Transcript for:
Trigonometric Integral Techniques

well let's start section 7.2 we're going to basically continue where we where we left off right here so it's a really nice segue in the next section you see we just did a trigonometric integral right yeah yeah and we did it with a reduction formula what we're going to tackle now is what if it's not just sine to the fourth power of X what if it's sine of the fourth power x times cosine to the fourth power of X or something weird like that well we're going to talk about at that exact example and I'll I don't know if we're going to get there today but we're going to talk about it eventually probably uh next time probably tomorrow we'll get back at it we're going to see what happens in these cases when we have trigonometric functions being multiplied certain powers inside of our integrals it's going to get kind of fun it's going to get a little weird going to get crazy but we're gonna have a good time with it 7.2 we're going to talk about tree integrals you know the one we're going to start off with first is a very common one we're going to start talking about what happens in your integrals when you have assigned to a certain power and cosine to a certain power being multiplied together so sine and cosine don't necessarily have to have the same power they could but I want you to consider the two cases that that we have okay so here's two cases case number one case number one is the case where we have an odd power on at least one of these functions so either m is odd or n is odd or both are odd do you guys yeah I know it sounds silly but you know what odd means yes odd is like one three five seven nine those those Powers so case number one is where m and or n is odd so they get both be odd what's the only other case they're both either yeah see if one of them is odd naturally the other one's going to be eaten hello if they're both odd none of them are going to be even but if none of them are odd they're both easy so m and and are even of course I'm not going to leave you hanging we're gonna talk about both cases here what I'm going to do for you I'm going to give you the the case by case to tell you exactly what to do first and then we're going to follow those cases with a series of examples does that sound okay with you yeah so let's talk about case one first case number one let's suppose so case number one either m is odd or n is odd so either sine has non power or cosine has an odd power or they both have an odd power if sine has an odd power if power of sine is odd what we're going to do we're going to strip off one of the signs so that what that means is like if it's sine to the third power of X we'd strip off one we'd have sine squared times sine does that make sense we're going to strip off one power so we're going to keep one factor of sine before and then with the other one you see if we have an odd power of sine and we take one sign away we strip off that factor then that odd power now becomes an even power doesn't it if it becomes an even power we can do a couple things with it we can no I'm sorry not a couple things we need one thing with it no we're going to do so keep one factor sign and use the Pythagorean identity to change sine squared x into 1 minus cosine squared x we'll see why this happens why this works I mean just a minute why we're doing it okay so if the power of sine is odd no problem strip off one power one factor sign keep it use pythagorean identities you change your even the rest of the even into this using Pythagorean identity if it's sine to the fourth well hopefully you can do something different with this if not you'd still do it you'd have sine squared squared and you'd still use this then you distribute that would be a very long tedious problem which we're definitely going to get to yay uh if it's the power of cosine that's odd you do the same exact thing only with cosine does that make sense so if power of cosine is odd do this basically this 4 cosine now but they're both odd what would you do if they're both odd I don't know well use the one that benefits you the most for instance pick the smallest power let's pick the one that gives you hopefully a sine squared or cosine squared by itself does that make sense to you so we we use the I'd say the smaller power or use the power that's going to give you something squared okie dokie so if both odd pick the one that will give you a power two if you can thank you if possible okay you guys all okay with case one probably not actually because we haven't even done an example you know what I'm talking about right now except that we're going to be stripping off a sine or a cosine you'll see it in a minute uh let's talk about case two first before we get to an example we definitely want to do at least two of these things before we're done today so case number two case number two that both the powers of sine and cosine are even if both powers are even so cosine squared and sine squared we've got both of them what we're going to do is we're going to use our half angle formulas and you're hopefully going to see why in just a minute you use half angle formulas do you does anybody have to remember the half angle formulas do you remember what they are some that you might I don't know if I don't know they're actually really really really similar for sine we'd have a one-half for cosine we'd have a one-half per sine we've got a one or cosine we've got a one excuse me for sine we have a minus for cosine we have a plus but for either of them we have a cosine 2X now I'm wondering if you see why this helps us chain rule works for derivatives what are we doing so can you do this derivative by itself no but if you change it can you distribute that easily can you do this integral yeah integral here not so much unless you want to do the reduction formula you can do that I suppose integral here piece of cake very easy because that's just a simple little substitution you'd probably do it in your head and then you're done with that okay with that one good would you like to see a couple examples let's bring it on right so so let's consider that let's go through the whole process okay first things first um does that fit in our integration table no well you do a substitution no maybe I don't know yet we don't know what this is designed to do is make your integral possible okay so we're going to manipulate this thing until it's possible so notice we're not doing hardly any calculus at all right now all we're doing is is using some identities breaking this stuff up and then trying hopefully to use the substitution so let's see is it definitely going to be integration by parts all the time no not necessarily so check it uh what do we have here do we have a case one or case two okay one okay we have some odd Powers if you have an odd power you're in case one no matter what okay so case one in this case they're actually both odd so what we're going to do is we're going to look down here and overall I pick one that we're going to power two so what I like to think of it I think the smaller one that's going to be a little easier to deal with because you can substitute out with the other one so okay we got cosine to the third we got sine to the fifth fortunately for us they have the same argument if they did it is he crazy all right so we're gonna leave that the way it is we're gonna look up at our powers what one would be our appropriate I almost said stripper uh our appropriate stripper here the stripper offer which one do we want to strip off the sine of the cosine cosine I'm not going to edit that by the way that's funny so so cosine of the third power no no what I'm going to make this I'm going to make this cosine to the second power of 2x what's the way I do this so hopefully this will make sense to you I'm going to write the sine to the fifth power we're going to leave that there I'm going to put the extra cosine that I stripped off over here don't lose track that this is actually cosine of 2x don't lose the argument okay it is 2x there okay I want to see if you guys understand that this is exactly the same as this these two things are the same show hands you feel okay with that good deal now why did we do it well check it out now that we have a cosine squared in there we change it from an odd to an even cosine squared we have an identity for that so instead of cosine squared we go no no well cosine squared it says for odd Powers try to use the Pythagorean identity okay well that's this or the reverse cosine squared equals 1 minus sine squared so cosine squared 2x no this is 1 minus sine this part is this for Pythagorean identity said so don't forget about the 2x I can't stress enough how important that is uh some of you people who don't do well in the chain rule you need to do well here with substitutions you can keep an eye on it after that we have sine to the fifth power of 2x we've got cosine of 2x DX okay I want to make sure you're still okay with that are you all right with it have we done any calculus yet no it's just true just some trick just identities actually now here's the cool here's the awesome thing as soon as we do this check it out this is why we stripped off one of the cosines and left it over here watch imagine U equaling sine of 2x what's the derivative of sine of 2x derivative 2 cosine 2X because you have a ah Jane rule that's right do you see a cosine 2X over here anywhere awesome that's why we do it so when we get this check this out see how this works you don't have to distribute this right now don't be crazy leave it just like this just like this watch here's one if I pick U equals look don't pit with substitutions with substitution you go through the whole process is in the the integration table heck no can you do a substitution yes don't go to integration by parts don't start doing crazy things if the substitution will work right now pick the substitution substitutions are typically on the inside of something remember we're talking about that instead of something so it's not going to be too many power it's just going to be sine of 2x if the derivative is there your substitutions can work so pick U equals sine 2X d u we all should be pros at this right now a derivative of sine of 2x is 2 cosine two x where's the two coming from DX I know many of you have done substitutions two different ways since I've been teaching in two different ways as well I'm going to stick with that there's two ways to do this you can solve for DX completely and get d u over 2 cosine 2X or you can just divide by two and get d u e over 2 equals cosine 2X DX hey do you see this anywhere in our integration that's why we put it last put Alaska that sticks out that sore thumb this thing is right here what that means for us is our interval is going to get really really pretty right now let's see what we've done do we still have an integral yeah yeah do we see level one hello are you with me yeah okay we still have a minus instead of sine of 2x squared what's this whole thing going to mean already it looks nicer doesn't it what else do I need up here bracket or parentheses I don't care which but you have to have them now check this one out I get sine to the fifth power of 2x sometimes it like in in your past substitutions you don't substitute everywhere you see the same thing because you're going to simplify it out sometimes you do here we're going to they're going to say hey if sine 2x is U that's also U this is U to the okay one more instead of cosine 2X DX we're going to have equal to I need to show off hands if that makes sense to you now that's beautiful here's why it's so nice what do you do with the two does it become a two or a one careful on that so one half integral what are you going to do here this is why we wait to distribute here we don't do it here this is crazy okay if you had to distribute here look at this if you distribute it here you have this whole thing and then you have this thing again right that's silly you do two substitutions and it would work but it's going to make it way harder all right we just wait till here so it's so much nicer now we can do our substitute or so no sorry our distribution and get U to the fifth minus U to the seventh du man that's like the first type of integral you saw in calculus one like that that thing is easy we get a one-half put your bracket there don't let yourself make a mistake of not Distributing this later we got what is this anyway uh you did the six over six perfect find the few to the eight over eight foreign here's our U so we're going to have here all the way down here we write it a couple different ways if you want to here's one way you can do sine of 2x to the sixth power over 12. minus sine of 2x to the eighth power over 16 and plus two of course you could do if you don't like it over 12 you could do 112 you could do minus 1 16 of 2x plus C you could Factor if you really wanted to but this is what I'm looking for from you show pants if that actually made sense is it that hard no but here's the thing uh the the math here it I promise you it's not hard it's just you got to follow the steps like precisely and you got to be able to identify what case you're working with the first step here is the hardest not the hardest it's the most important so the first step in doing this correctly is really really key for you to