| Start | I know a lot of you guys are having trouble right now. You're looking at the integral and you're saying "how do I know what to do?"
You know what to do but how do I know what to do?
Well remember, I've done more than 10,000 of these, probably on the order of say 20,000 of these over my lifetime. So, you know, I have a bit of a head start on the rest of you. The idea with the integration, oh, you start it Shannon? Yea. Good. Ok, so the idea of integration is, we're gonna give you a bunch of toolboxes. So you, as I've said before, you have a function. |
| 0:30 | You look at the derivative of a function
It's easy to find the derivative now. You should be able to find the derivative of almost anything.
But going backwards is tricky. When you look at the function and say what was the original function, what is this the derivative of? It's much harder. And there's not really so much theory at this point, so understanding is a little tricky. You really just have to start seeing the patterns. And that takes time. And that's part of why you practice. So first if you develop the mechanical skill, later the other skills will appear. |
| 1:04 | And you'll say ah, I'm starting to understand
why I use this technique for these kind of integrals and a different for other kind of integrals.
And I know it's frustrating, so that's why you just do lots of practice. So, as I said I'll put up stuff from my book to give you more to practice. And I'll see if I can find other practice problems for you. And then at some point, you've really done them all. And there really aren't any other types left. So one of the things we're gonna do today is gonna involve lots of "I never would've thought of doing that" kind of stuff. |
| 1:33 | So in the beginning you look and you go "well how do I know to do that?"
And the answer is well this is what you do.
And then as you get better at it, you'll develop more technique. And then at some point you'll be showing somebody and they'll go "how do you know to do that?" And you'll say "well I'm a genius." And, but really you've just done it more times than they have. Any questions before we get going? No. One of the things is you're gonna wanna remember a bunch of your trig stuff. |
| 2:00 | So I'm just gonna put them down here again for everybody's use.
So sin^2 of an angle plus cos^2 of an angle always equals 1. That could be rewritten, I'm gonna do this quickly and then it'll just sit here all class. As either this or this. Those are all saying the same thing. Furthermore, 1+tan^2(x) = sec^2(x). |
| 2:30 | So if you see sec^2 you can replace it with 1+tan^2.
If you see tan^2, you can replace it with sec^2(x)-1. We're actually not really gonna need that top one. So I'm gonna save that space. You can have it in your notes but it doesn't matter. So if you see cos^2 you can replace it with 1-sin^2. If you see sin^2 you can replace it with 1-cos^2. But sometimes you're gonna want a different substitution for this. |
| 3:01 | Which I'll show you in a second.
1+tan^2=sec^2, 1+cot^2=csc^2. Now, I can derive the next two or I can just show them to you. Who wants to know where they come from? Yeah, there's always a few. So, there's the double angle formula. |
| 3:30 | Which says cos2x is cos^2-sin^2, but is also 2cos^2(x)-1
and 1-2sin^2(x).
So you could rearrange this with a little algebra to get the following. sin^2(x) is 1- cos2x all divided by 2, or times 1/2. |
| 4:02 | And cos^2(x) is 1+cos2x, I'll rewrite this so it's a little neater.
Ok? So if you just, if you take these and you play with them you get these. So sometimes you're gonna see cos^2 and you're gonna wanna do this substitution. And sometimes you're gonna see cos^2 and you're gonna wanna do this substitution. And you say, how will I know? And the answer is well one will work and one will not work. |
| 4:32 | That's really what it is. And as we do them I'll show you and I'll say, well now we're gonna use this one, now we're gonna use the other one.
So what's gonna happen today is we're gonna do a whole bunch of trigonometric integrals. And integrals that have trig inside of them, and then you're going to rewrite the integral to be something that you can now do. Ok, by playing with the integrals. So we need a couple others for our toolbox. So do you remember how to do the integral of tangent? |
| 5:01 | I think we did it in class.
But let's do it again. You wanna do the integral of tangent of x dx. We're gonna make that sin/cos. And now we could do u substitution. So, how would I know to do that? Well otherwise, what are you gonna do with tangent? |
| 5:31 | You go I have no idea, so you might as well try something.
