, right. Now we'll correct plus and minus signs, but for the most part, if this is the derivative of something, that something better be a cosine, right? And unfortunately, it's not a cosine, but it can be with the Pythagorean identity substitution. What can we throw in there for sine squared? Cosine squared minus one. One minus one. So we're going to, that's our kind of our basic Pythagorean identity that interrelates sine squared and cosine squared. So if we're going to solve for sine squared, it's going to be one minus cosine squared. Is that, are we looking familiar? Reasonably comfortable? So then we actually, and I think as we go through this class, you're going to be writing the u's and the du's down less and less, but it's been a while. So let u equal cosine, right? Kind of already decided that. And if u is the cosine, that always bothers me the first couple of times I say that, u is the cosine. It should be u are the cosine, right? But u is the cosine of x. What's the derivative of u? So this particular integrand, well, let's do corrections. Do we need anything? Any numerical thing? Any sine thing that's not there? S-I-G and sine. We need a negative one, right? Right there? So we can have negative one times the sine of x times dx, so we can't just make our own negative one there without also doing the same thing out front. So we've got a negative one out front. One minus cosine squared is now what? And all of this other stuff sent out there for what purpose? D-U. D-U. Does that look like a simpler problem? Not simpler because it's still the original problem, but it's simpler looking, right? It's a simpler looking problem than what we had just a couple steps ago. So we've got this quantity. It's not two-way power other than one, so we can integrate each piece, right? So we want to integrate one with respect to u, which is u. And we want to integrate negative u squared with respect to u. It's an indefinite integral. We'll put a plus c there. So we've got negative u plus u cubed over three plus c end of problem. Let's move on to one that's actually going to challenge us, right? Not the end of the problem. Did we start with a u problem? No. No, we should not end with a u answer. So we started with what kind of a problem? X problem. It had x's in it. So we need to use these again, right? So wherever we see a u, we'll replace it with what we substituted in order to make it a somewhat easier-looking problem. So negative u is what? Negative cosine x plus u cubed is cosine of x cubed or cosine cubed of x plus c. Now, if you do have time, you can always differentiate your answer and work your way back to the original integrand, right? I don't know that we're going to take the time to do that because we're going to try to review several things here just this week. Questions on that? So that's if we had one function, sine or cosine, in this case sine, and we had it to an odd power. What if we have two of them in the integrand? Say we have a sine to the fifth and a cosine squared. The thing I would look at first is are they both even? If they're both even, then we've got an issue different from what we had on the original problem. But the fact that one of them is odd, then we can kind of use the same process. So we've got a sine to the fourth, got a cosine squared, got one of those sines that I pulled away from this, and again, the goal is to make that du, that stuff that we send out here to the right. So if that's going to be du, then my u is once again going to have to be a cosine, right? So this is okay, that already is a cosine. This one requires some attention. So sine to the fourth is really sine squared, didn't really accomplish much other than kind of prepare for the next substitution. So for sine squared we want to do what with it? One minus cosine squared, that gets squared, that quantity gets squared. So let's go ahead and square that quantity. First term is one. What's the next term? Middle term. It's a binomial squared. Twice their product, right? And the last term, the trinomial. Last term of the trinomial. Cosine to the fourth. I'm going to try, especially early in the semester, I know we're reviewing, but I'm going to try to write out every step. Occasionally I'll skip some steps, but I know that causes problems for some people. But if you have fewer steps than this and you're comfortable with that, that's fine. I'm also comfortable with that. So this cosine squared can be distributed to every one of these, and we have a very similar situation to what we had before, what needs to be done before we actually do the integration. So if this is going to be du, then u is going to be cosine. Again, negative. So we need a negative one there and there. Now we're in business, right? I need to see the relief in all of you, just so glad to be back to this stuff. I mean, that's humdrum vacations, just bowl games, you know, presents, new games and stuff. I mean, let's get back to math. Goodness. Okay, we've got a negative one out here. I see that look. I see it in my own children at home. You are an idiot. That's the look. I see that. You're foolish. You're a foolish man. What do we have here? u squared. u squared. u squared. Plus u to the sixth. And all this stuff out here is... du. du. Everybody feel comfortable that you can integrate that? Mm-hmm. Right? Integrating u things with respect to u. Mm-hmm. Mm-hmm. So this is going to be what? u cubed over three? Yes. Bring the minus two along for the ride. You know, it's day one. Let's go ahead and write it down. The two comes along for... Actually, the negative two comes along for the ride. What do we get? Two u to the fifth over five. U to the... Seven over seven. That should be... Is that right? Mm-hmm. So everything gets negated. And then when we're done... Which we are technically done, right? With the integration. Mm-hmm. Everywhere we see a u, we replace it with... Cosine. Cosine of x. So it's going to be negative... cosine of x cubed over three... Plus... Two... Cosine of x to the fifth... Over five... Cosine of x to the seventh over seven... Plus c. Can I take care of my signs right? So that really is subtly or slightly different from our first example. The thing that is the same is that we had one odd power. So we take that kind of one out to the side. Then we're stuck with even powers. We do the transitions on the even powers. We do the kind of substitution stuff there. What if we're stuck with an even power? So let's make it simple. Sine squared of x. We can't pull one of the signs out to the side and then change what's left into the other function, cosine. Because we need that cosine squared or sine squared in order to be able to do that. So this requires a... Double angle identity. The best way of kind of resurrecting that formula is to go all the way back to the cosine of the sum of two angles. How's that work? Isn't it like sine of a cosine of the... Yeah, it's that direction. We've got to get