get to the correct answer here if you strip off the wrong thing it's going to take you either 10 times longer or it's not going to work out for you all right so let's get started so what we're working on we're going to continue working on some of these trigonometric integrals and yeah they get pretty hairy but we have some cases some some ways to deal with these things that make them a little at least a little bit nicer we have a technique a method to go through this so let's consider some of these we'll gradually make them a little bit more difficult so you can see some different ideas that we have to work with let's look at the integral of sine cubed x cosine squared x DX now I gave you some cases what I said was there's really two of them either you have at least one odd power or you don't if K if you have case one if you have at least one odd power the goal is going to be split off either a sine or a cosine of that odd power and then use a Pythagorean identity if you can use a Pythagorean identity you change it from Sines and cosines to either all signs or all cosines does that make sense that's the idea here so I want you to get familiar with what you're doing don't just follow this like a pattern really understand what we're trying to do we're stripping off one of these things firstly so that we can make a Pythagorean identity substitution oh well substitution and then we'll have all of one type of function either sides or cosines with the exception of that one thing we stripped off the reason why we need that one thing we stripped off is because then a regular U substitution will work really really nice so let's look at our problem do we have an odd power are they both odd no okay so if one's odd that kind of makes our choice easy as to what we're going to do if there's one odd we're going to strip off one of those factors and use the Pythagorean identity if they were both odd we typically pick the smaller power because that will usually give us a Pythagorean identity easier does that make sense to you so for us what are we going to pick for for the thing that we're stripping off here sign or cosine and why are we picking that it's odd and if I strip off the sign I'm going to get sine squared that's the Pythagorean identity I'm looking for so here we're going to write this and I typically write it this way I'll write my sine squared x first and then my cosine squared x and then I'll put my sine X the very back whatever you strip off right there you put in the back and the reason is is because what's going to happen again the idea here use this use this to get a Pythagorean identity that way signs become cosines these will all be cosines and then I can make a substitution hey what's the derivative of cosine sine yeah so that's the idea is we're trying to get to that we're trying to get back to this going away so uh by the way some people ask at this point well wait a minute why don't I use the Pythagorean identity on this one right here well if I do then I'm going to get sine to like the fifth power somewhere and that's that's not a good thing either we want to be able to make that substitution and get rid of this so as soon as we get our odd power we strip off one of the factors then we go ahead and say well the method I gave you said change one of those things by the Pythagorean identity whatever you stripped off that's what you're you're left with so this thing is going to be what is it one but yeah the rest of this stuff is still here the rest of that is right there quick head now if you're okay with it so far yeah if you're going another one now why does that help us well why it helps us is because right now I don't want you to distribute that be silly okay because this whole thing would have to go here and here you'd have two integrals instead of our one we want to make this easy right here if I do a substitution on cosine I make this U and make this U what's the uh what's the derivative of cosine again sorry close what is it that's going to go away yeah my integral is going to change to a negative but that thing's going to go away does that make sense this is the idea if you have an odd power break off one factor use an identity make a substitution that's how it works often with this stuff so here we go well this is really nice as soon as you see this don't worry about Distributing or anything because if we do a basic substitution for cosine do I want cosine squared or just cosine what do you think cosine yeah just cosine if I have cosine squared I got to do the chain Rule and that's really annoying because I'll have cosines and Sines coming up at us and we don't want that so d u equals negative sine X DX when I'm doing my uh my substitutions on a lot of these trig integrals I won't solve it for DX I know I showed you that sometimes sometimes I don't what I typically do is just move this sine or any constants over so negative d u would equal sine X DX because and this is why we put this at the back end do you see how this piece matches up perfectly with that piece you see what I'm talking about that's what we want so I move this at the back end because this is now really nice easy substitution right there okay so why don't you tell me what's going to be on the inside of our integral now squared okay good we got the one minus what number cool cosine is U so cosine squared is U squared excellent what else times zeros times sine X times U times U squared okay so we got one I get minus I Got U squared I get another U squared du there's something a lot of you love missing what are you what are you missing right now oh besides that parentheses missing parentheses that's important why is that important for some reason some of you guys have the tendency to miss parentheses and still distribute because in your head you like know they're there or something it's really important for you to put those you need them to say this is what I'm doing otherwise this math kind of falls apart you need to show that so these parentheses are not trivial and then now sine X DX is the same thing as what okay we don't want to forget the negative d u so times negative du what are we going to do with this thing right now what do you think hold the negative value okay so negative integral what else are we going to do that will be U squared minus U to the fourth don't forget the DU don't get sloppy with this integration you got to show all this stuff none of it is trivial you guys okay with this so far yeah the integral I go bam how am I going to do that something really really easy this is brilliant this is nice so we'll have negative what's the integral of U squared is well you tell me what am I what am I having wrong here okay so if I don't have my brackets then this is wrong if I do have my brackets then we have negative U cubed over three plus u to the fifth over five naturally you could write this in the reverse order might be a little bit nicer to do U to the fifth over five minus U cubed over three now we don't want to stop there what's the next thing we got to do your u back in there that's very good so whatever our U is we'll have and I'm going to reverse this right now so I'll have cosine to the fifth power of X over five minus cosine to the third power of X over three and that's the idea I wonder if that makes sense to you by showing hands can you show me if that does easy medium hard what do you think kind of medium right this part as soon as you see this this is easy this part is the part that people struggle with how do I do it you got to get used to seeing these things following the pattern you know what I mean that way you go okay hey look at this odd power that's an even power I'm gonna strip off one of those things that's the idea here shall we do another one yeah so so far I think I've given you two odds and I've given you one odd let's see how it works with this example you know what I'm gonna need some more room do you have any questions on this because I'm gonna erase it some of these things take a take a while so okay so now we're going to do sine to the fourth cosine to the fourth so as soon as you see this problem on a test yes you're going to have something like this on a test as soon as you see this on a test you all right cool uh how in the world are we going to do that well go through the pattern do we have at least one odd hour okay so they're both even if they're both even the idea of Pythagorean identity goes out of your head because if I do this if I strip off either a sine or a cosine I'm left with an odd power does that make sense yeah the Pythagorean identity is not going to work with the odd power your substitution won't work because you'll have an extra sign or an extra cosine somewhere that's not a good idea so if you have both evens the idea is you're going to manipulate this thing a lot it's going to be a lot of work but what we're going to be using here is the half angle formulas where we have sine squared equals one half one one minus cosine two x and cosine squared equals one half one plus I do very good so we're going to be getting down to those things here in just a minute so let's work on how we might go about doing this and there's a few ways to do it okay so here's what I do if I'm working on this problem the first thing I'm doing is I'm going to try to make this either all sine or all cosine if I have the same power like 4 4 we can typically do this check this out well I could make sine x cosine X all to the fourth is that true yeah sure I could do that my idea right now is to combine my Sines and cosines with some sort of an identity right now if you don't know it I can actually do that so if I'm looking at a sine x cosine X I'm going to use this identity over here do you know this one that's the right one do you guys know that one double angle okay we'll check it out we can manipulate identities all the time if we know they're true we can manipulate them so if I take this this thing looks a whole lot like this Peach donut yeah well I'm gonna make this be exactly like this I'm basically just going to divide by two then sine x cosine x equals one half sine 2x oh look at that we okay with that yeah children's feel okay with that one well what that means is hey look at that I can make it sine x cosine X to the fourth of course this is a special case Okay where they're both exactly the same power if they're not exactly the same power then naturally this won't work but I want you to see this at least once where they're both the same power you got me yeah so we can do this okay then this piece is one half sine two x so our integral is integral of one half sine two x all to the fourth power the X y'all still okay yeah you sure let's keep on moving what's the next thing that I need to take care of um I don't care really about this use substitution I really don't because right we it'd be very trivial U equals 2x you'd have a one-half sure don't worry about that we can do that basically in our head what I'm worried about and we're going to be changing things so if you do this if you do a u substitution on this now you're gonna have to do it later again too so don't worry about it until the very very end when you're actually evaluating the integrals does that make sense don't worry about right now what I do want to get rid of is this looks kind of nasty to me can I take this to the fourth power and this to the fourth power yeah you can okay let's do that so I'm going to have the integral of what is one half to the fourth power you know what people do when they're just learning this or when they're going too fast on a test or their homework is they'll forget about this guy and they'll give me one half again do you see how common that might be when I'm grading papers I see that a lot where are they getting 1 16 from I'm getting one half all the time my goodness well you might not be taking that to the correct power so be careful with this stuff this whole thing is being raised to the fourth imagine if you understand that concept okay okay then I've got sine to the fourth power of two x DX looks a little bit nicer what's the next thing I'm going to do 1 16. integral sine to the fourth two x DX now stop for a second if you were so inclined thinking about what I taught you back in the last section what could you use here you could use the reduction formula it would absolutely work right now do you guys see why oh yeah it's well there's no why I know but there's a sign look at the sign it's through the fourth power that power is greater than two greater than equal to two you could use reduction formula right now in fact we did sine to the fourth in class just like two days ago remember well the two days ago you took a test but three days ago you remember doing that yes last thing we did you could do it right now you can just make a substitution this is where you do it U equals 2x D over two this would be 1 over 32. you got me then you get sine to the fourth of Udu and you do production formula like I think it's two times if you do reduction formula twice and you'd be done with this thing did you guys get what I'm talking about look back in your notes or watch the video from the last section if you don't remember how to do reduction formula now what if you don't see reduction formula is there another way to do it the answer is yeah which is really interesting