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Use an appropriate substitution and then trigonometric subslitution to evaluate the integral: J+v6-xdxJvrvei-xdx=...

Question

Use an appropriate substitution and then trigonometric subslitution to evaluate the integral: J+v6-xdxJvrvei-xdx=

Use an appropriate substitution and then trigonometric subslitution to evaluate the integral: J+v6-xdx Jvrvei-xdx=



Answers

Evaluate the integrals using appropriate substitutions. $$ \int \sec ^{4} 3 \theta d \theta[\text {Hint}: \text { Apply a trigonometric identity. }] $$

So be solved here. The goal, of course, is cubic square. Did a fight about spur there? So I want to do this using the production. We don't know. However, it in the form it is currently can't use those because we discovered that a crew can't Operator? Yes, a practice where they that in the denominator. And this car is a problem for us. Uh, would you change this before unify our performance? So to do this won't just your simple new substitution. And we will let you. You Teoh square with Nina. You hear this very theater. So do you. You see 2 1/2 squared. You think so? Rearranging this in terms. To isolate data, you have to to square a theater. You so we can play this into working here. So we now begin to grow Cosi cute of you over scribbler times two square root of you. So we get a cancellation here. This is a non here. You can pull this two x factor too. Right here to the funds. What we're left with is to integral, of course isn't cubed. Do you? Now there is a reduction continent that applies to this although it doesn't quite look like it's in this form is a different form of just one over sign that you that's flexible. We have BST How's he can't u is equal to one over sign you So they were not formula that we're going to use it looks like this. Let's use extra now That's how you normally but integral the X over Sign Our end of a X constant is equal to negative one over and minus one. Nice. How sign of the X. Divided by sine the power and minus one. But the X US and minus two over and minus one ties the same thing, nor has seen that formula except instead of signed with power. And we didn't have sign to the power and minus two, the X So we confine this to our formula here. Khost is because this is equal to into one over science. Cute. You you you hear. So in this case we have in you call the three more A is equal to one That's just one times you this you So you apply this to our equation just like to your arrest this line to make some room so we now? God, you and your fellow do you? Over. Sign Cute. You We have to. Is you two guys? Negative. One over three, minus one that dispenses with three. That's co sign you over. Sign to the power to you. Plus three minus two over three, minus one. That's integral. It's very you over Sign, have you? No. In this case, it's signed with power one because it's through my eyes to one. So you have this beautiful. Now the whole lot site here has been solved that we don't need to do anything more. What would you have this integral to solve? So start off problem. We'll just do a little bit of rearranging. Simple. Fine. So this factor right here three minus one is to so one over to what happened. So I cancel witness to And this over here three minus ones. Three wise to one's right through once ones too. So we have over two. So that will also cancel with this too. So what we're left with is negative. Cool. Sign you and I'll bring this science square into the numerator, Jr D Costa. That was so we have cause I knew times. Cool ck square of you. Us. Did you go? Uh, who's he? Can't you? You So we cannot solve this much more easily Because this is a common in tickle that we know from our integral tables. And we know the people. You, you you a natural rhythm. You can you aren't you Plus c So what? This into our formula, Our solution or with this church here? Oh, yeah, This negative hosting you I was cool. C can't swear you plus national I agree you can of you over to plus c almost like to do now is playing our former back into this woman back substitution. So we have negative co sign of the square. Earthy. That was cool. Seek and square the square here. Us the natural light room. Yeah. In this square data over to plus C. That is the final solution. Question