So you say well let's, let's try this. Let's see if it works. You say well now, I could let u=cosx. du is -sinx dx. So if we substitute in |
| 6:02 | we have cosx in the denominator. Now how do I know to make u cosine
and du sine and not the other way around?
Well if I let u be sine, du would be cosine, and I'd have u in the numerator and du in the denominator. And you have no idea what to do with du in the denominator. So, I mean you can't have du in the denominator. du has to be up in the numerator. At least for this stage of your life. Ok? So, I substitute the other way. And I put u down here. |
| 6:30 | And since du is -sinx, -du, positive sinx.
And now that is just log of u. So this is -ln(u) + a constant. And now you substitute back. Ok? One piece of chalk down. See how many I can go through today. |
| 7:06 | So far so good?
Alright, so cotangent would be the same way. Instead you'd make is cos/sin. You'd let u=sin and du would be cosine. So I'll let you practice cotangent on your own. Cause once you do tangent, you should be able to do cotangent. Agreed? Now for a fun one. How would you do secant? |
| 7:35 | The integral of secx.
So let's see. Well let's say if I make it 1/cos. The problem is if I make it 1/cos, right, I'm gonna have cosine See if I do this, this is wrong, so don't copy this. If I'm gonna do this now what? I'm not better off, right? I can't let u be cosine. |
| 8:02 | So we don't want that one.
If I make it cos^-1 it's the same thing. Cause I need a sine somewhere. So if I'm gonna have a cosine I need a derivative of cosine. So here's something that maybe you would've thought of, but probably not. Why don't I multiply the top and bottom by secx+tanx? And you go, you've gotta be kidding me. Well, I didn't think of it the first time either. Probably not the second time. But somebody came along, you know, 200 years ago and said I don't know how to do the secant of x. |
| 8:33 | And then some annoying math guy said hey here's a clever trick you could do.
Because this will now become and the answer is, how do you know you could do that? Well, I just did. That's how you know you could do it. Because it works. And a lot of these, the answer is because it works. Yes, there's a reason why. And as I said as you start to do more of these, you'll start to understand why. The answer is you can always multiply by 1. |
| 9:00 | So, secant times secant is sec^2.
Secant times tangent is secant tangent. The bottom is this. Ok. What's the derivative of secant? Secant tangent. And what's the derivative of tangent? Sec^2. So now you can let u be the denominator. And du will be the numerator. And I know, you're gonna say I never would of thought of that. I know, but now I taught it to you. |
| 9:33 | There's lots of things you never would've thought of.
So if I let u be the denominator. The good news is there's really only two types of this. So the odds that it shows up in any real way are very small. |
| 10:02 | We really can only give you secant and cosecant.
So I'm gonna show you secant and you do cosecant for practice. Ok? But the thing is you may need integral of secx for something. Or you may need the integral of tanx. Like when you're doing integration by parts. Sometimes one of the parts that's left is secx. So then you just know what this is gonna become. So now you got du on top, and u on the bottom. |
| 10:32 | That's log of u.
And now we substitute back. So that's ln(secx+tanx) + a constant. So these, you should be able to derive these. And then you memorize them. So mathematicians tend to fall into two groups. Ones who, every time they see it they have to figure it out and derive it. And others who they've done it so often that they just say well I just know what it is now. |
| 11:03 | Ok?
And you'll be in one or the other. You're gonna see this when you do, I mean you guys are doing science. You'll see this in science all the time. At this point in these, I've seen it too many times. But in the beginning, yeah I would sit down and refigure that one out. And then if you do it 3 or 4 times then you know the trick. So, so don't despair. So I'm gonna copy those over here so we have those written down. And as I said, and you should practice those. |
| 11:32 | And you should practice doing cosecant.
It's the same trick. You just multiply by cscx+cotx over cscx+cotx. So this is -ln(cosx) + a constant. A small number of books will write this as positive ln(secx). I'll leave you to figure that out on your own. And this is ln(secx+tanx) + a constant. |
| 12:11 | Ok? So feel free to memorize.