because what's going to happen here is we're going to get our integral evaluated and it's going to look different than if we had done this with reduction formula reduction formula will work it will be equivalent but it's going to look different does that make sense now I don't care how you do it if you see this I've already taught you reduction formula right you can do it that's fine anytime if it's signed you some power go for it some of us have a few of you have proved the reduction form for cosine go ahead and use it your answers are going to look a little bit different than if you do this next technique so Technique One you work it all down you go hey I see reduction formula I can do that it's assigned to a power greater than or equal to two no big deal remember that one it's the one over NS and you know Sines and cosines and a minus and one n Over N minus n minus 1 over n integral and it reduces it for you well let's say you don't see it is there a different way and the answer is yeah if there's not a reduction form impossible or you don't see it we can still do this with those half angle formulas and here's how you'd work it instead of thinking of this as sine to the fourth power what we need to do is we need to find something that will allow us to make this substitution we need this basically so when we get down here we go man how do I do it reduction formula if you don't see reduction formula you're going to have to rewrite this again so how we're going to rewrite it is 1 16 integral let me rewrite sine to the fourth power how am I going to write it as something like sine squared what do I do sine squared 2X I'm going to change what you said just a little bit I want sine squared 2x to sum power what is it squared squared you all right on that one yeah now as soon as you do that as soon as you go well wait a minute uh there's some even powers I'm gonna I know I'm gonna be using half angle somewhere unless I do reduction form that works out that nicely we can do that but if we can't then we're going to have some sort of a half angle formula coming up somewhere and here's where we go go with it so from here we still get a 1 16. sine squared let's do sine squared 2x let's make it into the cosine here's why we do it if we take sine squared and we change into one half one minus cosine maybe you're not maybe you don't know why we're going to do this what's the power here two what's the power here it basically reduces the power for you does that make sense you do you can do the integral of cosine to the first Power you can't do the integral of sine the second power of the wave is you need reduction formula or you need to make the substitution so we're going to do that use the identity now let's be real careful with this don't lose your stuff what's the power going to be up here okay and then inside let's do that what's going to be one parentheses one minus one minus okay I like that one half one minus and then cosine for sure now be very careful don't lose pieces of this thing it's not 2x explain why is it to it why is it not 2x okay because it starts from what's the argument about sine function here and this says whatever your X is also support it whatever your X is you're going to multiply it by two so if we have X yes 2x if we have 2X it's going to be do you see how common of mistake that's going to be for some people in here yeah okay don't make that mistake because then when you do your substitution you'll have a one-half when you should be having a one-fourth does that make sense or 1 8 or something so this is going to be 4X be careful be careful with your math okay with that so far good deal okay now what goodness gracious well if we're doing it this way we can't do a substitution because if we substitute for cosine is there a sign anywhere to be had we can't do it the only thing we can do here distribute all this junk so what I'm going to do same thing I did here I took one half to the fourth we're going to take one half to the second how much is that going to be so I've got a 1 16 out front already this is going to give me a one-fourth don't forget about that guy you need that so 1 4 you know what I need the bracket hope you follow the notation here all we did was we took one half squared it's one-fourth we took this thing now this thing now it's this thing squared 1 minus cosine 4X to the second power quick hit not be okay with that one can you guys distribute this are you just going to square this and square this at least we're having six don't do that remember that what this means is 1 minus cosine 4X times 1 minus cosine 4X would you please spend some time to distribute that for me what I'm going to do is I'm going to pull the 1 4 out front we have a 116. we now have another 1 4 what's 1 16 and 1 4 how much are we going to get there 164. then what so this now combines from 164. you guys are Distributing this right now for me so hopefully you get this it's not usually not very hard you guys know how to foil this is one minus 2 cosine 4X plus cosine squared 4X how many people got that perfect that's what goes here let's take a real close look at our integral and see what we got going on um if I asked you to could you do the integral of one yeah no substitution that's piece of cake could you do the integral of cosine 4X yeah yeah what's the integral of cosine now you'd have a 4X that would be over four you'd need a little substitution I typically do those in my head U sub on that would be over four does that make sense to you now can you do this one okay let's try this again integral of this could you do it yeah easy integral this could you do it without a substitution a little substitution but that's easy can you do the integral of cosine squared is that anywhere in your integration table no no we got to go over again so we start from the very beginning with our trig stuff what power do we have odd or even power with even Powers we use what Pythagorean identity or half angle which one do we know a half angle for cosine squared yeah that lets you break that down again there's lots of steps here lots to this be very careful write these all out I know some of you don't like to write steps and I'm going to say two stinking bad do it anyway you got to write this you got to be able to do this and make no mistakes on it so we have one we got minus cosine 4X we've got a plus let's look inside here remember that cosine squared x equals one half one minus cosine oh sorry plus cosine 2X hopefully we're familiar with that one let's let's apply it here I've got the one I've got the minus cosine 4X I've got the plus what am I going to write next one half got it keep on going come on tell me one plus cosine 2X without 8. okay now you all need to be careful on what the argument of that thing is going to become listen we're starting with cosine squared of 4 x if cosine squared x gives me cosine 2X cosine squared 4X gives me cosine remember are you taking an argument you multiply by two what is it perfect okay I'll tell you what we're almost done I want to do about a 10 second recap to make sure you guys are good with this stuff uh you okay with the Megan I'll do the fourth power yeah it's a special case and I get that you're not always going to have the same power there but when you do man try something like this it's kind of nice use identity are you gonna have to know your trig identities yeah otherwise this stuff doesn't make sense you gotta have that down people take calculus two to finally fail trigonometry people take calculus one to finally fail algebra all right but this is this is trig I mean my goodness we need to love tree then we go okay well that's cool you know what we can uh just take it all to the fourth power we recognize this one as either a reduction formula we could do which probably would be a little bit quicker than this to be honest with you sometimes you won't be able to do reduction form this eventually will or we go well I didn't remember that so let's practice doing the uh the whole even power thing the even power thing is make it a sine squared or make it a cosine squared somehow then use a half angle cool then distribute it look for any more squares you're basically reducing this with a half angle form it's like an inherent reduction formula the half angle formula always reduces your power you get what I'm saying so we look again okay well we got to do it one more time half angle again reduces the power one more time now we have everything to the first Power we get one no big deal cosine of the first we're going to distribute this in just a bit cosine of the first let's pretty this thing up a little bit what are you going to do here let's do it I think rule as nice as possible before we actually integrate so we got 164 we're going to have 1 minus 2 cosine 4X don't lose track of that 4X plus one half plus one half cosine eight X DX okay I'll tell you what whenever you distribute an algebra the one thing that you should probably do after you distribute is check for what that's right like terms do we have any like terms here yeah yeah I just combine those numbers this is one and one half that's three halves that'll make it integrate one less term here which is kind of nice combine your terms if you have any can I combine cosine of 4X and cosine of 8X no that's a No-No so 164th we've got three halves minus two cosine 4X plus one half cosine eight X DX my goodness that's a lot of junk I want to make sure that we're all all good to go do you guys feel okay on what we've done so far yes no okay can we integrate that little piece this one this one it's the same thing it's integrated when you do it just be real careful with your constants that's typical I mean almost nobody almost nobody gets the integral of cosine wrong y'all know it's sign it's the constants that are going to get you here so don't lose track of stuff we're going to have 164. we're gonna have a big old bracket here let's integrate piece by piece integral of three halves three times there three half sex very good minus here's how integrals work if you want to do these in your head this is the way I do it I do little substitutions in my head because I don't want to waste my time so watch watch here's what you do um the two's a constant correct yeah integral of cosine it's sine 4X what's the derivative of 4 x 4 over 4. that's how you do it so derivative of this thing that's a basic substitution when you just have constants it'll typically work with those arguments you should put it over whatever the derivative is if you notice this is coming from the d u over 4 that you would get does that make sense so that's where that's coming from let's try the next one uh what's the next thing I write one now we're able to do those without actually having to write a special substitution on that because that takes a little while right if you have a substitution here you have another substitution here that takes some time and that two and four cancel out well we're going to talk about that right now I want to make sure you guys are good on this so far short hands if you want okay so yeah a few things are going to happen of course we're always going to look at simplifying two with four done this becomes what 16. okay and then maybe we'll distribute maybe you won't it kind of depends I'm going to distribute because I want to so we'll have 3 over 128x oh goodness we'll have minus when we distribute notice this is a one-half yeah 1 over 128 sine of 4X plus wow what's this going to be so basically you have 64 times 2 times 8 or 64 times 16 I don't even know you tell me sounds good I'll take your word for it sine 8X and finally plus 3 plus c yeah wow wow that was a lot of work could you have left the 64. absolutely no problem you could just let the fact that you could factor out uh something else Steve question totally should have just factored out in one half yeah you could you could have factor to one half and made it 128 and then 3x should just send reduction formula that's right but you know what there's times when you can't right so I want to get you really practicing this uh the next one you see that we're going to be able to do that as well but I want you I want to see you practicing this this method does that make sense to you yeah so I get it the Russian formula works that's what I would do notice that this is gonna be a little more concise than the reduction formula production has Sines and cosines in it so this would be a little bit nicer in my opinion but you could do it with reduction form are you guys ready for the next one yeah okay the next one's gonna be some more question if we were to use the reduction formula is it possible that we could get something that looked different but was it equivalent to that that's actually just what I was saying is that when you do reduction formula it's going to have Sines and cosines the integral will be equivalent and you can get the same thing with identities but it's going to look different okay what happened you know it's the same thing you do the identities to prove it like all those times Intrigue where it said prove this identity that's how you do it that's why they had you do those so anyway uh let's continue let's do this one hey there's no DX there so I drew this one wrong you just say you can't do it oh yeah did you have any questions on that we're