[Applause] okay so time for new chapter time for new section section 7.1 um so far what you know about integrals is that you can basically do two things if it fits your integration table you are good to go if it doesn't fit your integration table the only thing that you can do with it is what sub to make it fit that's right substitution to make it fit the table right now in chapter 7 we're going to learn a whole lot of other integration techniques that you haven't learned before unless you've had this class before that you haven't learned before so have you ever wondered um there's a substitution that's kind of like the chain rule right yeah substitution is like the opposite or inverse of chain rule have you ever wondered why there's not a product rule there's a product rule for Drew is in there yeah is there a product rule for integrals not that I know not that you I want to show you something uh do you know what that is a product rule from product rule I'm going to prove something for you and we're just going to start with what the product rule is and we're going to try to undo it so notice that we've got a derivative and the derivative equals derivative of the first * the second plus the first time der to Second that's your your basic old product rule now what's the inverse of a derivative let's integrate both sides we can actually do this you can integrate both sides of an equation so if we integrate this DX and integrate this DX then what we end up getting is the integral of the derivative of f ofx * G ofx DX and over here we got the integral of fime of x g of X Plus FX * G Prime of X you guys okay with that so far yeah just take an integral on both sides now let's see what happens here what is going to happen here if I take the integral of a derivative going to do it so yeah calculus tells us that they are inverses so if I take the integral of a derivative DX DX that's gone all I'm going to have is f ofx * G ofx does does that make sense to you are you sure now the next thing I'm going to do I'm just going to separate this notice that with integrals you can separate by addition or subtraction so I'm going to do that I'm going to have the integral of fime of x g of X DX plus the integral of f ofx time G Prime of X DX so we had product rule I take an integral both sides no big deal derivative is gone integral and derivative they basically do each other their inverses I have just the functions themselves f * G here we know this was product rule if we break up our integral after doing integral on both sides we have one integral we got two integrals you guys with me I'm going to solve for this one I'm going to put this on one side basically I'm just going to subtract that guy does that make sense yeah okay so if I do then we've got integral of FX * G Prime of X DX equals I'm going to keep this and if I have subtracted that I'll have frime of X time G ofx are you guys okay with the algebra on that one I want I'm going slow that I want to make sure you really do follow do you follow it there's only one more thing that I'm going to do what I'm going to do right now I'm just going to switch these two guys around so by the by the commutativity of multiplication this is not a problem I'm going to put G of X before frime of X you okay with that one so integral of f ofx time G Prime of X DX equals let's see F ofx got that time G of x no problem minus integral of G ofx * fime of X DX okay I want to show man feel okay with that so far now that's just some algebra after doing a little bit of calculus all we did was do an integration on both sides what in the world does it all mean well I'm going to show you because right now it's a little hard to understand we're going to use some symbols right now to represent some of these uh these functions so what we're do is we're going to call U we're going to call F ofx U and we're going to call G of x v please keep in mind u and v are going to represent functions in your integral right now they're going to represent some functions here so we're letting U represent the function of x x v uh V represent the function G of x a function of X you guys okay with that so far yeah now we got some other things besides just f ofx and g of X don't we I mean we could change this to a u right now are you with me we could change this to a u we could change this to a v we could change that to a v you with me however we've got an F Prime and we've got a g Prime so what we're going to do is go okay cool well tell you what let's just do like a du let's do a derivative on both sides if I do this just like your substitutions do inte sorry derivative of U is R du derivative of f is frime of X ah DX do you remember doing that yeah just like a substitution well here we'll get DV and we'll get G Prime of X DX you still okay with it now look every one of these pieces is now part of this this little box of knowledge let's do a substitution instead of f ofx what do we call it no f ofx is just U what is this piece G Prime of X that's our D so this is DB F ofx was u g Prime of X DX D DV you guys okay with that so far yeah equals equals uh what's our F ofx again U what's our v v so f ofx is U time G of X is V minus integral uh what's our G of X now this is kind of cool what's our frime of X DX this whole piece so if we integrate the product rule what we end up getting is this thing right here what this thing is is how to do integration by parts just like the product rule is a Formula right it's just like the product rules of form it's been proven but just like the product rules of formula integration by parts is also a formula and even though it looks a little like what in the world are we doing I'm going to give you some easy ways to do some of these integrals you guys want to see how this stuff works yes first did you see how the formula comes about I don't want to just give you that CU like what in the world this stems from the product it's an easy proof but that stems from the product rule let me show you why we do this so so integral of x * Eed X DX so this right here What's this called again integ maybe write that this is integration by parts so when I say do integration by parts or when the book says do integration by parts or your best friend says at one of your parties hey can you do some integration by parts like yeah duh I know what that means that'd be the best party ever by the way so my kind of party so let's look at this for a second think through all the things you know about integrals does that fit your integration table the way it is this would that'd be easy that would that'd be easy but this can you use a substitution no if you pick this to be you the derivative is one does it get rid of the X no that's a problem so will a substitution work here no there's no way to do this integral for us right now we had to create something new that would take care of it and this is what we have you see when you have a product here we use the product rule sometimes when we have products here we can use for integrals integration by parts which is really really cool let me show you how how this works and let me show you what you should be picking for U and for V by the way it's it should be really apparent whether you pick the the correct thing or not uh within about the first step or two all right so here's how in general you choose how you're you and how you choose your be so couple little notes I want you to choose U so that you can take the integral of whatever V is basically the integral of the rest it'll make sense in just a second I'm going write out the notes right now write them down I'll explain them so choose U such that or so that the integral of V is actually possible and try to pick U this doesn't always work but this is a good indication try to pick U so that the derivative of U is better so that's smaller or easier to work with so choose you so that the integral of V firstly is possible because you're going to have to take the integral of V also again try to pick U so and so that the derivative of U is better bet I'm I know that's not mathematical but better as in you reduce the power uh it get becomes smaller it goes away completely something like that so here we go I want you to look at this this integral with me there's two parts to this there's an X part and there's an e to the X part you with me okay if I if I pick U to be e to X is the derivative of e to the X better no no it's still e DX I could take the integral of x that's okay but if I pick X what's the derivative of x is that better yes yes one is better than x isn't it I think so I'd rather work with one than x seems better also can you do the integral of e x then that's our appropriate uh pick appropriate choice for you so here's how we do this it's going to take some getting used to but here's how it works pick U so that you can take the integral of the rest your V and pick U so that hopefully the derivative of U is going to be better easier to work with do you follow that so in our case we're going to pick U equal x here's what b equals it's really easy to find there's an easy way to do it uh what's your U cover up with your hand V is all the rest of it so V is the integral of e tox DX the reason why we have that is because we're going to have to deal with the integral at some point we have to have our V equals the integral of the rest uh we need to be getting let's see what are we trying to get there we want to be able to find our V for this guy we got to have that so can you do your just like you're doing a substitution can you do your du how much does your du equal yes okay V was the integral of the rest can you do the integral of e to X how much is it don't worry about the plus C right now don't worry about the plus C right now we're going to deal with the so what no we're just right here we're we're not doing a derivative we're saying V is the integral of the rest okay we're not doing derivatives here we're actually doing the integral of of this thing so V is the integral of the rest that's what V is so okay do the integral of that V is now e to the X you with me okay cool um man do we have the parts do we have What U equals do we have what V equals do we have what equals same thing yes do we have what du equals put in the formula so what our integral is then is U * V minus the integral notice there's another integral you do integration twice here one to find the V and then again in your Formula E to the X I'm sorry what was what was the DU DX that's the idea I need to make sure this makes makes sense to you so