Nothing wrong with that. After all you've memorized the Pythagorean Theorem. Yes. Would you have to show work if this was on the midterm? I would show the work for one simple reason. If you mess it up, ok, you won't get any partial credit. |
| 12:33 | Um, there's not much you could do with this for a midterm.
Because it's just this, right? So, more likely there will be a problem where part of the problem will have you figure that out. I was showing somebody this earlier today. Where you get to a stage where this is part of the integration by parts. And there, you can just say well I know that's log of secx+tanx. Sound good? So now, let's have some fun. |
| 13:13 | I know you guys, you thought you were done with trigonometry forever.
But you're not. Just when you thought it was safe. So we did something like this already. Where you said I would just let u=sin and du is cosine. |
| 13:34 | Right?
We know how to do that. What if this was cos^3? Now, all of a sudden it's annoying. Before it was easy, because if it was just cosine, you had u as sine and du as cosine. Well, look at those. So leave the sin^4 alone. |
| 14:01 | And break the cos^3 into cos^2 times cos.
I personally think these are kind of fun. They remind me of the trig identity stuff back in advanced algebra trig days. If you thought those were fun then you're gonna have a good time with these. Course you might not have many friends. Or any friends. I'm not gonna go out tonight I'm just gonna sit home and do trig identities. That didn't make any friends. |
| 14:34 | Very sad.
Alright, now cos^2 is 1-sin^2. Now I could do u substitution. What am I gonna let u equal? |
| 15:00 | How do I know u is gonna be sine and not cosine?
Well if u is cosine then du has to be all of this, which is not gonna happen. Ok, but if I let u be sinx and du is cosx dx, then this integral is now u^4(1-u^2) du. |
| 15:30 | Isn't it?
Now you just distribute the u's to make that u^4-u^6 du. And that is u^5/5 - u^7/7 +c. |
| 16:00 | By the way, comment on your midterms, some of you guys were very, sort of, confused/sloppy/got in kinda the wrong spot.
With the phase from when you have the integral to when you don't have the integral. You'd leave a du in here. Or you had a plus c in here or something like that. So be careful when you're doing the steps. Ok, because you don't wanna get the TAs too confused. Now we substitute back. So this is sin^5(x)/5 - sin^7(x)/7 + a constant. |
| 16:40 | And you would've never have looked at this and said oh, it's gonna turn out to be that.
Ok, so as I said, part of what makes integration so difficult is there's no intuitive way to look at this and turn it into that. Now once you take cosine and you get to this step then it might be more obvious. |
| 17:01 | But not, not on the first step.
Ok, so that was a big thing with trig altogether, is you see something in trig in one form and then you play with it, and now it's in a new form and it was much easier to work with. Yes. Ok, so u^4 times 1 is u^4. u^4 times u^2 is u^6. Ok? |
| 17:30 | Yes.
How do you know when to do u substitution? With these problems, most of the time you will be able to get to u substitution. Cause what'll happen is after you rewrite the integrand, you'll have something with sine or cosine or whatever. And something with the derivative of it, and then you can play with it. Sometimes it's gonna be integration by parts. And I can't give you a, you have to play with them and do a bunch of them until you start to see. |
| 18:02 | Alright?
Let's do some others. I'm gonna erase this in a minute. Yes. Can you do, u^4 - u^6 is not u^-2 or u^2 or anything like that. Ok. Let's do |
| 18:38 | Alright.
What if I just have plain cos^2? I'm gonna use an identity. So if you have cos^2, I might use this identity right here. |
| 19:02 | So yeah I, well these you should definitely memorize. I mean you're supposed to know these.
These two I recommend you memorize. Ok, or you can derive them from the double angle formula. Ok, but I recommend that you memorize these. The only difference is sine is 1- and cosine is 1+. Why is that gonna be helpful? Well right now you don't know what to do with cos^2. You can't do anything with parts. With integration by parts, what happens is, if you haven't noticed, you have two terms in the integrand. |
| 19:35 | So if one of them is x to a power, each time you do integration by parts x goes to a smaller and smaller power.