good yeah okay please watch the video that's right watch the video again okay there you go questions too bad just joking let's make this sign to the sixth DX now what would you use here I would use reaction formula what if you didn't know the reduction formula forgot it I just missed it or just want to practice like I do I'm doing it differently can we do it yeah of course we can because he can take a lot of work yeah yes it is so how would you do it if you didn't have the reduction formula yeah sine of x to the side you're close but no you see if I want if I have even let's look at the powers look at him do I have odd Powers no I actually have two even Powers if you don't think there's two of them I know cosine is zero so it doesn't really count but I have sine to the sixth power that's my only Power it happens to be even so what are you going to use ladies and gentlemen when you have all even powers where you use a Pythagorean theorem or half angle have angles so we need to make this a half angle half angle only works if you have sine squared or cosine squared so somehow I want you to rewrite that with a sine squared go for it and don't write out sine squared times sine squared times sine squared okay you could I guess but it's going to be the same but it just looks sloppy how am I going to write out the sine of the sixth Tower as sine squared to something did you all do that one yeah okay I wanna I want you to I want to see if you understand the reason why we just did that what are we trying to use when we have even powers even powers only work when you have signed to the third second or cosine of the second we're going to have any cosines so that's the only thing that's going to work for us is to do sine to the second and then do the third power are you guys with me yeah do the next step can you use the half angle formula here yeah really okay to it let's see if the old guy did you get one half times one minus cosine two x and then all of that raised to the third power with the brackets yeah cool next thing you're going to do let's do it what are you going to do yeah you are this is going to become how much 1 8 is going to be pulled in front of my integral you follow yeah now what do you have to do with this thing uh just keep it that way foil it I know what I forgot something sorry you're going to have to distribute that so if that was to the my gosh if that was to a higher power you'd have to distribute even more so yeah you have to if you're going to do this method without reduction formula you actually have to distribute that so on your own right now off to the side do one minus cosine 2X times 1 minus cosine two x times one minus cosine two you're gonna have to distribute that I'll let you do that for about 10 minutes I'm right on the board here you're welcome I think that's right you want to check there are easier ways to do binomial expansion that's one of them if you know the pipe the Pascal's triangle one three three one and you know alternating signs with a minus minus plus minus and you know Powers will use zero sorry Powers getting zero one two three you can distribute things like that very quickly very quickly so if I did like to the fourth power basically be the same it'd be one four two one four six four one and then you can if you know the the binomial expansion you do that so anyway this is what you should have if you distribute it and combine like terms did you all get that uh yeah are you just gonna trust me that that's right yeah probably okay that's right let's continue uh now now look after you distribute after you do that you really do got to look at your problem see what's going on here folks check this out can you do the integral of this yeah and you're going to leave that don't bother changing that that's a good thing remember that you could separate this into one two three four more integrals we're not going to we're doing our head but you could so leave this alone leave this alone can you take the integral of cosine yeah leave that alone can you take the integral of cosine squared no so this is something we got to work on okay let's see if you're paying attention with cosine squared would I use Pythagorean identity or half angles bam that's exactly right so this is going to be a half angle right here now stop we're gonna do one more thing cosine to the third can you do the integral of this right now no no change that one too so there's another one ugly little star isn't it another little star if you get cosine to the third okay let's think about are you going to do half angle are you going to do Pythagorean identity oh it's exactly right so let's go slow I want you to just do the half angle right now on this guy just do that let's see three but I know this thing is one half one minus sorry one plus cos oh let's see if you got it this is this part right here is one half 1 plus cosine 4X did you all get the 4X good minus let's talk about this one now if I've got a cosine cubed of 2x you're right that stands for that's an odd power so what do we do with odd powers pull one off very good we strip off a we get the stripper we get the uh that's right so we strip one off here cosine 2X DX oh look at that we're gonna do a couple things at once okay so don't get lost on me here I'm going to leave all this stuff the same I'm going to distribute this because that's nice we'll distribute that I'm also going to start changing this guy show pants feel okay with this so far all right so we've got a 180. we've got an integral we got a one we've got a minus three cosine two x we've got to check this out this is three halves times one is three halves three halves times cosine four x is three halves cosine 4X so this is exactly the same all I'm doing is Distributing three halves so we got those two pieces quick head knocker out with that one now when you're working with this stuff inside of an integral you can do two things you can do this if you want to you can go wait a minute this is too confusing for me right now I don't want to deal with this here you can write a DX and split off another integral does that make sense yeah do it off the side just don't forget some minus you're gonna have to have a bracket around all the junk that you do right now does that make sense to you yeah you can do it in here too and do the substitutions and or distribute and make things a little bit nice for you that's what I'm going to do so either you're doing this off to the side no problem or you're doing it the way I'm going to show you you guys okay with this so far okay so this is like its own little problem right here so I would choose to do I say okay well you know what uh this right here this is the same what are we doing here we're doing half angle or Pythagorean because connected with this you do half angles so again if you would have this integral so Pythagorean off to the side if you chose to do it this would become 1 minus sine squared you want Pythagorean not half angle that is cosine 2X that's supposed to be a little two there and then you do a simple substitution and you're practically done with that you guys okay with that one okay maybe I'll do both just so you see that it works out the same over here what would your you become sun it's usually inside of something sine of what do you those people who hate to do the chain rule don't mess this up uh what's this going to become or d u over two equals cosine two x DX exactly what we have here therefore we get I know I'm working quickly but we've already done something very similar to this this is one minus U squared this guy is d u over two or in other words this is one half integral of 1 minus U squared d u one half U minus U cubed over three one half U minus 1 6 U cubed and lastly we do our association back in we'd add one half what is it what was our substitution anyway minus 1 6 sine cubed 2X okay I know I did it fast but I want to know how many people could do this on their own at this point it should be pretty much all of you because we all know how to do a substitution this was the only big thing is seeing just really have you noticed how the the hard part about this is seeing what to do it's not actually doing it it's going oh wait I got a cosine cubed is that half angle or is that Pythagorean it's odd so it's Pythagorean this was even therefore it's half angle so well separate it so that you can do Pythagorean so Cube no no no strip it off we've got Pythagorean we've got that Pythagorean identity do a little substitution it works out pretty nicely show offense if you have the same thing on your own you're okay with that one cool now notice something this integral is done so this is what you plug back in right here when you're done with the integral does that make sense it'd be minus this whole mess of crap now if you do it the other way I don't even know if I want to show you the other way now that we've already done it stick take some time uh if you do it the other way basically just put a bracket here you do the same thing sine squared 2X cosine 2X DX you would distribute it so you would have a cosine 2X here I would probably combine that cosine 2X somewhere over here and then you'd have the same substitution that you just did does that make sense to you so you can do one another way I'm going to leave that so I like that way I like doing off to the side so this is going to be minus this stuff one half the sine 2X minus 1 6 sine cubed 2x I'll put the plus C on the very end of my problem I don't need to deal with it right now okay so I don't want to lose you I don't lose anybody so let's check it out you guys are okay down to here correct this one we said no no let's separate this let's call this one big big fat integral that's why we get the DX right here where there wasn't one before then we're subtracting the integral of this guy so we said now let's make this a DX and we'll have the integral of this so we move that over here we do it off the side it's a problem within a problem so 1 minus sine squared 2x times cosine 2X do the integral then put it right back we get the same thing so are we going to integrate this right now heck no are we going to integrate this stuff right now yes of course we are what might you want to do before you start integrating that final returns yeah that's right and in fact if you would have done this the other way that I was about to show you you would have some more like terms to combine so you have less terms it's whether you want to combine them now or combine them later we're going to do it at some point what like terms do we have how much is it so this is this is gonna be five knots okay so 1 8 integral five halves minus three cosine two x this is gone plus three halves cosine of 4X notice DX calculus ends here this we've already done distribute yeah you can do that you can distribute that right now if you wanted to you want to do that yeah so minus one half sine 2X good plus 1 6. sine to the third power 2x now let's work on the integral of this thing almost done you're forgetting the DX on both of those what DX the whole integral oh you split it that looks kind of the whole idea was right here we split this and we did this integral separately that way we use substitution and go yeah yeah you'll want this to do this separate we have it back in terms of X now we can put it back so this was all one integral this one I said no this is too hard for me to do right here or you can you actually could not a big deal or you could do it separately I just want to give you that option so either way and I don't care how pull that off make sure you get all the way done then you plug it back here but notice we've already done the integral there is no more integration to do here I really do need you to understand that show fans if you do you feel okay with with that question someone looked like they had one question you sure okay almost done so 1 8 times some nasty stuff let's do this integral can you do the integral of five halves yeah minus be very careful with these guys this would be minus three integral of cosine two x please sorry six times two no it's not times two divide by two when you do integrals you're dividing by this derivative because you're setting this equal to U derivative of this equals d u so DX would equal d u over the two plus three halves let's try this again okay derivative this one perfect again you're kind of doing those substitutions in your head minus all this stuff oh dear this is fun right foreign so working from the very bottom 5 16 x oh goodness minus 3 16. sine 2X I know right sine 4X minus one half sine 2X plus 1 6 sine Q 2x oh goodness all right well almost done what's in the last thing that we're going to do distribute the 1 8 on both sides we did we're going to be eight times it also goes here yeah you're right this is a big bracket yeah you're right yeah absolutely right so if I distribute that uh what What's this change too and 42. how much 40. it's only one more thing we got to do we're looking for any like terms so like terms do you see any uh the sign I don't know sine 2x yeah this one no this one yes these two are like terms and thankfully because Patrick got this redistributed that they have a common denominator right now which is really nice so we're going to have I was just testing you Patrick you passed 5 16. how much is that going to