we picking we're picking you so that the derivative becomes easier and you can do the integral the rest the easy way to do this identify what your U is cover up with your hand V equals the integral of the rest that's what V is for us it's that's the easy way to do it after that well find the derivative of U Dera of X du equal DX in this case is a really easy one do the integral of whatever you had left over so that's e x now put it in the formula integration by part says integral of UV = U * V minus the integral of V du DX in this case now can you do that integral yes how much is it so we get x * e tox minus what this when you finish that last integral that's where you do the plus C you see you could get a constant here you could um but we don't need to because you're saying integral of constant plus a constant just get another constant so we'd have a plus c here guys right with that one tell me one more thing that we could do change it again we could Factor an e to the X if you a lot of times e x you can Factor it so what we know is the integral of x e to X DX actually equals x e x - e x + C or E X * x -1 plus C either one of those things show hands if you can follow that one you're probably need some more practice right yeah okay cuz right now this is only the first this is about the easiest that we can do uh they get a lot more intense I should say not really any different ideas same idea if you can't do a substitute by the way when you're doing integrals don't jump immediately to integration by parts go through your list if it fits integration table that's the easiest and you just do it if it doesn't you try what first subtitution please try substitution because if you can do a substitution that'll be easier after that then you try stuff like integration by parts then you try a trig sub trigonometric substitution which we haven't covered yet we're going to cover that in this chapter but that's later on so let's try few more examples we're going to start building these up to more and more intensity that way you get used to doing some of these problems I don't want to give you just like spoon feed you and then on your test just Hammer you with some hard ones so we're going to do some hard ones here in a little while but I want to get you used to this first so how about I'm also going to try to make some mistakes mes um that way you can you can see the appropriate choices and when you will realize that you've made the wrong choice does that make sense to you you go oh man that's what happens when you make the wrong choice here so okay we'll take a look at it now the first thing we always check for what is it when we check integrals what do we check for we check to see if it's actually in the table first is that in your table heck no most of them aren't going to be okay otherwise it' be really easy uh after it's not in your table what do you try to do now you try to do a Uso is a u substitution going to work here if I pick this the derivative is X squ you see X squ anywhere no if I pick this that's 1 /x you see a one over X anywhere no the substitution is not going to work okay well if a substitution doesn't work our next go-to this is the the the model of thinking that you do okay does it fit the table if it does awesome if it doesn't Okay substitution does it fit substitution if yes cool that's that's pretty easy if not dang it then we got integration by parts it'll still work it's just a little bit more work so go through and try to pick the appropriate thing the appropriate factor for your integration by parts so we got two choices again for our U our U could either be X cubed or it could be Ln x what would you pick for your U uh 3 x why not X Cub let's think about I want to show you though check this out let's say that I picked uh you can write this down or not I don't I don't really care let's say you picked = X cubed you with me if you pick = X Cub can you please tell me what your V would be no it's not just LX it's the don't forget that integral my goodness don't forget that it's the integral of lnx DX it's all of the rest of the stuff does that make sense you're basically just covering up the U the U is one part this is your u v equals all of the rest of the stuff so V equals quote unquote the rest show hands if you feel okay with that it's a big deal all right now could you take the derivative of this yeah we can take the derivative darn near anything and that actually makes it easier for this one uh can you take the integral of Ln X you can what is it okay I'm not asking for the derivative I'm asking for the integral the derivative of L and X is 1 /x what's the integral of L and x e no it's not [Music] e well I don't know what is it can you do it can actually then if you can't do it then we' picked the wrong thing does that make sense you need to pick listen listen listen this is not a trivial Point here uh you need to pick U so that hopefully the U becomes better the DU makes it better but if not okay the main thing here is that you can actually do the integral of V because if you can't then you're stuck none of you right now unless you've had this class before know what that integral is do you know what the integral of Ln X is no you don't so we made the wrong choice this is a bad choice well if that's the bad choice don't beat yourself up over it just erase the problem and pick the different thing what should our U actually be okay what you said all along genius so if U equals Ln X can you please write down what the V is going to equal oh that' be 3 no it wouldn't be 3 too much too we still got an integral don't we uh listen I'm not lying to you all you do is cover this up uh sorry cover that up your you and write the rest whatever you see that's what goes for your V even Rhymes don't you love rules at rhyme whatever you see is what goes for your V whatever stupid I should edit that I'm not going to watch yeah I feel Victorious right now can you take the derivative of this guy okay what's the derivative of Ln hey the only thing where the derivative equals the integral is when you're deal with e x okay that's that's it l and X is different so we get 1/x ah good now with the V we're not doing derivative here we're actually solving this integral can you do the integral of x 3r then we picked the correct Choice here okay the other choice didn't work this choice we get X to 4th over 4 or 1/4 x 4 I don't care how you put that we don't need a plus c cuz we're just using this in our formula right now we have the plus C at the very very end should be okay with that one and do you understand why we picked the U the way we did good that's important that's the hardest part of this is picking the U correctly honestly it is the rest of it is a Formula which you know how to do you know how to do easy integrals you know how to do you know do all sorts of derivatives you're good to go just use the formula so what the formula says is that the integral of U DV equal U * V minus the integral of V du just plug in the pieces watch this real careful though make sure you plug in the pieces correctly can you please tell me what the U is good right here that's our that's r u times RV do the V that you actually solve for don't integral again what's the v x you guys okay on the U * the V we got a minus sign put a minus sign we got an integral sign put an integral sign put the V here again the V goes in two spots V goes here V goes here now stick with me Ln x x 4 4us sign got it so Ln x x 4 4us sign integral integral x 4 4 again that's our V now look at your du your du is not always DX your du depends on what you picked for your U what's your du so we need 1 /x DX it's different than substitution it's different you don't start solving for the DX here the formula says you're going to take this and put this whole thing where this is you don't start solving for that like like some people do like I do for substitutions so if your U is Ln X put that if your V is x 4 four put that minus no problem integral no problem V is X4 4 put that du is 1x DX show hands if you understand the concept good now simplify it solve it this should be solvable if it's not solvable listen carefully please if this isn't solvable it's the only piece of calculus you have left right if that's not solvable for you if you can't do that try a substitution we'll try another integration by parts if this thing explodes way bigger than this thing then you've picked the wrong choice and try the other substitute try the other U okay so you shouldn't be making these way way worse they should start getting better smaller powers and things like that okay or a doable integral this one actually get the same power back but we can do the integral so in our case we got Ln x * x 4 4 minus the integral what are you going to do with all this stuff tell me something you can do combine combine how much we get x 4 4 not X to 4th x to the 3r yes and then we can also do what bring the4 out maybe 1/4 out front would be a good idea hey does that look like something you can do yeah I sure hope to goodness that you can I mean that's that should be easy one you guys okay with the algebra on that one yeah 1/4 comes out x to the 4th you should know how to simplify here yeah so let's uh keep after it this one I'm going to write a little bit different CU I don't like having my lnx first so I'm going to write 14 x to 4th Ln X yall okay that that's the same thing 1/4 x to 4 Ln xus 1/4 hey look at that now we can do an integral of something that's really really really easy what's the integral of x to the 3 in fact we already did it yeah I now is where you put your plus C is the plus C important yes so don't forget it okay so don't do all this work and then go I'm done no you're not done put the plus C uh let us know that that that happens so 1/4 x 4th Ln x no problem minus 116th see where that comes from yeah yes no everybody x 4 let's see could you do some other things with it factor out X do a lot of things with this okay you could Factor an x to the 4 you can actually Force this thing to factor out a u a 116th or 1/4 or a six you could factor out a lot of stuff here so if you did it's going to look a little different so keep in mind when you look in the back of your book if it looks different they might have just started factoring things so don't give up on it um if you're if you a man I think I'm right try factoring something see if you can see a common factor that they have okie dokie so if you feel okay with that one can we build it up just a little bit Okie doie okay I'm going