And then at some point x is gone. When you have e on the other hand, or sine or cosine, e just keeps staying e. So the derivative of e^x is e^x is e^x, that doesn't help. We're integrating it. And sine and cosine just go back and forth, from sine to cosine to sine to cosine. So, you're not really getting much value out of that. So here, |
| 20:01 | you can't do u substitution because there's no sine in there.
And you can't do integration by parts cause, what would you do? So you could do this substitution, however. And make this, put the 1/2 on the outside, 1+cos2x dx. So that's you're substitution. Yes. Ah, why not use the other one? Cause if you made a 1-sin^2 you wouldn't be any better off. Ok? |
| 20:30 | So how will you know which one to do? Well one clue is if you just have sin^2 or cos^2 alone
and you don't have anything else in there, then you're gonna use this.
These. If you have both sines and cosines you usually will use those. But not always. It's tricky, it kinda depends also on whether the power is even or odd. There's sort of families of these, and I don't wanna do too much at once. And I don't wanna get too intense because that's really for 132 not for 126. |
| 21:03 | But ok, this is now relatively simple to integrate.
This is 1/2 and what's the integral? Well the integral of 1 is x. And the integral of cos2x is sin2x/2. Plus a constant. Somebody's having fun. So far so good? |
| 21:35 | Alright so now how do we make this more annoying?
Once you can do cos^2 of course you should be able to do sin^2. What if I gave you cos^4(x)? So this is gonna be the first one you get to do on your own. The way you're gonna do cos^4, I'll give you the first step. Is it's cos^2 squared. And then you're gonna use that substitution. |
| 22:01 | So you're gonna have to FOIL it out.
And you're gonna have to do a substitution a second time. So this is gonna take you a couple minutes. Yes. How do I get- ah, well the antiderivative of cosine, right, is sine. And then you have to divide by the derivative of the term. Ok? Cause it's sort of the opposite chain rule when you're doing integrals. Right, if you're doing the derivative you would multiply by 2, since you're doing the integral you're gonna divide by 2. |
| 22:31 | Alright so that's gonna take you guys about 5 minutes.
See how you do. Let's give ourselves some room. This will take a bunch of steps. I'm gonna erase this so I can give myself some room. As I said, it takes practice. It's very algebraic. It's lots of manipulation of stuff so it takes- be careful. |
| 23:03 | So we had this.
Ok. So what was our substitution? Well the substitution for cos^2 is 1/2 of 1+cos2x. So this is gonna be [1/2(1+cos2x)]^2 So first thing is you're gonna have to square the half. |
| 23:36 | So 1/2 squared is 1/4.
And it's gonna be 1/4 times the whole thing, so you can put it inside the integrand, you can put it outside, it doesn't matter. And then you're gonna get (1+cos2x)^2, so that's 1+cos2x times 1+cos2x. |
| 24:00 | So you're gonna have to multiply that out.
So far so good. Ok. 1 times 1 is 1. Always got that one. Cos2x and cos2x is 2cos2x. Plus cos^2(2x) dx. Ok, so when you multiply that out |
| 24:40 | Alright, so now we have to integrate this.
So this integral is gonna be easy. That's just x. This integral is gonna be easy. So let's split this integral up. We're gonna have 1/4 times the integral of 1dx. Plus 1/4 times 2 the integral of cos2x dx. |
| 25:01 | Plus 1/4, the integral of cos^2(2x)dx.
Ok, so I distributed 1/4 and I'm making this an integral, this an integral, and that an integral. Yes. If you did 1-sin^2 in the beginning, you would be in the same spot. |
| 25:30 | Cause all that would happen, if you used 1-sin^2 here
and you square it, you're just gonna get 1-2sin^2 + sin^4.
You're just gonna have the same kind of stuff. So what you're trying to do is get rid of the square. The squaring, and you try to reduce the power down to just a sine or a cosine. Cause you know how to integrate sine. You know how to integrate cosine. So, if you can get the powers from the 4th to squared to no power, the power of 1 Then, then you can do it. Ok, so that's what you're trying to do. |
| 26:03 | Alright this is gonna require another substitution. So let's just do this integral.