be 4 16. and lastly let's see okay I want to show you one more thing with this if you didn't if you didn't do the integral so I do want to prove to you that it does come out the same so imagine you didn't do this right here what you'd have is 1 minus sine squared 2x times cosine 2X all DX you'd have a minus cosine 2X Plus sine squared 2X cosine 2X DX we distribute this thing this is what you would combine with this thing it would be minus four so when you look at that minus this become minus 4 cosine 2X if you do the integral of the minus 4 cosine 2X you get minus 4 sine 2x over 2. correct that would be the two you get 2 sine 2x when you multiply by 1 8 you get the one-fourth right there does that make sense to you because it matches the coefficients are going to match this so this one would get combined with this one right now this one is what you do over here you do the um the use of on sign if you get du equals 2 cosine 2X DX you get a d u over two equals cosine 2X DX this is what would match right here give you a little substitution kind of off to the side and what you're now getting is you get U squared over two does that make sense you squared over two so when you do U squared over 2 check that out U squared over 2 would give you U to the third over three times two that's six six times the one the one-sixth 1 6 times 8 gives you the 148 here's the U to the third power of the sine 2x oh my gosh that's a lot of work either way it works I just want to prove to you that it actually does so it means if that one made sense okay that's hard right what would be way easier formula would be easier but if you don't have it or you can't use it then this is what you have to do so what you should know at this point odd Powers odd Powers you use which identity please everybody odd Powers Pythagorean even Powers very good all right let's do our last example about integrals that involve sine and cosine after this we'll talk about some other types of trigonometric functions like tangent and secant cotangent cosecant that stuff I'll give you the rules on that after this example but I want to go through one more maybe when you don't have some whole number exponents so you get a feel about what's going on now when we talked about our rules on integrating sine and cosine there's two cases there was the case where both powers are odd where both powers are even or where they have one even and one odd when you have a fraction like this look for the other one this would be pretty hard to deal with so we're going to look at sine to the third power what case does this fall under this fall under both even both odd or one of them being odd that falls under one being odd now when we have that one being odd do you remember what we do with that break one off that's right we strip one off remember we strip it off because what we're going to do is oh I hope you know we're either going to do half angle or we're going to do Pythagorean what's the case here it's gonna be Pythagorean because we're going to strip one off and change the sine squared as soon as you change the sine squared to a cosine the reason why we strip off that side is because now a substitution will actually work the derivative cosine is basically sine with a negative the derivative cosine gives you the sine it's gone it's out of there and we're able to integrate the rest of the simple substitution you ready so let's try it so the first thing to look at here we have odds or evens that counts as an odd so whenever you get an odd power of sine or an odd power of cosine strip off the one that gives you a square power so I don't want sine to the third I'm going to write this as sine squared x I like it and cosine one half I like to write that one first because I like to strip off this sign and put it here because it makes our substitution look a little bit better okay so again here's the idea we got an odd power strip one off and then what are we going to do with the sine squared what identity is that is that the half angle of the Pythagorean use that one so here we go okay cool I like this we're going to do 0 to pi over two we're not going to touch this till the very end the idea again behind doing this is manipulate your integral so that you can use a substitution the way that we do that is we try to make these be the same trigonometric function you have to make it the same and you have to make it the same so that when you take a derivative it's this guy over here so use Pythagorean to change sine into a cosine then we'll have all this all the same stuff um 1 minus cosine squared x times cosine to the one-half X sine X DX so when we have an odd strip went off now we use Pythagorean identity to make everything besides this guy into cosine once we do that do you see a substitution that's going to work don't worry about Distributing you don't have to do that don't worry about that your substitution what's your substitution going to be okay then d u equal oh derivative of cosine X what is it negative sign very good so we know that negative d u equals sine X DX that is what we have here so let's keep on going can you all tell me what's going to be in my integral once I make that substitution so we'll strip it off we've got all cosines we're making our substitution what's this change too one minus what perfect keep going what next oh good so we still have U cosines u u to the second U to the one half yeah don't forget that negative okay the sine DX is negative d u okay tell me what we do next okay I love that so moving up here negative integral of we'll move the negative out front we'll keep our bounds of integration tribute yeah let's distribute the U so right now once you make the substitution it's a lot easier to make the substitution than distribute rather than Distributing then make the substitution there's less to have to deal with so when we do our distribution we get U to the one half minus one minus oh when we multiply Powers together do we add them or multiply them you add them so you're the first let's get that one right two over let's see four over two plus one over two that's five over two you get bypass I hate to say but use a calculator if you have to I don't care get it right okay don't multiply those and give me one oh my goodness uh add the powders you get two plus a half well that's two and a half five halves that's what that is you guys okay with that one finish that off go ahead you would if you added one plus a half that'd be one and a half that's three halves but if you add two words did you get three halves over you did three halves over three halves did you get U to the seven halves over seven halves what am I missing do I need a plus c here oh no we're doing it no because it's definitely okay so good what am I missing front of the whole thing in front of the whole do you guys have brackets yeah yeah you should because we're subtracting this whole integral okay so we're subtracting this thing what's going to happen here let's simplify this what's it look like okay so negative negative one two three two thirds U to the three halves okay I like it then what plus two sevenths beautiful would you want to change your balance on this one right that's a good question that depends on you if you want to change the bounds then you should have already done it correct yeah if you don't let's let's say this some of you guys made a little bit of mistake on that last test let's say you don't change the balance do you plug in 0 and pi over 2 here no no what do you put here you'd have to put whatever you did it should have to be code yeah whatever you is you have to put cosine back in there then you plug in 0 to pi over two does that make sense to you if you go ahead I like changing bounds personally I think it's easier or at least it's more straightforward so right here what I would do is it's okay we start with x's so if x equals pi over 2 then U equals and if x equals zero then U equals and we just plug that into cosine so if we do that we've got cosine of what's cosine of pi over two zero and if we plug in pi let's see uh if we plug in zero what's cosine of zero one okay the only problem that we're going to have here if you do it this way please be careful this is probably good for you guys to see this is why we're doing it right now where's the zero go the bottom top the top goes on the top because pi over 2 got mapped to zero does that make sense to you okay where's the one go on the bottom left zero got mapped to one show fans should feel okay with that one okay if you do it this way it's hard if you do it this way what happens right here so other way if you had this you'd have to do cosine for your U and then plug in the actual values and it's fine you go ahead and if you someone who had that on the paper go ahead and do it make sure you do the same thing we did if you do it this way if you actually change bounds you should have remembered from your Calculus one that when you flip your integral become your sign changes do you remember that property yeah if you flip this you go okay let's make it from zero to one let's put it in the correct order negative becomes positive it changes that to eventually okay we're done okay and now it becomes well something we can do we don't have to re-substitute that cosine so this would be two-thirds U to the three halves minus two sevenths U to the seven halves we already did that now let's practice this one if I do this if I change my balance do I have to put cosine X back in for my U no I'm going to change them these are now in terms of U from zero to one those are used now they were X's now we've they're used we just showed that we get to plug in one we get to plug in zero we don't have to plug them into cosine because you already did over here that's how we changed it so we'd get two-thirds one to the three halves minus two sevenths one to the seven halves I always plug these in in case something funky happens okay we get zero okay that's going to be zero minus zero so nothing's going to happen over here one to any power is still one so two thirds minus two sevenths how much is two-thirds minus two-sevenths eight twenty four okay eight over twenty one Michael got to be negative zero to one we had one to zero with a negative so when we flip that the negative changes to a positive so we are changing you caught it we are changing signs though this would be negative correct but if I have a negative out front already it becomes positive that's the idea I want to show fans if that made sense to you if you did the other way you wouldn't do this you still have zero to pi over two but you'd be plugging it into cosine here and cosine there I want to make very very sure that you're clear on that one are you guys clear on that yeah yeah you would um if you didn't if you left it with the original balance you would get negative 0 to pi over two like that with all this junk and then you have negative two-thirds U but the U would become cosine X 3 has plus because you would distribute the negative seven halves cosine X to the and then you plug in not zero to one you plug in 0 to pi over two well if you look at that it's going to be the same cosine of pi over two this is zero zero cosine of pi over two this is zero but the zeros aren't zero cosine of zero is one cosine of zero is so this is negative two-thirds plus two-sevenths oh look at that it's exactly the same time is something huh because we should also subtract them so it'd be uh shoot negative zero plus zero minus negative two-thirds plus two-sevenths is how it would really look that's the same thing this if you distributed zero two-thirds minus two-sevenths that's the same same idea exactly makes sense now even more sense I know it made sense before but now it's like extra sense anyway we're going to talk about one more thing in this section how to do some integrals that maybe don't have just Sines and cosine but I'm practicing with Sines and cosines all day long all less involved let's look at something else oh by the way did you have any questions okay all right good too late so what happens if you have something like a sine cosine but tangent and probably something else like secant we we grouped sine and cosine together because the derivative of one gives you the other one does that make sense yeah and we group tangent and secant together because derivative one kind of gives you the other one especially with like secant squared and things and cotangent cosecant so those ones are grouped together because we it's possible to do if you don't have that scenario you have to change things to sine and cosine and work in a different way I'll give you that example much later so like uh at least another day so what about integrals that involve tangent and secant or cotangent I'm hoping that you understand the the ideas that we use here we're going to use the same exact ideas here because the code the tangent and the cotangent secant and the cosecant they work really similar and the identities are very similar so let's take a look at what we do with this one and we'll use a corollary for this one so here's new just like we have rules for sine cosine we're going to have rules for tangent and secant if tangent is odd so if tangent's an odd power if the power of tangent is odd what we're going to do is something kind of fancy we're going to strip off not