to change this problem slightly in just a moment but I want ask you a question first uh what do you check first when you're doing in your table table Okay good does it fit your table no next you check for will a substitution work here look again will a substitution work here absolutely cuz the derivative of this is 6x2 you see it your X's are gone so if I gave you this problem this is this is my whole point which some of you obviously didn't listen to my whole point is do not jump automatically to that that's silly okay try a substitution if it'll work here don't do this problem I told you I'm going to change it in a minute so this this would be great for you U equals this Dera of 6x2 x s are gone you pull out a 1 16 you're you're basically done you're going to get Negative cosine 1 16 negative cosine of 2x Cub them right plus C done easy will the substitution work now no no no so what are we going to do uh the next one you always check a substitution before you check integration by parts I hope that I'm getting through to you right now I'm getting through to you right now always check that because it'll make it easier if it'll work so oh man uh let's see here what's a good choice what do you think 2 sin 2x well 2 if we pick say what now if we pick s 2x uh as your U does the derivative get better no no and the integral grows right s thir okay so in your next process you would have you have a COS sign somewhere and you'd have an x to the 3 your integral is going to explode a little bit does that make sense I'll show you if you want I think we have time for this if you did don't don't write this in well I'm going to leave it up here but you can if you want I guess if you did this oops that would be it wouldn't it U integral the rest be so that would be D equal 2 cosine 2X DX do you know why I have a two out front X 3r over 3 then i' have UV minus integral of V du this is my du yeah do you see what I'm talking about right when you do this you should give up not calculus all right but right when you see this look what happened look look at honestly look this is basically the same thing but my power grew it didn't get smaller it grew is that a good thing no then this is the wrong thing to do so don't try to manhandle this problem and pound it into submission because it's going to kick your butt all right this thing's going to get worse if you pick this again well then that's going to go to a four and then that's silly so we don't want to do that so this is a bad choice you understand why it's a bad choice integral got worse not better even though you could do the integral of this one this derivative didn't look better you want you want both you want both really you want the derivative to be better and you want to be a to do the integral so let's try the other one instead of this being U let's pick that one being U then V equals cover integral of the rest du equals this is easy part dual 2dx you guys okay with the D equal 2dx yeah now listen you should be able to do this other thing this other integral uh can you do this integral you might have a u substitution in there so if you do a simple U sub I wouldn't pick you because you already have a u i pick like a w or whatever you want uh anything that you want besides x v and U right now now so if you picked a w substitution = 2x DW = 2 DX DW / 2 = DX and then do your substitution what you're going to end up with after you do this a whole lot is you're going to understand that really what this V is going to be is negative cosine 2x/ 2 is what that's going to be did you guys see where the over two is coming from yes the over two is coming from when you have your U or Well w = 2x DW = 2 DX DW oh look at that over two that's where the over two is coming from no I blow your minds did you follow that yeah do I need to do the whole thing for you no 12 cosine W ah W is 2x that's that it's open okay now wees so we've done our U we've done our V let's put this thing together in the formula the formula should look do you tell me what's the first thing I should write here good time what I'm going to eras this yeah show should be okay with that one so far yes yes guys in the middle oh yeah abely clean it up before we go any further that's nasty all right let's make this uh - 12x^2 cosine 2X - 12 x^2 cosine 2X let's do a couple things here let's pull out this negative and make that a plus let's pull out this two and simplify it with this two can you guys see that yeah twos are gone let's make it plus integral cine 2X time x DX you could even if you want to move the X out front right here x cosine X that might look a little bit better let's change that so x cosine 2X okay want your hands you okay with algebra on that one now calculus is done just like you follow your DDX with derivatives follow your integral form for integrals so here can you do that inal yeah you can is it in integration table H is a substitution going to work yeah yes okay what's the derivative of 2x so you need another Parts you need an X right would this work yes if it's a square is a substitution going to work here no oh my God okay so we did is it in the table will a substitution work what's your next choice you can do it again now don't screw this up pick the same thing for you that you did here the same thing it should be reduced by a power do you see that this is basically the same except instead of X squ you get X that's how you know you done it right okay this should at least go down a power if it goes up a power you pick the wrong choice if it goes down of power you pick the right choice you just got to do it again so let's do it again lots of practice so we're going to hang on to this okay we're going to look just right here uh what's the and we got to move quickly what's the appropriate U check this out when that's X see how when we take a derivative the power drops doesn't it 2 to one 1 to zero that means that our next integral would be very easy or next integral would be possible at least so V would equal the integral cover it integral of whatever's left cine 2x DX I'm going to do this inter real quickly I'm not going to show you the substitution like here do you see that you're going to have another substitution here U or W or whatever is going to be 2x that means DW is going to be 12 DX that means we're going to get an over two so this would be S of 2x over 2 can you verify that for me sin 2x but then over two by the subtitution rule okay so I would not write this again right now I'm just going to hang on to it I'm just going to focus right here so I'm going have a plus big old Stuff Plus well we're going to have U * [Music] v u * V minus the integral of V du now what's our DU in this case that's nice that means that we're not going to have any extra work to do I want you to see if you're okay okay with this so I know I'm going about a minute or two over but this is kind of important for us you okay with the U the U the U is DX now you okay with the v v is integral the rest pretty easy integral you need to be able to take these integrals otherwise you made the wrong choice now we got U * vus integral of v and then du du is DX make that substitution again tell me something we're going to do with this okay so if we do that we get some stuff plusx sin 2x minus the integral oh yeah 12 integral sin 2x DX only we've already basically done that one look at this this integral is that integral so just look back at your table we've already done it so we have all the same stuff minus 12 this again was cine 2x/ 2 exactly what we have here this integral is the same as this one so cosine 2X over 2 so F be okay with that one now we're just going to clean it up okay just cleaning this thing up we've got all that nasty stuff was NE 12x^2 cosine 2X plusx sin 2x what happens here cancel not canc plus cosine 2X and last bit okay I want to make sure you're clear Before I Let You Go this is the first little part it's right there do you guys okay on that one from the first interation by Parts this was the first part of our second integration by parts the U * V this little piece this plus comes from when we do our last integral minus and negative becomes a plus 1/4 cosine 2X plus see oh my gosh look what happens to the powers Square to the first Power no X at all that happens a lot when you have this if you add x to the 3 power you'd have to do that one more time you had to the fourth power you have to do it one more time ultimately we're going to find a reduction formula for that we're going to create our own formula for this what I want to know is if that actually made I know this is kind of sloppy I didn't have room over here but do you understand that this is the plus one for so P if this made sense to you good all right so for our today for our day today we're going to practice more integration by parts so if you don't remember from last time no we've got a weekend in between this uh except for people watch online they have like 5 seconds in between this so for us though what we're talking about uh is integration by parts and the way we do integrals is first thing we check for is do we have something that's in the integration table if we do it's really easy it's in the table if it's not then we try a substitution once we try a substitution well hopefully we can do one if we can't we can do it because it's easier if we can't then we have now integration by part so we check this is this in an integration table for us no of course not will a substitution work no no if I substitute this part there's no other e to X in there if I substitute this well there's no it's not going to take care of the E tox if I do sign there's no cosine up there remember with substitutions the derivative of your U has to be somewhere in your integral that doesn't happen here so what we're left with is an integration by parts now these last four examples that we're going to do today um they are they're like the the more difficult Concepts the more difficult ideas the ones like wow I didn't know you could actually do that with this stuff I'm going to show you some really cool interesting things that you can do with inte ation by parts are you ready for it yeah so here's what I'm going to do I'm going to pick U to be Sin Sin 2x now as soon as you pick U what's v so I pick you we got to be able to be pretty good about finding out what V is what's v very good everything else so you just cover this up with your hand write exactly what's left over in this case integral of BX DX quick head notot if you're okay with that one so this is coming