the right terms. Cosine first. Cosine second. There's my help. Plus or minus? Plus, minus one line. One of those two? I'm not quite sure which one. Minus? Sure. Sine. Is that ring a bell? Yeah. That's a distant bell, right? Yeah. Way back down the road somewhere. Well, we don't really need this. We need a version where these are the same. So a plus a or b plus b. So let's just say it's a plus a which is really the cosine of two a. That's what we need. We need a double angle identity. So we'll take this cosine of the sum of two angles. Well, if it's a and a instead of a and b, it's cosine of a, cosine of a minus sine of a, sine of a. And what's that really? Cosine squared a minus sine squared a. So that may be something that looks familiar to you. Cosine double angle identity. So eventually on this problem, we want to make a substitution for sine squared. So we want to solve this for sine squared, but we've got some other baggage in here that we have to get rid of. So for cosine squared, let's make a substitution. So we've got some different versions of this that we want to use. This is one of the versions, but it's not really going to help us just by itself. So for cosine squared, we'll replace that with one minus sine squared, and we end up with this particular double angle identity. What is that? What's one minus sine squared minus another sine squared? One. One minus two. Yeah. Sine squared. And now let's solve that for sine squared. So I'll move that term over here and make it positive. And I'll move this term that's positive over here to the other side, which makes it negative. Divide both sides by two. I'm just going to take half of both sides. That's not an A. What should that be? That should be a two A. So there's one of the, you may want to call it a double angle identity. You could actually technically call it a half angle identity because I don't like what I have here. I've got too many letters mixed up. Correct me here. From day one, I need correction. I need help. What's wrong with my development here of this formula? Your x is an A. Yeah, x is an A. I better clean those up. Cosine squared of A should be one minus sine squared of A. So this should be an A. This should be an A. And this should be an A. Now are we all right? So sine squared of A, so we've got this angle A in this formula, but we've also got this angle twice A. So A is half of twice A, or twice A is double A. So depending on your viewpoint, it could be either way. So let me clean that up again. Sine squared of A. So let's make that substitution into our original problem for sine squared of x, which can you think of something that has sine squared of x for its derivative? What that is? That's kind of what we're trying to do. So that by itself is not going to work. So let's make a substitution for sine squared of A where we have sine squared of x, which I kind of was a little early with my x's anyway. Let's put in one half. Well, I don't really want one half inside the integral. Let's move it outside. One minus cosine of 2x dx. Can that be integrated? That'd be a shame, if it couldn't, because all that work was wasted. So here's what we have. One half. So we can integrate each piece, one with respect to x. Gosh, I can remember that. So we don't have something to a power. We just have this quantity, so we can take each piece of the quantity with respect to x. What's the integral of 1 with respect to x? x? What's the integral of cosine of 2x with respect to x? Negative 2 sine of 2x. Okay. That's mostly right. It's been a while since we've done this. Let's go ahead and write something down. Our u is really what? 2x? 2x. What does the integrand need if we're going to let u equal 2x? It doesn't need, but yet not half. What's du? It's 2 dx, so we need a 2, which means we have another one half, right? So isn't this now cosine of u? Is that correct? Is that right? This is cosine of u, and isn't this other stuff out here at the end? Isn't that du? And what's the integral of cosine of u du? Positive sine. Positive sine of u, right? Isn't the derivative of sine cosine? So what's the antiderivative of cosine should be sine? So we've got the integral cosine of u du, so it should be sine of u. This is the kind of stuff that I think early in the semester, this little written down u and du is going to be done up here, which was probably either the case or almost the case at the end of 141. Let's try to get back to that point. Is that right? If you took the derivative of sine of 2x, you'd get cosine of 2x times 2, right? And then we've got that little one-half out in front. So one-half x minus one-fourth sine of 2x. Gosh, I have no idea where we could check this to see that it's true. Actually, I do. I wouldn't have said that. Where could we check this? Table of integrals. So if you looked at the table of integrals, gosh, there's something we haven't even addressed at all, hyperbolic forms. Probably should do that in here. Trigonometric forms. So I'm looking at the table of integral integration formula number 63. It should be in kind of the back, inside cover of your book. Does anybody have that? Does that look right? We come up with what we should have for the integral of sine squared. I think we did. We don't need that table. We need it to check occasionally. We'll come up with our own table. Any questions? What if we're stuck with... We've got limited time here. What if we're stuck with cosine squared instead of sine squared? We'll make this our last one. We've got about two minutes. We had our double-angle identity was cosine of twice A. What was it? Cosine squared of A minus sine squared of A. Is that right? Earlier on the other problem. The other problem, the integrand was sine squared, because then we kind of pursued it so we could get rid of the sine squared. Let's pursue this so we can get rid of cosine squared. So we're going to leave this alone. Try to keep my A's and X's straight here. What can we replace for sine squared? So cosine of twice A becomes what's cosine squared minus a negative cosine squared? It's the same thing as cosine squared plus cosine squared. Two cosine squared. And then we've got this minus one. We want to solve for cosine squared because that's what we're stuck with in the original integrand. How can we solve this for cosine squared? Move the one over and take half of each side. So there's what we're going to use for cosine squared. Looks kind of similar, by the way, doesn't it? To what we plugged in for sine squared. So for cosine squared of A, we're going to plug that in so our problem becomes. I'll just slide that one half out in front. We just integrated a problem very similar to that. I think we can integrate this in much the same way. So we'll do a tiny bit more with trig integrals, but not much. And we'll go on to review trig substitution, partial fractions, and table of integrals the rest of this week. 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