only a tangent but we're also going to strip off a secant so we're going to keep a tangent and a secant I'll explain one second so the power of tangent is odd foreign keep one factor of tangent X secant x in fact I'm going to write it backwards so we're going to keep a tangent we're going to strip up for the tangent as secant keep one factor of secant x tangent X I'm going to write this way because we're going to make a little bit more sense right now remember the idea behind this I hope that you do the idea is that we're going to break up a little piece right yeah we're going to break up a little piece so that we can take the derivative so if tangent is odd if tangent's odd like three we're going to strip off a tangent that's going to make the remaining power a 2 tangent to the second power it has an identity it's going to change into secant hey what's the derivative of secant secant that's why we're keeping that because this will allow us to do a substitution so if the power of tangent is odd keep one factor of secant tangent and use tangent squared x equals secant squared x minus 1. what that does that changes all but this tangent and to secant then you can use the substitution the derivative of secant is secant tangent and you're pretty much good to go now let's say that the power of so we check first the power of tangent okay if it's odd you're good then if it's not well we're going to check secant let's say the power of secant is even if I receive it is even what we're going to do is what we're going to do keep or strip off a sequence squared and use secant squared x equals let's see what would that be tangent squared x plus one if this is the case if we have an even power of secant well then what we can do is strip off the even the rest of the even and keep a secant squared if we keep a secant squared and use this to translate everything else to the tangent hey tell me something what's the derivative of tangent secant squared that's going to allow us to make a substitution tell you what I want to try at least one of these right now just to get it in our heads real quick so the stuff makes sense Okay so oh you'll find a lot of similar patterns to cyanosa same idea it should be a little bit easier now that we have this stuff down question oh so those those two definitions those work for either both of those cotangent or second two like this right so if and I'm kind of putting this up here for this one too um you could read this way if the power of cotangent is odd keep one factor of cosecant cotangent and use cotangent squared equals cosecant minus one I'll use that one does that make sense to you if the power of cosecant is even strip off cosecant squared and use the appropriate identity you guys with me yeah okay so make sure your identities are are good to go and will be absolutely fine very similar ideas here which is why I'm not going over this one completely separate they're the same it's the same thing let's try one I'll give you one in a while too but let's try one with Tangent and CQ first thank you all right let's go through the whole process here we're going to look at the power of tangent first and if it fits our Paradigm we're good to go if it doesn't well then we got to look at the power of secant so let's look at the power tangent what do you notice about it sometimes and that's great that fits right here so what we do with the odd power of tangent we're going to make it even we're going to strip off a power of tangent but we're also going to strip off a power of secant so what am I going to write as far as the tangent goes tangent to a forum yep tangent to the fourth and I'm also going to do secant to the second now here's what we're going to do with that we just stripped off the tangent we just stripped off the secant I'm going to have secant x tangent X DX I want to make sure that you guys are out with the so far original okay with that it's just some algebra it's just understanding that this is the same thing I'm just doing this on purpose because right now I'm working to make a substitution that's the idea now do I want to change secant squared no no something I'm thinking wait a minute that that's even should I do that no the idea here is you're trying to make this so your substitution has this derivative this is the derivative I want if I change this to a tangent I'd have a substitution of a tangent what's the derivative of tangent squares is it this no if I change this to a secant what's the derivative of a secant this that's why we're going to go about it this way okay this not so much maybe we manipulate the tangent a little bit I don't want tan to the fourth I want to make it look like my identity here so I'm going to do a tan squared x squared so the whole idea is identify what identity you're going to be using then use it if we have odd power of tangent strip off a tangent and a secant then make your tangent into a secant in order to do that we've got to have our identity just right so setting to the fourth now I'm going to do tan squared squared how much is Tan squared x what is that so we're going to have secant squared x minus 1 all squared then a secant squared x times a secant X tangent X DX so 10 second recap before we actually go I'll go on and do this thing are you guys okay the idea that if tangent is odd we're stripping off at that secant tangent yeah this is because we're going to work towards making the substitution we do that by making it look like it fits the identity doing the identity and then doing a very simple substitution if we do our simple substitution U equals what don't distribute now please don't do that you'll have to do it later you go secant x d u equal what's what's uh do you equal this ladies and gentlemen is why this works when you strip off the secant x tangent X you're actually stripping It Off so that you can do this so that secant x tangent X DX will equal your du you'll change this to a u squared minus 1 squared you'll change this to a u but this whole piece why we stripped it off was to allow this to happen the derivative of secant x is this therefore d u equals secant x tangents DX I want to show Advantage if that makes sense so you can follow that uh what I want it's very simple after this what would you do you spoil and then distribute everything that's right so distribute all this crap distribute everything and then do your your integral go ahead and do that at home it'll take us maybe a minute next time we come in uh just make sure at the very end you're gonna have to substitute back in for whatever you whatever you use so more torture I mean fun uh more calculus so what we've learned is that if we have some integrals which follow some trigonometric patterns such as we have Sines and cosines where sine is odd or cosine is odd or they're both odd and they're both even we can do stuff with that we can work with them with following the patterns that I've given you earlier in this section we're also moving on to what happens we have tangents and secants of course we use sine and cosine together because the derivative of one gives you the other that's why they work we use tangent and secant together because the derivative of one kind of gives you the other they're related as far as their derivatives so what that means is we can use substitutions a lot of the time if we do a little manipulation same thing worked with cotangent and cosecant so here's the ideas if you have a power of tangent or cotangent that is odd we're going to keep a secant tangent or likewise a cosecant cotangent and use tangent squared equals secant squared minus 1 or cotangent squared equals cosecant squared minus one do you see the similarity between the COS and the the other functions it's very similar again the idea is is this you're stripping off a piece because you're going to be doing a substitution so basically we're stripping off the derivative of what we're trying to get in our integral so if we strip off look at this if we strip off a secant tangent we better be substituting to making everything else in the integral a secant does that make sense I'll say that again because some of you guys weren't really listening here if we're stripping off a secant tangent everything else in our integral better change to a secant because the derivative of secant is secant tangent this is what we're doing we're stripping off the derivative if we strip off a secant squared if we keep that one then everything else in our integral better be changing to what tangent because the derivative of tangent is secant squared so we're kind of taking a piece of this a wave of our integral just stripping off a little bit that way we can make a substitution that allows us to do our integral just a basic U substitution I'm going to try one more with uh tangent secant to show you what that idea is like I think last time we did one where tangent was odd I haven't shown you one where secant is even so we're going to do that so here we go let's suppose we have an integral from zero to pi over four of the square root of 10x times C great to the sixth power of x to the sixth power let's see if this follows our one of our patterns see if it fits so what would this fit as would this fit as sine and cosine of course I don't even have those as far as tangent and sigan goes what does this fit is tangent odd or secant even which one oh it does I don't care if they're both even we have one or the other either tangent's got to be odd for us or secant's got to be even which one do we have okay so we don't fall up here we're falling right here the power of secant is even no problem well then we can do this what are we supposed to do if the power of secant is even we're going to strip off a secant squared so basically here's what has to happen with this if you're gonna if you fit this pattern we're gonna strip off a secant squared so this is going to become secant to the fourth power you with me the reason why this works so nice is because if this is even I can strip off that secant squared and I have something that can be raised such that I have sink into the second power in there then I can use my identity change everything into tangents and make my substitution so so check this out we're going to follow my my ideas here we'll have zero to pi over four not really any calculus happening we're just we're fitting this to our pattern we're making this stuff work we'll have tangent to that one half power we'll have secant tell me what I should have secant two there fourth very good times what squared this is what I mean by stripping off a secant squared we're taking this at well is that still secant to the sixth of course it is I'm stripping this off here's why if you didn't catch me the first time I know it's kind of out there when I'm just talking it through but in the example hopefully you see this because this was even when I strip off that secant squared it's still even does that make sense that means I can write this as secant squared to some power if I can write this as secant squared to some power I can use this identity to change everything into tangents this is the idea if I have this tangent I'm good if I can change this to tangent I'm good because then the substitution hey what's the derivative of tangent that's why we strip that off so the idea is strip up a piece use the rest of it to change it into whatever this one is the other trigonometric function so if I have tangent and secant and I have a secant squared I need to change this into tangent so that my substitution will actually work does that make sense you would strip this off and then do something with Tangent that'd be silly because the derivative of secant is not the secant squared the derivative of tangent is secant squared so even good deal we'll strip off a secant squared it's still even that's the point here if it's still even then what we can do is keep the tangent to the one-half power that's great seeking to the fourth no no how are we going to write secant to the fourth power what do you think yeah perfect and we'll keep this exactly the way it is I want to know how many people understand why I'm writing this as secant Square to the second how many people understand that why what's that going to allow me to do yeah that's right we're going to use our identity so this one do I want to change that this one do I want to change that yeah this one no this is going to allow me to make that substitution do I want to change this one oh yeah 0 to pi over 4 we're going to have Tan x to the one half in here well this is why we did this because this was still even I can write this as secant squared squared it's it's our our secant squared which we want because we're going to do that that change here we're going to use this identity and it still raises some power that's fine we can deal with that but I had to have a secant squared to be able to do this to change everything into tangent so this becomes 1 plus tangent X squared let's see if that makes sense to you even cool strip one off it's still even that's why we did this if it's still even I can write this as secant squared to some power if that was to the sixth power that wouldn't be a problem that would just be a three that was an eighth power no problem that would be just a four the idea is if this is even I can strip it off I still have something that's