back to you I hope right now the way that we pick U is typically U becomes easier and you can do the integral of the V well in our case this integral is going to be really easy isn't it this one doesn't get any better but you're going to see something really unique that happens on these types of problems where we get when you think about it we're going to get repetitiveness here aren't we the integral of e DX is e x integral or the derivative of sinx is cosine X and the derivative of that one is is sinx we're going to get things back again no matter what we choose does that make sense to you so watch what can happen here if we do this then du = 2 cine 2x DX can you please explain to me where this two comes from hey and V equals do I need to plus C right now no no we're going to wait to the very end to put a plus c quick head now if you're okay with that one now what we do with our our integration by parts is we pick our U do a derivative we pick our V we do the integral of whatever's left over that's the V is the rest then we do integral of vdal U * V so in our case our integral is going to equal U time B minus the integral of B now now this is what's interesting minus integral of V but then we got to have du how much is our DU in this case so we need this whole thing right here okay so let's make sure you guys got it U * v u * V minus okay integral of V integral of V du is this whole piece quick head now if you're okay with that one yeah now check it we did this last time we saw we saw something similar to this last time I'm going to move my e x to the front by the way so e x sin 2x the last example that we did by the way what else can I do here two out two out yeah we're going to take that two out because it is a constant being multiplied e to x cosine 2X DX if you think back if you watch back on the video for the last example that we did do you remember the last one we did we had something kind of similar to this one we worked it down except instead of sin 2X and edx it was like an X squar remember that and we had to do integration by parts twice do you recall that example well look at this one doesn't that look almost exactly the same as that except instead of sign we're going to have cosine could we do integration by parts again yeah let's do it again now if you're thinking wait a minute wait if I do integration by parts here and it gets this Plus something looks almost exactly the same as that why are we doing that and aren't I going to get something like this but something looks just like this again yeah but that's the whole point and I want to show you why so we're going to leave this thing alone and we're going to look right here here's the whole thing if you pick U equal s of 2x what are you going to pick for U here pick the same thing so now U = cosine 2X not this I'm sorry not the exact same thing but pick uh the the same part that you picked so if you picked s of 2x for this one pick cosine of 2x for this one V again is the integral of e to X DX that's pretty easy so U cosine 2X no problem V is just the integral of the rest du help me out with the DU what's the DU going to be why negative because the good negative sign so this is going to give us negative sign the chain rule is going to give us a two so we're going to have -2 sin 2X and V is going to equal that's pretty easy the integral of e x is e x show hands feel okay with that one so far good deal now we're going to wrap this thing up so here we go we got e x we got s 2x we got Min -2 we got min-2 you guys okay with that so far now please watch carefully when you're doing this integral this is- 2 * this whole thing isn't it so if we just start writing out stuff willy-nilly we're probably going to make some sign errors here so what we're going to do is hey if this integral is going to be U * vus integral of V to U put a bracket here because what's going to happen is we're going to be Distributing that -2 throughout this this problem does that make sense yeah okay so minus two no problem now our integral what are we GNA write first e u * V so yeah you could write cosine 2X * e x or e x time cosine 2X it doesn't matter it's U * V so I'm going to write e to X cine 2x U * V we just have it in backwards order minus the integral of what what's going in here that's our V and then oh V 2times -2 sin 2x I forgot something do you see what I forgot my DX DX sorry I forgot my DX let's double check to see if I'm right we go like a 10-second recap to make sure that you guys got it so we're looking here it doesn't fit our table we can't do a substitution we're going to try integration by parts we picked something all right cool we'll pick this one u v is the rest derivative of U is this guy by Chain rule integral of V is this guy no problem we do U * vus the integral of V du and we start over again we simplify see if we can move anything outside of our integral this part calculus is done this part Min - 2 integral okay we get something real similar well let's pick the same type of idea here let's pick cosine 2X is our U okay V is the integral of the rest remember we're not worried about our minus 2 that's outside of our integral V integral of e X DX okay derivative of cosine 2X we can do that -2 by the chain R sin 2x DX V is exactly the same as it was before so again we have it we have U * vus the integral of v and then another du you guys okay with that one let's simplify some of this stuff let's see what's going on so what can I do here let's do that so I'm going to get e to X sin 2X x- 2 big old bracket I'm going to get e x cosine 2X plus plus 2 integral e to X sin 2x DX lots of X by the way I like to go slow on these ones because the worst thing you can do here is mess the sign up that would really screw this problem does it make sense to you so Don't Mess the signs up here so we're going to go slow we're going to make sure this is right um what else can I do here I don't want to factor anything I want to get rid of these brackets how do I do that dist if I do that if I distribute that then what we end up getting is something really long e to X sin 2x let's see - 2 e x cosine 2X yeah minus 4 that's very good because we're going to distribute Min -4 integral e to X sin 2x DX no more brackets I want to know if mathematically you can make it there in your head you feel okay with that one show hands if you do now did we do anything great on this problem what it looks like is that do you see what happened do you see this we started here right you started there mhm we did all this junk in fact we did the jump twice and we got down here and this is exactly like what we started with so you go dang do it what are we GNA do we can do it again if we do it again keep getting that that's what was that Einstein's definition of insanity or something like that I don't know who it was but said doing the same thing over and over again expecting a different result uh if we do this again unless your math was completely wrong you're probably going to get the same thing again and again and again does that make sense to you so this is a problem so what are you going do about it we're going to be smart about it check this out here's what we know we know that this this integral equals this part do you believe that yeah check this out so what we know is that the integral don't lose me here this is the important part this is easy you can do this everybody can do this right now this is the interesting part this is why we're doing this problem right now so we've got integral e x sin 2x I'm so excited I'm almost shaking going to be so much fun not really but it's it's going to be fun I'm not shaking now this thing equals we have equals the whole way down equals this whole enormous piece of junk okay you know you've been being recorded right s it's okay you should apologize to the people I don't think they care they care I care maybe they'll get excited to B in India you're Bing my people in India goodness okay that's long enough what I want to know is do you understand this is true we did the integral of this it equal this whole mess of junk let's make sure I didn't miss anything do I have all the numbers and stuff right pretty sure okay do you believe that we have equation here do you believe that you can do anything you want to as long as you do it to then what if we do this we go wait a second uh this part looks great to me this part looks not so much but what if I were just to take that and add that oh man I know right oh my God here's the whole cool part about this if you get a piece of your integral back again like the same thing you guys see how we got the same exact thing back again what can we do shoot just add it to both sides that's going to get rid of that integral so let's see uh we have one of them here I just added four of them how much do I have I have five of them so we're treating this thing like the equation that it actually is so 1 + 4 = 5 * integral e tox sin 2x DX equal well this is awesome I have e to X I've got sin 2x- 2 e to x cosine 2X what happened to this gone I added it over completely gone so the whole idea here is this well maybe you're not on the wrong track maybe you're actually on the the completely right track you just get the same exact thing back and go wait a minute man I blew this thing up no no you did it right so if we have all but the last integral and the last integral is exactly what we started with just add it to both sides like this shoot hey we have minus 4 just add four add four times the integral of this whole thing it's going to get rid of it just like if you had Min - 4x and you add 4X hello it's gone add it to both sides here's five we only have to do one more thing we got to get rid of this five how do you do it divide five divide by five so I'm going to do a couple things right now first thing I'm going to do is factor out my e to the x no if that's possible so over here if I factor that out I got e to X sin 2x - 2 cosine 2X so just factored out my edx and then if I wanted to get the five over there I do what now I divide by five I divide by five then I end up getting 1 so our integral of e x sin 2x DX what we wanted to find out we know since we divided by five this is gone we get 1/5 over here5 e DX sin 2x - 2 cosine 2X one thing I'm missing what am I missing plus do the plus c oh my gosh wasn't that cool is that kind of exciting it's interesting isn't it well how are we going to do it well if you get the same thing back again addable sides then all you have to do is divide this addition gets rid of the inle that you