It's just 1/4x, that one's done. 2(1/4) is 1/2. And that's sin(2x)/2. Alright so let's do this one. Now cos^2(2x), well we did cos^2 a minute ago. That's gonna be now (1/2)(1+cos4x)dx. |
| 26:35 | How do I know it's 4x?
Because it's this formula. Whatever this variable is, this is double it. So if this was, if this was 10x, this would be 20x. Ok, if this was 100x, this would be 200x. It's a double angle formula. Ok? So that is 1/2 of the integral of 1+cos4x dx. |
| 27:07 | And you know what that integral is.
This is gonna be times 1/2 of integral of 1 is x. Integral of cos4x is sin4x/4. Plus c. And then you could distribute the 1/4 and there's some simplifying you can do and all that stuff, but that I'm not concerned about. |
| 27:30 | Where'd the cos4x? Once again, this is the double angle substitution.
So when you see this, ok, when you see the cos^2, it's 1/2(1+cos) of double the angle. So notice you went from squared to non squared and the angle went from x to 2x. So here you had cos^2(2x), and you get to 1/2(1+cos(4x)). So, this is squared and 2x, this is not squared and 4x. Yes. |
| 28:01 | Well when I'm all done, right, the 1/4 is out here.
Ok, so this is now (1/8)x + (1/2)x, this is actually (3/8)x. Ok, (sin(2x))/4 and (sin(4x))/32, so if you put it in WolframAlpha that's what it's gonna spit out. Ok, it's gonna give you the final form of it. Which you may not come up with and frankly I don't think you need to. Obviously, what if you had a cos^10? |
| 28:30 | Ok, you'd have to do this like, many times.
So we would never ask you that, cause that's just sort of sadistic. Ok, but by the way there are patterns to this though. So if you look at tables of integrals, and now the computers have them programmed, ok. There's a formula, and if you do this enough times you'll start to see and then eventually you don't have to actually do these out anymore. But this class will never go there. I'd say fourth is as annoying as it gets. So far so good? |
| 29:01 | Yes.
Ok, so we did this last time. So if you have cos2x the antiderivative of sin2x divided by the derivative of 2x. Which is 2. Ok so it's the chain rule for integrating. It's going the other way. Yes. This one. This is this. Ok, so the antiderivative of 1 is x. |
| 29:30 | The antiderivative of cos(4x) is sin(4x) divided by 4.
I'm sure very few of you are able to do this. So you practice it with sin^4. So I'm gonna give you a similar one to practice. Ok. Question. Right. Remember, this angle is double that angle. Ok, so when I have cos^2(2x), ok, it's (1+cos(4x)) times 1/2. |
| 30:04 | Ok?
Cause it's two 2x's. Alright, I'm gonna erase this. It'll be on the video. Take a picture. Copy it from your friend. Give up. Any of those. I give you one very similar. And let's see if you can execute it this time. Then we move on to another type. |
| 30:45 | Now, there's actually two different ways you could do this.
Ok, but let's practice doing it this way. So now, you're gonna wanna substitute for sin^2 and for cos^2. |
| 31:03 | So it's the same substitution except you have two different ones. You have one for sin^2 and one for cos^2.
And then the technique should be very similar. You can't have very many of these in a test, they're 15 minute problems. Ok? And nobody wants to grade them. |
| 31:31 | I just write 0.
If I don't see the right answer I just put 0. It's alright. Don't want you to be my doctor, anyone. Yeah, you throw 'em in the air. Face up, A. Face down, F. That works. We'll give you the first step. Give everybody the first step. Ok. |
| 32:03 | Ok.
Alright, so now we could distribute this. And (1-cos)(1+cos) is 1-cos^2(2x). This one's actually kinda easier. |
| 32:40 | Do we agree? 1/2 times 1/2 is 1/4.