even which allows me to do this idea once I have this the secant squared now I get to use my identity to change everything into tangent except for this guide because I want to use this thing for my substitution you have to have the derivative of whatever you're going to substitute that's the basics of U substitution is the derivative has to be there so if we're trying to change everything into tangent we've got to have a secant squared otherwise our substitution will not work so it has to be okay with what we've done so far cool now should I distribute right now or should I do our substitution right now I would you can distribute but there's lots of other stuff right so if you change everything to you now be a little bit easier so right now we're going to do our substitution can you please tell me and I'll make a little area here for this what is our substitution going to be then d u equals secant squared x DX this is why we had to have this that's why this whole thing works so now this part is equal to U that's brilliant one more thing I'd recommend to you right now is if you have definite integrals which we do here sometimes it's really nice to change your balance so I would do that if this was my problem I'd be changing bounds right now which I'm going to do so changing bounds and look over here when X in the cell I showed I said when X is pi over 4 then U is whatever and when X is zero then you is whatever we get so when we do it next to our substitution makes a lot of sense to do that if x is pi over four can you tell me what's tangent of pi over four one then U is one right there that's already a little bit nicer when X is zero how much is tangent of zero zero now what happens is to calculus One Step but I'll make sure you get it now that we've changed bounds when we get down to the integral will we be resubstituting Tan x for U no no you change bounds you don't need to do that that's why I do it now so let's see what happens uh why don't you tell me bless you don't tell me bless you I said bless you but you tell me I I love to be blessed bless you too um you tell me what I'm going to write what's my integral now one zero and one which one one to zero zero to one one so zero to one zero okay what's inside U to the what yeah perfect yeah U to the one half and then what one is we really want to make sure we don't miss any of those Powers because that would really blow this thing out of water so we have U to the one half we got it tangent is U so U the one half we've got a one we got a one we've got a plus we got A plus we get a tangent squared so U squared whole thing being squared what happens to my secant squared x DX what does that mean this has to match your variable it does we're good to go can you guys do the rest of it from here yeah go for it do your simple integral you're gonna be Distributing a lot of course and then evaluated let's see if you guys get to work on it on your own let's see if you guys get the same thing I get two okay let me get a work check from you did you distribute the same that I did the same thing I did yes good fractions use your calculator if you have to I hope that you don't but make sure your fractions are correct foreign keep on working are your fractions right ah the one half five halves nine halves you know what the hard part the hard part is really getting down to here right is using this stuff this I'm hoping you guys are okay with this is this is old stuff this is just distribution making sure you know how to do fractions adding fractions and then doing some basic basic integrals U to the one half integral U to the one half is U to the three halves over three halves two U to the seven halves over seven halves either the 11 halves over 11 halves and then all of these bottom denominators just move up to the numerator so in our case we would have two-thirds U to the three halves plus four sevenths U to the seven halves plus two elevenths U to the eleven halves two moves up two moves up is multiplied by two two moves up no big deal quick hitting that if you're okay with with that one so far okay now this is what I was talking about earlier I want to make sure that you get it do you at this point have to substitute back in your tangent here if you didn't change bounds would you have to plug in your tangent here okay so what are our bounds of integration where are we going to again so we're going to plug in one we're going to plug in zero we're going to do it right to our U's that makes it kind of nice so I'm going to do this in my head because we have some fairly simple numbers this was nice for us if I plug in one one too many power gives you one so basically we get two-thirds two-thirds plus four sevenths plus two elevenths if I plug in 0 I'm going to get 0 and 0 and 0 but I really want to check that and make sure you're checking that so this would be minus zero so essentially we just got to add up all of these fractions which of course you can do in your head and get 328 over 231 I mean easy yeah obviously I did that in my head I didn't do that in my head I'm not going to waste some time adding fractions right now I had that done beforehand so uh 328 over 231 is what you would get if you added up all those things together should pinch feel okay with this one I'm not going to waste time doing the the basic stuff this is really it right this is the stuff you guys should know already so hopefully we're okay are we okay you want to do one more I don't think we've done a cotangent coaster here let's let's take out one of those see how that works uh are there any questions on this before I erase it this time at all so you're saying that if an original problem I've been like seeking to the tent then we would just if this is the same thing with cp2 the 10th you can go about it exactly the same way the only difference is this would be a secant squared correct and you have seeking to the eight this would be secant squared and this would be a four the only problem with doing this is that when you get down to your U sub you have to distribute a lot because you'd have one plus u squared to the fourth so you'd have to distribute all of that stuff you have lots of terms there yeah right so that's it can be done but it's a lot of distribution more no you sure it's doable it's just no just wondering if you could just split it twice and then do two at some point you will be Distributing somewhere with this stuff because if you do split it right you'll have multiplication and when you do um when you do your your identity right here from here to here you get a plus after you get a plus you will have to distribute all that stuff that's the idea okay so last one before I give you some other ideas on this section let's work over here this time okay so let's check our pattern out we know one for sine and cosine we know one now for secant and tangent secant and tangent or actually I should say this way cotangent and cosecant were really really similar to uh tangent and secant so the same thing that we just did with Tangent and secant we can basically do it with cotangent and cosecant of course the derivative is a little bit different so it's got a negative in there we'll have to be aware of as long as we don't mess up that negative it works really similar even the identity is pretty much the exact same thing so it's really close so let's check it out which one does this fit which pattern is this the one where power of tangent or cotangent is odd or the one where our secant or cosecant is even which one yeah the cotangent is odd so that fits this top one says the power of tangent or cotangent is odd we're going to do something what are we going to do with this okay so that's what I want you to do right now do the first step don't just wait for me do it yourself you know what I'm gonna need more room than that so if cotangent is odd you strip off cosecant cotangent that's the idea you strip one of those things off let me see if you wrote what'd you write what'd you write down what does cotangent become foreign X cotangent X DX I would do that because that's the way it looks in my head when I do derivatives that's the way it looks show fans be okay with that one so far now I need you to understand that don't just follow this as a pattern okay really understand the process the idea here is that this is supposed to be a derivative of something is this the derivative think about this when you're doing these problems is this the derivative of cotangent or a system derivative of cosecant so the derivative of cosecant you with me yeah what that means is that I want all of this to become cosecant am I going to change the cosecant no am I going to change the cotangent that's why it's written this way for you use cotangent with this identity right here make it into cosecant you're stripping off these pieces as a derivative so you want to change everything else so that when you do the substitution the derivative exists if you do this and you start using an identity on cosecant and change it to cotangent the derivative of cotangent is not cosecant cotangent it would be using this one it would be uh the derivative of cotangent would be the cosecant squared so we could do it that way so again the idea is when we're stripping it off we're actually getting a derivative here change everything else to the trigonometric function that gives you this derivative is that clear so change cosecant no the derivative of cosecant is this change cotangent yeah yes so in our case well probably got to write a little bit different cotangent to the fourth is not our identity how can I rewrite that please I don't want to change cosecants those are those are great for us that's going to give us our substitution in just a second so we write it so that we can actually use our identity our identity says if we have cotangent squared that's the same thing as cosecant squared minus 1 squared believe it or not we're almost done almost people ladies and gentlemen you're okay so far I want to know if you understand the reasoning behind what we're doing here we're trying to get all of these to be cosecanted because this is the derivative cosecant so once you have that as soon as you have this downward they're all the same and the derivative is right here what should you be doing that's like the definition of when to use a u substitution is when you have one piece that if you take it out with a substitution and you take a derivative and the derivative is in there it'll work now here's the deal you do got to be pretty good about your your derivatives of these things we don't use the derivative of cosecant a lot but you still get to know it derivative of secant is positive secant tangent derivative of cosecant is negative cosecant cotangent let's switch over that negative so negative d u equals cosecant X cotangent X DX and now this is exactly what we have here and that will let us do our substitution I'm going to move up here can you please tell me what this integral will look like after I do all of this substitution integral of what's this become e squared yeah good so this is u u squared minus 1 squared cool we got U to the fourth power no problem this whole thing is equal to negative to you do not forget about that negative that's a big deal so negative d u and now let's do a simple integral tell me some things that you're going to do here negative up front very good what else you're going to have to distribute so negative comes out front if we distribute We Got U to the fourth minus two U squared plus one if I did it right did I do it right yeah yeah okay times U to the fourth now what do it again negative integral U to the eighth minus two U to the sixth plus u to the fourth b u can you guys do that integral yeah a piece of cake negative just don't forget that you either need to distribute the negative now while you're doing this or have it up front because it will distribute eventually one ninth U to the ninth minus and lastly tell me some other things that we're going to do here please substitute we do have to substitute so we don't want E I don't want U's we start with x's you better do it give me some x's and then distribute negative so I'm gonna do all those steps at once we're gonna have negative 1 9 U becomes cosecant to the ninth power of x we're going to have plus because we're Distributing the negative 2 7 cosecant to the seventh power of x plus one-fifth cosecant to the fifth power of X and finally let's see yeah that means we're done five bags and one over nine oh yeah you're right I'm sorry I missed that okay good catch that should be negative you can Factor this if you want to that's fine um I will try to get you to get the basic idea of getting down to here once you get to here really we should know how to do that right so if you get down here and go I want to factor that go ahead and Factor it you want factors for fractions factors for fractions I only care I want to make sure that you are good this is really what I'm trying to teach you is up to this point this stuff should be your Basics after that nice show if Angie feel okay with this one with this one okay those are the basic three if we have Sines and cosines you have patterns for that if you have secants and tangents now you have patterns for that if you have cozy and cotangent is no problem what happens if they don't fit