can't do you can't do that by itself you got to do it this way so if you just add to both sides you're done just divide by five and we're good to go so feel okay with that one isn't that kind of interesting that's cool right any questions or comments I got to erase this whole thing right now because it's these problems uh besides the next one are going to take up the entire board so uh each every time the next one might even take it more than the entire board question so the only time you would do that like add to the other side is when you get the same thing back twice you do Parts twice I don't know if it's twice but if you get it back again and there's one integral and it's exactly like what you started with why wouldn't you do that you know if it's exactly the same you go wait a minute I just got back the same thing either I did it completely wrong or this is perfect because I'm just going to add that to both sides and divide by the whatever the the constant is the coent do that because you have two uh looping terms would it matter if you switch the Ed the X and the coine I don't think so and I haven't done it I think I did it a long time ago I don't think it matters uh my point here was to pick the same thing though pick sinx pick cosine X if you did e to the X you could do the integral of sin 2x no problem uh you just have to pick that again I think this is a better way but I I haven't done it uh the other way I I like this way because it worked out really nice I have to worry about integrals uh that's why I picked did because integral ofx is very easy question if this was on if this was a homework problem is that what our answer would look like that is your answer okay other questions all right can we try another one yeah I asked you a while back when we were doing one of these things it was integration by parts and I said uh hey what are you going to do if I give you a problem like I don't remember what it was I think it was something like like uh X Ln X or something like this remember that so what are we going to pick and some of you guys said well let's pick U equal lnx said really so uh or sorry um U equal x and V would equal the integral lnx and said well do you know what the integral of Ln X actually is do you know what the integral of Ln X is no we don't well now what I'm going to do is instead of asking you this problem or something similar I'm going to ask you this problem what is the integral L I'm asking for integrals not derivatives Hulk angry just joking I'm wearing a black shirt right shorts yeah and jeans so uh do you know what the integral of Ln X is no do you know what the derivative of Ln X is yeah that's one of Rex so here's what we're going to do make it a little bit more interesting too uh I'm going to ask you how much is this from 1 to e now of course the 1 to e part that's really easy we're to take that care of that at the very end okay so when we're going through integration by parts though I really want to figure out what's the integral of Ln X okay well let's start picking uh let's start going through our process is that part of our integration table no it's about to be is it right now uh will a substitution work if we do a substitution of = Ln X the derivative is 1x do you see 1X anywhere up there no substitution work the only thing we have left is integration by parts the only thing we can pick is what okay so U equal lnx now if U equals lnx see if you guys are thinking outside the box here how much is V it's got to have an integral integral DX um if you don't know what integral DX means integral DX means integral of 1 DX so remember how X is 1 X integral DX is really integral 1 DX a lot of people don't write the one but it's there so let's see let's start with a V how much is the integral of DX or 1 DX it's X yeah integral of one is just X you guys okay with that one we know that du equals 1X DX that's actually pretty easy we know what the derivative of Ln X is 1x DX so van if you're okay with that so far let's start putting this thing together by the way you don't start evaluating this right now you wait till the very end to evaluate uh from one to e okay you don't have to change bounds in this because if you think back to integration by parts well what happens is you're using U * V they're already in terms of X you don't have to do a substitution here so don't worry about changeing bounds don't worry about that stuff you just evaluate at the very very end does that make sense to you okay so we know that we can pick you the only thing we have the pick is L and X the rest of it is integral DX that's X so v = x we're going to do U time V minus the integral of V du only du is 1 /x DX now that's kind of interesting you guys okay with what we've done so far u u v got it minus the integral of v no problem so U * V us integral of v and this part is our DU look at what happens in this integral what's going to happen what do I have in there one so I'm going to reverse these because it looks a little funny so I'm going to have X Ln x minus the integral of hello 1 D or just DX again can you do that integral yeah it's actually really easy that's X Ln x minus hello now if I didn't have my one to e plus C if I didn't have my 1 to e put a plus c that's actually the integral of Ln X the integral of Ln X is X Ln x - x or x * L and xus one if you wanted to factor that that is the integral we we actually just did it it's really easy if you do integration by parts kind of cool though right we can invent our own integration tables with some of this stuff like with integral L and X show F should be okay with with that one now if I'm asking you to go from 1 to E I don't have a plus c because now it's a definite integral how you write this you don't continue to write your integral because you actually just did your integral correct this would be silly to write you've done it how you write the evaluation is you either put a bracket like this or the line or even a bracket and we go one to e and then we just plug it in so here we'd have e Ln e minus E minus 1 * Ln 1 - 1 let's see what happens here most of this stuff should be kind of kind of nice uh how much is Ln e this is one so we'd have e * 1 - e minus okay how about how about Len one this is zero 0 - one that's still there you don't just get zeros everywhere when you when you plug in zero or when sorry when you get one zero so this is 1 E * 1 is e minus E okay big old minus ln1 is 0 so 1 * 0 is 0 minus one how much is this whole thing it's one now that's interesting I think that's really interesting you remember what integrals actually mean a reverse derivative it is an anti-derivative well what I mean is a derivative is a rate of change or the slope of a curve at a point do you remember that that's what a derivative is what's an integral area under the curve that's right so a definite integral is the area under a curve between two points the area under the curve lnx between 1 and E if you know what Ln X looks like it looks like this it crosses at the point one here would be the point Point e the area under this curve is exactly one is that not weird so the area under lnx between one and an irrational number e 2.7 blah blah blah blah blah blah blah forever if you find the area under there it is exactly one I think that's just amazing I don't know why but I think that's so cool anyway we can find the the integral of lnx now and we can even evaluate it with different integrals show be okay with with that one interesting stuff today that's interesting it is right I think so now are you ready for the hardest one that we're going to do today this is going to be ridiculous all right we're going to have a lot of work it looks so simple to start with but it's not going to [Applause] be okay okay here we go okay I'm going to add one thing to this problem that's going to make it from easy to hard okay so if I had the interval of secant x if I had the integral of secant x that's actually in your integration table do you understand that so you look there first so you want you wouldn't do anything with this problem except go okay this is uh Ln absolute value of see X Plus tangent X plus I think that's it plus c I want to look that up that's in your table if I did this that's even easier that's tangent you're done if I did this oh man we got a problem cuz that's not in my table I can't do a substitution because if I pick U equals see X the derivative of secant tangent that doesn't appear again so I can't do that so we're going to have to find out some other way to do these sort of problems so can to the fifth or seeking to the third power I'm going to show you some techniques that we're going to get to a little bit later in section 7.2 this kind of like an introduction to some things that we are going to be doing uh I'll give you some better ideas later but right now I kind of want to invent something with you so here's the idea one idea is if you don't know the integral right off the bat and you can't use a substitution integration by parts is kind of defeating you here because there's really nothing to work with uh you get back exactly what you you have here or make it worse it's not a good thing we break it up somehow into a part where at least you know how to take the integral of some of it like that why I'm doing that is well I don't know how to take the integral of secant cubed but I do know how to take the integral of SEC squ does that make sense what you can't do you can't break this up asant time squ time SEC Square just go hey the integral of secant is this time integral of secant square is 10 you can't do that okay uh because that that's like a product that's like the whole idea of uh product means integration by parts so we can't do that so only one product only one product okay well let's start doing this if I do this I'd probably pick uh let's see let's see what you would pick you CU because I don't know how to take the integral of that part I can I can take theal of squ so = Cub X let's do the derivative right now if I do the derivative I'm going to do the derivatives quick right now because I'm expecting that you guys can can do this all the time okay the derivative of secant Cub would be 3^ 2 x * SEC X tangent X do you follow that by the chain rule three comes down becomes a two leave the inside alone derivative inside is see s see X tangent X you guys okay with that one okay if not go back and refresh your memory on how to do integral of secant Cubed on your own V would be the integral of secant 2 x DX this is nice I broke it up into