And (1-cos)(1+cos) is 1-cos^2. Which by the way is sin^2. If you wanted to you could make it sin^2. But you don't have to. At this stage you could do whatever you want. But I would make it sin^2. |
| 33:02 | Now if you're really clever you know that the double angle formula for sin(2x) is 2sinxcosx. You could've gotten there faster.
But that's ok. I can't expect you to remember all the trig identities. Now from sin^2 we use that identity. 1/2(1-cos(4x))dx. |
| 33:32 | Why cos(4x)? Because that's 2x.
And by the way you didn't need to do, you don't have to convert this to sin^2. You could've made this 1/2(1+cos(2x)). Doesn't matter. In the end you have to end up in the same spot, right? They all have to come out the same, otherwise you made a mistake. Or something funny is going on. |
| 34:00 | So (1/4)(1/2) is 1/8.
And the integral of 1, we don't even need this anymore. The integral of 1 is x. The integral of cos(4x) is (sin(4x))/4. And then you could distribute the 1/8 if you want. Aren't these annoying? Yeah. You'll get there. |
| 34:31 | We're gonna do an easier one now.
Yes. Where'd I get the 1/2 from? Which one? This one? That's from the sin^2 identity. Remember sin^2 is 1/2(1-cos(2x)). Yes. |
| 35:01 | You mean why'd I get here?
No because you can divide by the derivative. You divide by the derivative. When you took the derivative of sin(4x) you get cos(4x) times 4. Ok, so if you do the antiderivative you get -cos(4x) divided by 4. The reason is when you do the derivative of this, you're gonna get a 4 from the chain rule. That cancels the 4. You get 2 here. Ok? I know, you guys keep asking me that one. |
| 35:31 | Alright, let's do a totally different type.
These are fun. |
| 36:00 | Ok.
What if I wanted to do that? So now of course you could put that in sine's and cosine's, but that won't help you. It won't help to put it into sine. Try it. See what happens is, people came along and they said "I'll put it in sine's and cosine's." They did that, it didn't work. |
| 36:33 | So now you need a plan B.
Ok. And somebody said well, I know the derivative of tangent is sec^2 but I have sec^4. So could I do anything with that? I only wanna have sec^2, cause the rest is tangents. So what if I make this tan^4(x) and I break this into sec^2(x) sec^2(x) dx. |
| 37:10 | How is that gonna help?
Well because 1+tan^2 is sec^2. So you get tan^4(x) times (1+tan^2(x)) sec^2(x)dx. |
| 37:32 | Now why don't I change the tangent?
If I change the tangent, try it, and it doesn't really help. Now why did I change them both? If I made both of these 1+tan^2's, I'm not gonna have any derivative running around. Because this is the derivative of tangent. So now I can use u substitution. I let u=tanx du is sec^2(x) dx. |
| 38:00 | And I get u^4(1+u^2)du.
See what I did? There are patterns to these, as I said. After you do a bunch of these you'll start to see them. Yes. |
| 38:35 | That's what I did.
Oh, can you always do that? It depends on- can you always split the sec^4 into sec^2 sec^2? It depends on what else is in there. Ok, but the derivative of tangent is sec^2 so you only need a sec^2. Ok, so you wanna take the other sec^2 and put it into terms of tangents. So if this was sec^10, ok, you'd get sec^8 times sec^2, then this would be (1+tan^2)^4. |
| 39:07 | But then you'd just make it (1+u^2)^4 which you can multiply out with a little pain.
Because this, you have to be pretty sadistic to make it too big a power. But that's not hard. You could distribute that u^4. That's (u^5)/5 + (u^7)/7 +c. |
| 39:31 | So it's (tan^5(x))/5 + (tan^7(x))/7 +c.
As I said, you would never have looked at (tan^4)(sec^4) and said oh, it's gonna turn into this. Ok, so these techniques help you turn the integral from something that you have no idea what to do with to something that you do. |
| 40:00 | And there's gonna be families. As you start to play with these, you'll see it matters a lot whether the powers are even or odd.