those patterns so we're going to talk about that right now like if the cotangent you had one more cotangent in that last problem some of them can't I mean some of them there's no real good way to do them um someone like I'm going to show you the only way around doing this like like this one like if I were to give you that if it fits the patterns I've given you for goodness sakes you use the patterns and do them okay if it doesn't fit exactly sometimes we can change it so that it fits a good pattern sometimes we can't if it doesn't fit any pattern at all like this one does that fit anything it's got secants and tangents right but they're not multiplied together one's being divided this doesn't count this is more like tangent and that's actually a cosine if you move that to the numerator this tangent cosine so that's really not exactly what we're looking at so one big hint for you is if it doesn't fit the pattern if it doesn't fit so first thing you don't don't do this first okay check the pattern first if there's Sines and cosines you can you can work with it you can work with any sine and cosines I've given you if they're secant and tangent and it fits so that either tangent's odd or secants even you can do it if you have cotangent cosecant where cotangent is odd or cosecant is even you can do it if it doesn't fit those things then one thing you try after you check that see if you can change it to Sines and cosines using those identities and then see if it'll work so let's try to change it so the note that I that you probably should write down is if it doesn't fit the pattern try changing it to Sines and cosines but that's it though so for others not covered try converting the sine cosine foreign so let me show you some ideas that I would do if I was looking at this problem the first thing I would do is I look for an identity so that I could just nail right off the bat now this one's pretty close but that's not an identity one plus tangent squared would be an identity that'll be easy because 1 plus tangent squared would be secant squared secant squared of six squared to give you one you'd get X hey that'd be nice and we're not that lucky okay so that's not an identity don't create your own identity some of us like to do that in this class uh instead work with this thing with identities that you do know or if you don't try to break pieces apart so first thing I'd say is I don't really like secant squared secant squared isn't cosine so one thing I would try is I would split up this integral into one over secant squared x minus tangent squared X over secant squared x the reason I'm doing that is because I know that 1 over secant squared x is actually the same as what cosine squared remember convert to Sines and cosines that's one way we can do it so then this becomes cosine squared x and let's take a look at what this actually is what this actually would become this is tan squared x times cosine squared x believe it yeah this is cosine squared x that's what that is so 1 over secant squared is cosine squared understood yeah well what's tangible so tangent squared is sine squared x over cosine squared x times cosine squared x what happens here that's fantastic so just by doing a little bit of a change here by pulling those those fractions apart and by using our identities for what's 1 over secant squared is for what tangent is what secant one over secant squared we can simplify this fraction from this nasty thing to this really nice thing I guess your opinion should be okay with that one hey have we done any calculus right now yeah this is trigonometry purely tricky exactly the next step is going to be trigonometry because this is kind of awesome uh there's an identity and if you want to look it up right now you can look it up this is actually an identity do you know what it is negative one please don't say one or negative points this would be one that would be one this is this is not a negative one it's not the way that this stuff works uh no no it's not it's not but if you want to look it up go ahead and look it up I think I know what it is okay Rob no sine 2x perfect does it help to know your trig identities because if you didn't if you didn't know that look at this you know how to do cosine to evens don't you you know how to do that uh you'd have to so you'd have to do this one you have to do this one independently and then you have to put them all together but if we know our identity we go hey look at it let's go Santa can you do this that's now A Piece of Cake this would be like I would do the use up in my head I don't write out the U Subs for a lot of these trig ones because they're so easy and they take and we do so many of them especially like you found out into your last homework or this homework I'm sorry when you have like uh cosine squared of 2x well then you're going to be getting one half one plus cosine of 4X that angle is going to change if you do a use of right off the bat with that you're gonna have like eight U Subs per problem that is going to suck so I just do in my head I go okay what's the integral of cosine it's positive sign whatever's in my hand I don't change but I divide by the derivative of what is on my hand it's like the opposite of a chain of a uh a chain rule instead of multiplying by derivative you just divide by derivative and that works does that make sense to you it says doing U substitution that's a u Sub in my head I'm doing U equals 2x right derivative is two DX equals du over two that's where the over two is coming from plus C lets me know I'm done so if I should be okay with that one identities are nice try to use those and the ones that and the ones that don't follow the pattern you are going to have to change the sine and cosine for most of them once you do that please for your sake look at some identities and try to work with them just follow the correct identities down and you'll be good to go okay we move on now there's one more thing that we haven't talked about that will give you one example with and we'll call it a day um noticed in all of these patterns that these angles are always the same have you ever wondered what if the angles aren't the same have you wondered that we're going to we're going to do one like that right now so what happens where the angles of sine and cosine are different so for integrals where sine and cosine have different angles try using these three identities again of course with trigonometry we almost always go back to identities at some point even with our basic patterns we're using identities same thing happens here if you've got integrals with two different angles this is what you do there's three cases there's where you have two sides where you have a sine the cosine or where they have two cosines there's only three three cases so let's suppose I have sine times sine where the angles are to put different where the angles are different you know what I'm going to do more room let's write that down but I'm going to write it just over here a little bit or where we have a sign and the cosine before the angles are different or lastly where we have a cosine and a cosine where the angles are different here's some identities I don't know if you've ever seen these before you may have I'm not sure you probably have somewhere but we don't use them very often here they are so if you have sine of something times sine of something where your some things are different here should I do it probably something you're going to want to write down and have next to you when you're doing your homework at least I would probably just it's nice here thank you you're not allergic to this are you foreign yes minus cosine okay good and that one cool and okay write them all down look up here when you're done that way I can talk at you we're gonna do an example here so continue writing then let me know that you're done and then we'll do an example foreign you get this perfectly right sign for sign pluses and minuses sign for sign as well okay good so let's check this out if I hadn't told you this could you do that problem well you might you might be able to change this cosine 2X I'm sure there's an uh identity for that one but we have an idea that will change this correctly and make it a lot quicker a lot easier for us so let's check it out we got sine x cosine two x do you see what I mean about different angles yeah so what this says is this would be our MX it's just 1X this should blow away which one are we at top one middle one bottom one so this is our 1X this is our 2x it just basically says look at this angle look at this angle do this with it so when we're doing this with it we still maintain our balance of integration we're just going to do our identity here so what's the first thing that we're going to do one half's got to be there what are we ultimately going to do with that one half no we're not contributed you can pull it out okay so I'm going to draw my bracket here let's change this using this identity tell me the first thing I'm going to write inside of my bracket sorry sine of what uh good it just says this angle minus this angle x minus not 2x minus that x minus 2x okay what else there's a plus holding those together and then another sign and then what that's right perfect don't forget your DX's let's simplify a little bit what do we got here okay what now nature wait son of what sign of combine them Negative X you okay with sine of negative X are you first I want to make I get some weird looks on that one like what are you doing uh what's x minus 2x okay you can do this stuff okay they're inside the parentheses x plus two x's some of you are thinking so hard you're missing the easy stuff that happens a lot in this class quick hitting if you're okay with that so far the reason why this exists is so that you can actually combine the angles all right you want this this one's crazy I mean this yeah I can buy this negative X and now this is an issue because we don't want to end with a sine of a Negative X we want sine of x we want to end with that or whatever we have here so I mean cosine actually you didn't identity for this one sine of negative x equals good it's an odd function so when we do this we go right this is not a problem this will be one half zero to pi over two a negative sine of x plus sine of 3x DX let me ask you something can you do this integral that's a piece of cake it's not cosine what's uh what's the can you do this integral yeah it's a piece of cake too no problems let's do the integrals what's the integral of let me write this up here okay one half's going to be there no problem let's create a bracket here so I don't lose anything what's the integral of sine X can you tell me negative cosine negative cosine so much carefully I'm gonna do some math in my head here if this gives you negative cosine this gives you cosine X this would give you integral of sine X is negative cosine so negative integral of negative sine X is positive cosine X sure if needs to be okay with that one double negative okay double negative two wrongs make a right here so let's do the next one what's this going to give you we'll give it negative cosine 3x okay so instead of plus I'm going to have minus cosine 3x over three that's a little U Sub in your head for the over three so when you're doing these do the integral no problem negative so minus cosine negative so minus cosine three x over the derivative of three so it needs to be okay with that one good deal now what is definitely should to distribute that if you really wanted to you don't have to you can wait to the very end I would I like dealing with this and that separately you could actually divide out a one-third if you want you could do divide out of one third this would become three cosine x minus cosine 3x not a problem that would get rid of that fraction in there so whatever you want to do as far as this goes this is just algebra just plug in some numbers and doing it so I'll leave the one half just remember that in here when you when you evaluate you'll have cosine of pi over two minus cosine of 3 times pi over 2 that'll give you nice numbers should I bother to plug in zero yes for heaven's sakes yes minus cosine of zero minus cosine of three times zero one half over three uh I don't think I forgot anything but you let me know if I have have I forgot anything no okay so we got the one half we got the one half Bagel bracket you need the big old bracket this is the evaluation of pi over two so cosine pi over two cosine three times pi over two over three are you guys okay with that one minus no problem because you subtracted this cosine of zero minus cosine of three times zero over three let's work it out so this will give us one half big bracket don't lose things here what's cosine of pi over two how about cosine of negative I'm sorry cosine of a three pi over two zero zero zero over three big difference cosine of one cosine plus zero is one how about this this sine of zero will be one minus one third don't forget about three this is one third you okay if you're about three this whole thing goes to zero you go wait a minute it doesn't make sense you have an area that's zero you can complex analysis uh what happens here so this is zero one half zero minus two I did right I get Negative one-third did you get Negative one-third yes that make sense to you guys yeah