something I can actually take the integral up everybody what's integral of secant squ okay we're starting so our first part here is we know this integral is U time V minus the integral of V du I forgot my DX again V du let's clean this up just a little bit instead of secant squared what am I going to put we have lots of writing going on so I want to make sure you guys are okay on every single step are you okay getting see cubed x * tangent x * V yes no minus integral of V Tan x this is Du it's 3 SEC Cub Tan x and then our DX show hands feel okay with that one okay let's clean it up just a little bit what are we going to do here you know okay three out what else so - 3 integral of tan^ 2 x SEC Cub X DX okay this is fun now well this is fun what do we do here when you have something like a tangent and a secant what you're going to see in the next section that we do is often times we're going to be using identity to change one into the other for instance I don't want tangent X and see X because even if I did integration by parts of that it's going to be really hard to do so let's make it easier or maybe even impossible to do let's make it easier instead of having tan^ 2 x we're going to use an identity for that what you need to know is you need know your identities hav them written down tan^ 2 x is the same thing as 2 X - 1 did you know that this is true so tan^ 2 x is s x - 1 let's use it because now we can change our tan squ into a secant so instead of our Tan x tan^ 2 x we have sec 2 x -1 and then SEC Cub X DX so true that's an identity use it oh wait a second this is kind of neat you're going to see something we just dealt with actually what should I do here can you distri it yeah I'm going to distribute that because right now substitution isn't going to work because if I try this one I don't get exactly this thing back again does that make sense I'd have a secret tangent again that would be a big problem I don't want to do that so let's distribute and this is the same process that I'm going to teach you in the next section you're going to do this a lot so let's distribute this guy well that becomes secant Cub X tan xus 3 integral of tell me what I get from my integral sec to the 5th power of xus Cub X DX okay really do need to show up hands if you are okay with that so far are we all cool what can you do every time you have integral of a subtraction what can you do separate let's separate it now watch what we're going to do make sure that when you separate this this minus three goes to this whole entire integral does that make sense yeah so when we do this this really should have a bracket here what we'll get is SEC Cub X Tan x minus 3 integral secant to the 5th power x hopefully you're with me on this one let me have your eyes on the board here this would separate into two integrals correct it would be minus integral secant Cub this will make it plus three integral SEC Cub X DX [Applause] I want to make sure you're okay with algebra it's just algebra right now but are you okay with it yeah everybody show hands if you are you're okay with now there's something good about what we just did do you see it can you subract say something good about it what do we have good give it a compliment what do we have good about this thing we got the same can you subtract one of the interal from the other this one this is great that's great right now why is that great that's this that's fantastic and we're going to use that in a minute tell me something really bad about this that still has an integral so no matter what we're going to have to deal with this one does that make sense so what I'm going to do this is why this problem is nasty okay right now this would be awesome if there's no integral if this wasn't here you would just add that to both sides you divide by four and You' be done does that make sense this I'm going to give us a little asteris we're going to have to do this one off to the side now integral SEC cubed x DX do you understand that that's really similar to this idea can I strip off a secant uh square if I can do that then I'm going to do the same exact thing I just did so I'm going to do this kind of quickly here but you need to understand it's the exact same idea only in this case I'm not going to have secant Cube I'll just have secant that means we won't have to do this process again we'll get down to here and be able to add our integral so we're going to do this idea twice you're going to need a lot of practice right now so this becomes integral of SEC x * SEC 2 x DX same exact idea strip off something that you know the integral for well then U equal SEC X the derivative is SEC X X tangent X DX V would be the integral of the rest sec^2 X DX and our V is Tan x again you guys okay with that one do you see the same technique at work that we just did strip off the secant squar call that your V integral of your v no problem that's easy uh this one call that your U derivative of U you can all do that that's see X tangent X now if we do our integral that's going to be U * V minus the integral of V du we'll clean it up a little bit so U * V got it minus integral of V hey no problem du okay looks kind of nasty but we can deal with it got that same thing we'll simplify it a little bit tan tan tan squ uh now this looks a whole lot like that one very very similar oh instead of secant to the 3 I have secant to the 1 I'm going to use the same identity again so instead of tan s I'm going to write that as SEC s- 1 you guys still with me yeah are you sure I don't want to blow you out of the water here a lot of algebra a lot of trig here lot of algebra a lot of Tri so secant x Tan x minus the integral of yeah let's call that SEC s x - 1 * see X do the same idea again distribute it [Music] again that would be see Cub xus X DX do the same thing here that we just did here let's split off our integrals again well if we do that then we got SEC x minus sorry notus see X Tan x minus tell me what I'm going to have here what's the next thing see cubed x DX okay tell me what the next sign's going to be is it going to be plus or minus you guys want see why it's going to be plus remember we're separating this this will be integral minus integral but it's in Brackets so this minus would distribute it become plus integral SEC X DX oh goodness almost there let's see we got see X Tan x we got minus integral of nasty thing we can't get rid of now this one we can actually do that look back in your notes if you have to I think I gave it away earlier but uh the integral of secant x do you know what the integral of secant x is X yes you're almost right there's one little part you forgot absolute power so be natural log of the abute power x plus C I'm going to refrain from putting the plus C but yeah you're right I'm going to refrain from the plus C until I'm completely done with my integral I don't want that thing H around because it's really confusing you guys okay with that one yeah you sure have I blown your mind that yet like what the heck this crazy right we're about uh 75% of the way done we're almost done almost here's what we got we did basically a problem within a problem that was just almost as hard same exact idea though break off the secant squar make that your V because you can do the integral of that no problem Tan x You Got U * V minus integral of well V du clean it up tan S no no no C squ minus one distribute split it up again this this stays the same why do I want that to stay the same because it's this it's this right here okay so I gota I gotta gota get rid of this in a second this one I wanted that because I know the integral that's in my integration table now that's why we did that in the last chapter you guys okay with this so far yes no yes yes all right now keep in mind it was this that's equal to this tell me what I'm going to do next notice no integral no integral integral what am I going to do add that so if I add that treat this like an equation add this integral to both sides if we do if if we add that let's see what do I what do you want me to erase it's going to have to be something the left I can't see the right you know what I think I can write it over here too how much is this going to Beal how much that's going to equal is this thing is this still here this is this still here no got rid of it is this still here yeah and we're almost done to stick with it right now what do we know why do we need to divide by two is that two answer what I want I want this without the two so yeah divide by two so we're going to get C Cub X DX = 12 you could do a bracket around whole thing you do one do2 for each of them see X tan X plus 12 Ln absolute value see X Plus Tan x and absolute value crazy with my absolute values I know maybe your brains are hurting a little bit what I want to know is are you okay on just this piece right now show pants if you are okay do you see how it's very similar to the the first example we did today really similar you get down to one part you can't get rid of hey it matches up add it to both sides no problem in this case 1 + 1 gives you two just divide by two this piece is gone that's great that's what we want we have this plus this divide by two you get one half you guys right with that one now here's the awesome part we're good to go on this right this is awesome this piece we're going to be good to go as you pointed out we're going to be adding this to both sides of our equation you got me this piece we now have something for that so what I'm going to do is I'm going to erase this part what we know is this this is back to our asterics so are you guys have any questions on this got erasing as I need space okay so the integral of see to 5th power of X DX is equal to here's what we know we know it's SEC cubed x tan xus 3 * the integral of sec to the 5th again DX plus three times plus three times here's the cool thing here's what we know we wanted plus three times this piece this piece is this whole mess of junk over here you with me so instead of 3 * integral SEC cubed x we're going to have 3 * this whole thing right here so we're going to have 3 * 12 sec X Tang X plus 12 Ln absolute value SEC X Plus tangent X and absolute value and fr talk about probably the longest problem you've ever done huh yeah crazy we're almost done almost we've gotten rid of the integral right we've gotten rid of the integral we're have to probably distribute the three to get three halves instead of 1/2 but that's easy that's not a problem at all what are we going to do next this is the only integral that we