That helps you figure out which one you're gonna do. I'll show you why. As soon as we get this one copied down. Aren't we having fun today? This is your punishment for not coming on Monday. I know, some of you were here. And some of you were studying your biology, I hope it was worth it. But you hurt my feelings. |
| 40:32 | Yeah, there's that too.
You might just find this fun altogether. So this Friday when I'm staying home and doing trig integrals, you can join me. Yeah. Whoever gets it wrong first has to drink, so But it's only water. Cause you're under 21. Alright, let's do a similar one. But not the same. Instead of (tan^4)(sec^4), what if it was (tan^3)(sec^3)? |
| 41:01 | Yo seriously?
Haven't we done enough of these already? tan^3, you're welcome. sec^3 So what are we gonna do this time? Cause now if you look and you say if I break out a sec^2 I've got 1 secant left over and I can't turn that into a tangent. And if I take the tan that's not gonna help. So here's another idea. |
| 41:31 | What's the derivative of a secant?
Secant tangent. So let's make this tan^2(x)sec^2(x) times secant tangent dx. How do I know to do that? Cause these are odd powers not even powers. Lost one. |
| 42:02 | That's alright. We'll see her next week.
Now I can take sec^2, tan^2, I'm sorry and make it sec^2-1. Just like the last one. Why would I do that? Because the derivative of secant is secant tangent and now I can do u substitution. |
| 42:36 | So let u=secx.
du is gonna be secxtanx. So if u is secx du is secxtanx dx. Believe me, first time I went through a lot of these it was just... by the way we've only made them x. Wait until we make it πx. |
| 43:01 | Or (π/4)x.
Or if we really hate you, π/17. Or something really annoying like that. I would never do that. But, you know if you just throw in some constants it just gets messy. Ok, it doesn't test whether you know what you're doing. So now this integral becomes (u^2 -1)(u^2)du. Isn't that u substitution great? |
| 43:33 | Cause if the derivative, if 1+tan^2 is sec^2, then tan^2 is sec^2-1.
Alright, just move the 1 to the other side from this one. Notice I'm not doing the cofunction ones. They're the same. So that's u^4 - u^2 du. That's (u^5)/5 - (u^3)/3 + a constant. |
| 44:06 | Substitute back.
(sec^5(x))/5 - (sec^3(x))/3 + a constant. How do I know when I'm done substituting? When I really play with the integral, when I've got it to a point where I say "aha" either with u substitution or some of the rules I know, I can now integrate it. |
| 44:35 | I know.
I know these aren't satisfying answers. If you do these, there's families of them and you start to see. Well it's cause, why is it (u^5)/5 all the time? Cause I'm doing these cubes and squares. If I gave you higher powers you'd get higher powers in the answer. I know, they're really fun. |
| 45:00 | Alright.
You got this one? Let's do, should I do the one I did before? Where do I plug in u? Well I got to here. Ok, so if secx is u this is u^2-1. |
| 45:31 | That's u^2, and that's du.
So far so good? Alright, let's go back to a relatively easy one and have you practice. Everybody do |
| 46:02 | I don't believe I did that one yet, did I?
Ok. One clue. Is when you see even powers in sines and cosines to begin with, you're gonna use these. And when you see odd powers, you're gonna use those. Ok, so we'll give you a step for those of you who didn't figure it out. |
| 46:32 | You can make this sinx sin^2(x).
And then substitute for sin^2 and then you can do u substitution. Higher power would be u and lower power would be du. Like I said, if it's an odd power, you're gonna use the 1/2(1-cos2x). Because it's even, I'm sorry, even power you use that. |
| 47:01 | If it's an odd power use the other substitution.
Got that on recording, right Shanny? Yes. Hi. Hi. Alright, so from here sin^2 you can replace with 1-cos^2. And now don't take 1-cos^2 and put it back into sin^2 and get back where you started. |
| 47:36 | Let u be cosx, du will be -sinx.
So if u is cosx, the higher power, du is -sinx dx. So we get negative integral of 1-u^2. |
| 48:02 | Because sinxdx becomes -du.