don't like it's already matched up with that one just add it to both sides if we do if we add 3 integral see 5th power x DX is an equation just add it over here then I end up getting how much is here 4 over here I've got SEC Cub X that's right this is gone I added it plus I'm going to distribute this 3es x tangent X plus 3 Ln absolute value SEC X Plus tangent X what's the last thing divide everything by four so this gets divided by four this is going to be a 1/4 this will be a [Music] 38 does that make sense if you divide by four so this gets divid by 4id four ID 4 / 4 so we get the integral of see to the 5th power of x equals 14 SEC Cub X tangent X plus 38 anytime you divide a fraction like this just multiply the denominators and you're good plus C got to have a plus c oh my goodness do you get the idea here did I did I just lose you I don't want to lose you a lot it's a lot my goodness this is like a whole p on my piece of paper it's an entire page one entire page you have a lot of things going on here I want to show hands if you could follow it makes sense to you good so so the idea is this start with something that you can actually take an integral hang on start with something you actually take an integral up if you can't find it split it up do the integral of a piece then see what you get if it's trigonometric or something with an exponential or both often times you're going to get back a piece that looks exact what we start with if you don't have any other integrals you're good like we did over here no other integrals no problem just add to both sides if you do you might have to do the whole thing again take care of that integral then you can ultimately add it to both sides divide by your coefficient and you're done okay so the last little bit that we're going to talk about in section 7.1 I'm going to invent for you the reduction formula what this does is when you have like a a sign to some power inside of your integral and we can't take care of it just by the integration table we can actually reduce it to a lower power of sign I'm going to show you how to do that right now so this is called the reduction formula and it starts with this idea suppose you have some integral with sign to some power side of some power greater than or equal to two now naturally you probably wouldn't use this if you had s to the first Power cuz what's the integral of s to the first Power negative coine that's right it's negative coine that' be easy okay so hopefully you understand that this is going to happen for n is greater than or equal to two so when we don't just have sign that would be silly to use this but let's suppose that we do have something some power greater than or equal to two so if you had sin^2 X well probably want to figure out how to how to do that right how to how to find that out and one way that we're going to do these is with a reduction formula uh so basically what we're going to do is do this problem in general and see what it gives us so here's an idea the idea is we want to split this thing up so that we can use integration by parts a substitution is not going to work because we're going to have signs and signs does that make sense to you are you with me is if you okay I'll show you here's one thing we're going to do right now we can't do anything with the problem uh there's no substitution that's going to work because if I substitute for sign I get a cosine does that make sense do you see a cosine up there that substitution won't work so let's break off something now check this check this out if I have S to the N power well let's split off one power of s if I do that then I get S to the nus1 power Time s do the first Power I need you to verify for me that this is equivalent to this you guys with me on that one remember when you multiply common bases you add exponents so nus1 + 1 gives us our n does that make sense here's what I mean about a substitution not working right now if you couldn't visualize this in your head visualize it now if I try to do a substitution or sign what's a derivative of sign it's not sign so this is a problem so what we're going to do now is well let's go through the whole process does this fit our integration table exactly not even close uh will a substitution work we just talked about that no because derivative of this is cosine not sign that doesn't work so we're going to have to use we have to now remember when you're doing integration by parts and some of you haven't had very much practice I know it's a brand new section and you haven't ever done it before when you're doing integration by parts you might have to do it a few times it's not a one and done all the time okay especially if you have something like uh which we saw in the math lab T Cub e to the T you're doing that you're having four integrals there you're going to do it three times integration by parts three times so it's often times you have to do it more than once well here and and the way you pick it you pick it so that one of two things happens one thing has to happen you have to be able to do the integral of V does that make sense you have to be able to integrate that and what you also are looking for is secondary is that the derivative of the U becomes easier doesn't grow it gets smaller power wise okay those are the the two big things you're looking for so here let's look at it now here nothing's going to get better okay nothing's going to get uh easier as far as a derivative goes because s just like cosine or any other trig function it recycles so we're not looking to make the derivative any easier what we'll be looking for is something we can take the integral of can you take the integral of this no no because it still has a power to it can you take the integral of this then U's going to be this thing V is going to be this thing does that make sense yep okay so U equal s to the nus One X and V equals the integral remember how we do integration by parts yeah we cover this thing up V is the entire rest of the thing it includes integral includes whatever else you have it's horrible looking including the DX U some all are getting kind of sloppy on some of your integrals you're not putting the DX you're missing stuff you're not putting the DX over here or the DU make sure you have those little pieces that's kind of important okay so next thing we do why don't you tell me what's the next thing that we do do thetive derivative okay so du equals look at how this derivative is going to work the power is n minus one you with me this involves what it's going to be a chain rule that's a chain rule it's power rule power rule is a chain rule so power rule says watch watch carefully remember we're just proving this so you're not going to have to ever do this proof um I'm going to create a formula for you for you to use which is kind of nice but I want you to follow it at least so the chain rule or power rule says you bring this down you leave the inside alone you subtract one from this and then you multiply by the derivative of the inside which is cine X DX it's just a chain rule it says you bring down the power got it you subtract one from the power got it multiply by derivative the inside we're good to go show fans feel okay with that that derivative awesome now integral of what's the integral of sin how much cos negative cos okay let's see what this does for us so we've just performed the what was that called again the the whole process what is this integration by parts integration by parts okay we're doing integration by parts let's write out what it is we know integration by parts says once you picked your U and you found your V the integral equals u u v minus integral of V du or in our case U time B minus the integral of V du where this whole thing is my du I'm going to clean it up in just a little bit but I want to make sure you guys are okay with this do you see where everything's coming from we've got our U * our V got that minus no problem integral of here's our V and then this whole mess of junk is Du so we got exactly the same thing show hands feel okay with that one D got equals and say what now the the U the U goes here we have that whatever V is whatever you solve for we don't need this we need this the V follows that up so V goes here this is this minus minus integral integral plug this thing in again so cosine X and then du is this whole mess of junk that's it's right let's see where I found it right here does that make sense so look at this again here's V here's du here's V and here's U that's what's going on you find that indefinite integral of sinx we don't have to put um Nega cosine X plus C no the plus C is going to be one constant at the very end of this thing this is just a temporary deal that we're doing okay um in fact some people show this a little bit differently uh some people show it like this they say well where because remember it's it's UV right so they show this as uh DV equals this and then they go oh well let's get rid of the D let's make this V well that's the derivative of V correct how do you want to do a derivative inte they integrate everything we just ignore this inte of DV would be V integral of this is so we kind of we kind of skip that little part just to make it easy it's easier to write it question theine at the end of should be negative no because we're taking a derivative right so the derivative of s is positive cosine the integral of s is negative cosine that's where we got that from but the derivative of s is not negative cosine is posi sign get them straight in your head okay we got to have those right and some of you made that mistake on your last test we got to get these things right you guys all clear with this one I'm going to clean it up just a little bit so uh what we're going to clean up is we're going to do probably pull the negative out front sin to n-1 x cosine X tell me what happens here what happens becomes it does cuz minus and negative that's going to become a plus also one of these things is a a constant what is it cosine X is not a constant that is a constant yeah nus one that's going to be a number right actually comes right from here so we're going to have plus nus1 integral Also let's clean this up a little bit I've got a cosine X I've got a cosine X that becomes let's see what we've done negative comes out that should be easy this becomes a plus no problem this is a constant pull out in front of your integral cosine x cosine X is cosine 2 x and then sin nus 2 power of x n your head if you're okay with that one are you still okay okay it's just a little bit of algebra that we've done now here's the deal uh oh man this right here is a problem because if we do a substitution with either of them it's not going to get rid of the entire thing we don't even know what that power is you with me so what we're going to do is we're going to trick the problem a little bit uh we're going to call this something different we're going to call this using one of our identities remember that identity it's Pythagorean identity for S and cosine so here we go we know that this thing equals sign n -1 of x * cosine X plus nus one that okay so cosine squ no no no no let's not do cosine squar what we're going to do here is we're going to replace that with sign so we have SS and signs and you're going to see Y in just a little bit remember that you've done some of these uh integrals probably on your homework where you have to make it down remember adding the integral to both sides we get the integral back again you're going to see that here in just a minute we're going to get this integral back again remember what happens when we do that when these things recycle like s and cosine does when they recycle go oh man I'm getting the same thing back is that a problem no just add it to both sides and you're good to go you follow so we're going to do that in just a minute so cosine squ no let's change it this is going to be 1 - Sin 2times Sin N - 2^ x DX let me know if I missed anything I'm wrri a lot of stuff hopefully I haven't yall still okay yeah okay tell you what next thing we're going to do next thing we're going to do is we're going to distribute this because I don't like this right now I'm going to distribute this so we're going to have what's that going to become when I distribute this sin 2 x what do you do when you have exponents being multiplied let's just add them how much are you going to get okay and remember this is in here make sure I'm on the my right track here okay you still okay with that so far yeah we are almost to the point where we can start add other side you see we have it right here right but this is wrapped up in this integral so we're going to split up this integral just one big thing that you need to notice this is n minus one time this entire integral does that make sense which means if I split this up it goes to not only first one it also goes to the second one is that clear so when I do that we'll get Okay negative sign N - one x cosine X nothing changes Plus n -1 * integral of sin n - 2x DX don't forget that DX here's what happens so this it's basically just like you dro the parentheses because you're Distributing this to this first integral no problem what's the next thing I'm going to write minus not integral nus one yes in parentheses then integral what's in the integral s of I that's it now this is cool is weird right we're doing this in general uh we didn't start with any number nothing that's this kind of neat so we worked this all the way down we now have this there's no integral there here we got an integral but that's okay we're creating a formula okay this part what is this part it's what we started with yeah it recycles recycles okay so every time you soon this before I seen I think we've seen it like three or four times now in this class because I want you to get used to it this stuff happens especially on your type of homework that you're going to be doing what do we do to get rid of this piece add it to both sides it's an equation this we have proven that it equals this entire mess of garbage so if we just add this nus1 * the integral of s to the n^ X if we add it to both sides let's think about what happens let's do the the right side first on the right hand side this entire thing is gone we have negative s to the nus1 power of x cosine X Plus nus1 integral s of nus 2 power of X DX this is gone you guys right with that one yes no yes no you sure this is all the same right all the same we just added this to both sides therefore it is gonzo simplified now over here this is the interesting one how many integrals are here this just here how many are here so the number up front is one correct one how many are here n- one add them what's 1 + nus one n so then write in Black so we can see difference so that we know n * this integral equals this stuff now every time before when we've had a number out front remember n is just a number it's like two or three or four or five or six or so what do we do with with that let's just if it was three we divide by three if it's n just divide by n so if we divide by n what we end up getting is -1 n sin n -1 x cine X Plus n -1 / n the integral sin n- 2 x DX of course plus C because we're talking about an inte but that will come at the very end let's make sure I did it right do you guys see where the one/ n's coming from dividing by n here we had n minus one well now we're dividing it by n there's only two terms this one and that one this right here this is the reduction formula for sign now if you're thinking that's a lot harder than just doing the problem it's really not I'm going to show you do you want to see an example of how this stuff works I'm going have to race the board though so show up hands and feel okay with the proof wait just I'm just going to erace this side are you guys okay with getting this far didn't make sense on how we got that far you just you just have to use it I just want to make sense uh to you where it came from that it wasn't just magic okay so reduction formula I'm only going to give you one example on how this actually works I think one example will make it very clear on how it works works so here you go is it kind of a cool little proof though it's interesting right it's really nice little proof add both sides of you've been practicing divide you're good to go kind of neat and what this allows you do is this if you got an integral of sin to 4th power of X DX s 4th power of X DX you start thinking you wait a minute how in the world am I going to do that problem I I I don't know I mean does it fit the integration table the way it is right now does it will a substitution work no no remember if I pick sign the derivative cosine it's not up there so that would be a problem integration by parts will it work yes yes it will but we've proven that we can actually do this every single time we have any power greater than one or greater than or equal to two we'll check it out here's how it works uh what's your what's your n so this is sign to the fourth so our n would be four here's how you work it it's really nice what this interval is going to equal let's read it it's going to be negative 1 over what1 over 4 what's the next thing I write sign of where you getting the three from there 4 - x cosine X so let's see n was 4 so 1 over 4 got it sin 2 4 - 1 is 3 x cosine X and we're going to put a plus again yeah n is four so N - 1 that's 3 over 4 what's the next thing we have it's an integral you're still going to have an integral that's why it's called the reduction formula because it's going to reduce the power of the sign by two every time do you see that by two so instead of having s 4th now we're going to have an integral it says of s good yeah n is 4 4 - 2 is 2 and it just made it a little bit better isn't that kind of nice is it easy for you it's just a formula so it's kind of nice that way you don't have to waste a lot of time on integration by parts make sure I got that right questions so then once we get the integral of that we can still keep going by pluging in that's the question does this now fit this model shoot do it again as long as the n's greater than two greater than equal to two if it's one you I you're good to go so if this is a power three you'd be done right now because you'd have the integral of just s to the first right integral of s is just negative cosine hey you're done so we do just do this one more time so we rewrite everything U it's just this little piece remember this is in a bracket some of you are in the in the very bad habit of not writing parentheses parenthetical expressions in mathematics are absolutely crucial uh it allows you to distribute allows you to see what's being multiplied order operations wise what's being added subtracted and not being multiplied okay so they're very important so when we have this thing when we do this this little segment well we can do the same thing just be real careful that you know this positive free force is going to distribute across both of these terms does that make sense that's huge so for us let's do why don't you do it just do this little piece sin^2 X do it just like that okay just write out what that little piece is okay what's our n again people 1 what's n so just plug in the two everywhere you see the N yeah 2 - 1 is one that's cool cosine X I have to race this part now plus plus okay 2 - one that's 1 / two integral of please be careful when you do this uh how much is what's our n what's 2 - 2 what's anything to the zero power don't you be putting a sign there that doesn't make any sense at all if you put sign then this would have had to be a three does that make sense yeah hello yes no yes sign some of you it doesn't make sense like you're telling your eyes 2 - 2 anything that's zero one yes you don't even really need the one there but we can have it so this would become -2 sin x cosine X plus 12 hey what's the integral of one there we go now just keep in mind that this whole thing there's a 34s in front of it and then Plus in front of that and a cosine X and A S 3 power of X and A4 this has to distribute which is not a problem you just can't lose track of it because that would be a problem quick head not if you're okay with that so far do you see how much faster this makes integration by parts if it's just signed of some power that's kind of nice so here we go we've got Final Answer 1 1/4 sin 3 power of x cosine x - 38 hopefully you see where that comes from sin x cosin X plus 3 38 x and lastly what do we always have when we're doing indefinite integrals yay there we go cool kind of cool right work anytime you have just a power of sign to some power greater than equal to two that's the idea show hands feel okay with with that one awesome now we are actually done with Section 7.1 finally oh my gosh thank goodness it's taking us so long it's taking us like a week Michael is there a reduction formula for the other function for cosine yeah I think you can create one for cosign I haven't done it uh I just want to see that this was possible much I bet you guys could work on it see if it is possible