Yes. How do you know when to do the- Ok so I'll repeat it, because people obviously didn't hear it. When you look, it's ok, when you see the integral and this is an odd power, you'll end up breaking it apart into two parts of sines and cosines and using these substitutions. When this is an even power, you'll end up breaking it apart and using these substitutions. |
| 48:32 | So as I said, one of the things about the families, and if you look in the book
and I'll put some stuff up online, you can see that it breaks it down basically into whether it's even powers of sine, even powers of cosine
odds, mixes, tans and secants, there's about 6 or 7 families of these.
Ok, and once you see how to attack each family it's very straightforward and then you'll get very good at these. So, how do I know to do this? When I see sin^3 |
| 49:02 | I get a sine and a sin^2. Since I know I can now turn this into cosine, I'll be able to use u substitution because I'll have cosine and I'll have sine.
If it's even and you tried it it just doesn't work. So that's how you know to do this. This is -u + (u^3)/3 because I distributed the minus sign. And then you substitute back and get -cosx + (cos^3(x))/3 + a constant. |
| 49:41 | Yes.
Why do I let u be cosine instead of sine? Well if I let u be sine this would be 1-du^2. Which wouldn't be very useful. So in general when you see powers, higher power will be u and lower power will be du. Ok, in general. |
| 50:02 | Alright I got one more fun one for you guys.
Should I do it Cecelia? Should I do sec^3? Do I hate them enough? Alright, because I hate all of you guys, we're gonna do sec^3. This is an entertaining one. So And if you don't, if you wanna try to do this on your own, it's really fun. But we'll do it together, I won't start you. |
| 50:39 | See when you watch this on video you might wanna pause.
But this'll go up on video and then eventually it'll be closed captioned. Which is keeping Shannon very busy. She's having lots of fun with that. Ok, what do we do for sec^3? Well what did we do with sin^3? We broke it apart, so let's break this apart. |
| 51:10 | And sec^2(x) is 1+tan^2.
So How is that gonna help? |
| 51:31 | Well let's distribute that.
secx, because we can do the integral of secx, it's that. dx plus integral of secx(tan^2(x))dx. So I'm gonna rewrite this up here for a reason. What did I not do? |
| 52:02 | Nope.
Ok, I got two integrals there. |
| 52:31 | Wait, wait. I may have made a slight mistake.
I'm sorry. I'm sorry. I forgot. Just start over. Rewind the video. I left out a step in the beginning. Sorry about that. Yeah this is a confusing one, I promise. I know, tear that page out of your book and throw it at me if you want. I know you're thinking of it. Somebody will probably do it if I stand here long enough. Cause then I'll turn around and I'll say who did that? Alright. |
| 53:02 | Start again.
My bad. Integration by parts. Gotta do integration by parts first. Which is gonna be which? Well I'm gonna let u be secx. dv be sec^2(x). Why? Because if I let u be sec^2, the derivatives a mess. And if I let dv be secx I get this log thingy, that's not gonna make my life easier. On the other hand |
| 53:30 | here, du is secxtanx dx.
V is tanx. Ok, that means this is u times v secxtanx |
| 54:01 | minus integral of vdu, which is secx tan^2(x)dx.
So you look at that and go have I made it easier? Kind of, I went from cubed to squared. Now what do you do? Well This is I need to take the tan^2 and I gotta make it sec^2(x)-1. |
| 54:45 | So to rewrite this, I now have the integral of sec^3(x)dx equals secxtanx minus secant times sec^2 |
| 55:04 | plus, cause minus secant times -1, plus integral of secx.
And remember that trick with adding back to the other side that we had the other day? I got -sec^3 + sec^3, so now I've got 2sec^3(x)dx. Isn't this one a great one? Secxtanx + secx dx. |
| 55:34 | Now it's easy because we have integral of secx. We memorized that.
So 2 integral of sec^3(x) dx secxtanx + log(secx+tanx) + a constant, and then just divide by 2. |
| 56:00 | There's a fun one.
And as you saw since I had a false start, I, you know I sometimes have to go through these a couple times to remember the trick also. Alright, I think everybody's had enough for one day. We'll see you on Monday. |