Hello and welcome. We are looking at chapter five, Section five. This is Problem 23. So this is an ugly looking integral. We have Data Cube here, Times co sign of Data Square. All right, So you're instructed to first to make a substitution, then used integration like parts to evaluate. So whatever is inside that co sign is my first bet for my substitution. So I'm gonna call good need X. They just square. I am, um and that's gonna play nicely with my data outside, um, coastline as well. So d x then would be to theatre d theta. And, um, if I saw for De Fada, I have d x over to favor. Now, remember, I'm substituting this in, so I don't want any fae. This s o If X status squared, then data is square root of X. If I want to convert that this stated to a square two backs, which I do just be d x over, too. Root ex. Right. So now I can start my substitution off to do one more. Um, change. Now, when I have a definite integral on, I make a substitution. I got to make sure I remember to substitute and change my, um, my initial point in terminal point starting in, Andy. So luckily, all have to do is square those, which makes them a lot nicer to work. Right. So these are in terms of data. So to get them in terms of X, I just square them. So I have made a cube. I can churn, um, two of those status they squared into acts that I have one fatal left over, which is going to be by the same rationale. Here is gonna be a rude X. I'm gonna write that route. X Times root ex Just for starters. Here, we'll clean that up in a second times co sign of X and then D fada is, um, one over to root acts times, dx. This is a big mess. We can help ourselves a lot if we clean it up a little bit. So, um, X times rude X times one over to root ex is going to be 1/2. No, this is X times X to the 1/2 uh, times X to the negative one. Right? Let me clean that up. So it's a little clear for you, right? so x times 1/2 times x to the 1/2 times Excellent. Negative 1/2 is gonna cancel. That's just gonna become one ad. The exponents. Uh and so I'm left with 1/2 x times co sign it x d x. That worked out very nicely. All right, so I'm gonna take this to the next page with me. It's easy to remember. All right. So moving forward with this if I let you, we've done a few like these before, Or maybe you have in this homework section. I know I have. In this home ec section, um, I let you the X and Devi the coastline of X. I have u equals X see you equal d x have d v equals co sign of X And then don't forget my d x there and then the equal, um, we'll be side effects. So applying integration by parts I'm gonna have from high over to two pi. I'm gonna leave this for the end of empire. Over to two pi. I have X times sign of X That's you. 10 to the minus. The integral of Edie, you grow from pyre or two to pie. Uh, sign of X dx. So that worked out real nicely for So, um, I just need to take that anti derivative there, and then I can evaluate at pirate to impart Just be still x sign of X subtracting. Um, the drift of here. So sign of X. The anti do derivative would be negative negative coastline of X. So this would be negative. Negative coastline of extra make. That's a positive co sign of X on. Then Don't forget my plus C. I can actually, no, This is a definite nibble. I don't need my pussy. So now we can just plug in my values of, um, pie and then subtract by the plugging empire. Hi. Sign of pie. What's co sign? If I And then I'm subtracting, however to sign of pi over too. Plus co sign of Piper too. So evaluating using our unit circle here uh, sign of pie is zero. So this whole term's going to zero. Co sign of pie is negative. One, a sign of pie or two is one. So I'm subtracting high over two times one, and I'm also subtracting coastline of pie or two. But that's just zero so distant Oh, and I forgot one thing. So if we go back one page, I have a ah, 1/2 here and here. So I was talking about how it's easy to remember, but I forgot. So this 1/2 was outside the whole time. It's gonna be 1/2 times the whole thing here and then 1/2 times the whole thing here. So that changes our answer slightly. It'll be negative. 1/2 minus pi over four. All right, so that is our, um our solution or problem 23 we are done.

Section six out. One problem 29. So we're dealing with integration here That requires integration by parts is way to look at this. Let's just look at the indefinite Integral. First I can rewrite this as they just squared co sign of they that squared and then data dictator, that allows me to make a substitution. What if c is equal to Theda squared then DZ is equal to two theater the theater. So if I were to take this integral and just multiplied by two and divided by 1/2 then I end up with the interval 1/2 and this becomes Z CO sign of Z Easy. Okay, now we're back to this. Looks familiar If I'm looking integration by parts let you be the polynomial term Do you use equal to Deasy? Let Phoebe the trick and a mentor coat side of C d Z. Then we know that the excuse me, This with D v D. V thing. You integrate that V When you integrate the coastline, you get the sine function. So this is sign of C. So it tells me is that this integral is equivalent to 1/2 and that you're gonna have UV so that z sign of Z minus the integral of VD You. So that's just the sign of Z. Do you see? We know that when we integrate the sine function, we get minus co sign. So this turns into 1/2 z sign of Z plus the coastline of the sea. So plus, obviously a constant of integration. That's my anti derivative. Remember, our substitution was Z was equal to theta squared. So it tells me that if I look at my original, so the anti derivative, if they'd a cubed co sign of Data Square D theta is going to be 1/2 on, then it's they just squared scientific, that squared plus co science they that squared. And the original problem was to integrate this from square with a pirate to to square with a pie. So let's go ahead and do the definite integral now. So to integrate from the square root of, however, to to the square root of pi, they had a cube co sign Data Square D theta. It's going to be 1/2. They just squared thigh sign of that s squared. That's co sign data squared. All of this is evaluated from square root of pirate, too, to the square root of pi. So now it's just a matter of calculus. And now zubrin trigger over its into substituted these invert arithmetic. So this is going to be let's substitute. Um, and sorry about it. Will. Type of here is the square root of pi sinless Substitute the square root of pi. Into this formulas, you're gonna get 1/2. And then when you substitute the square root of pi, you would have pie times the sign up, I plus the co sign a pie. And then when you substitute square with a pirate, too, you would end up with pirate too. And then you would end up with a sign of pirate, too, plus a co sign pi over two and then evaluating all of this. You know that the sign of pie is just zero co sign a pie. Negative one. This becomes negative 1/2 minus the coastline apartment to a zero. The coast signing piper to is one so minus pi over two. So that's the final answer. Minus 1/2 minus par for two is the answer to this definite Integral Data Cube co sign data squared

Okay. So I would be solving this interview right here. Now we're asked to use the production for cues office. Did you go? However, Informant is currently can't use it. Production formalist. But yet so the justice interval. You can do that using your substitution. So moving a simple and you substitution. Here, Enbrel. What? You local too. Et the power t minus one. So that would make you equal to you. Have our tea. It's you. Do you want to isolate? The TV? Doesn't have it here. So e t is equal to you over e r a t so we could get us into our people. Do that happened? Changeable. Go into the power t place seeking cute You these you over the party. So we now get a nice cancellation right here. Marriageable now becomes the interval. Seek it. Huge you. Do you Now we get by reduction formula. So there's no standard, uh, direction forever that comes in the form of seek and to some power. There is one of the key news because it looks a little bit different. It's standard form already till here. So we have an integral you over cool. Same to the power in? No. Well, we needed normally have some constant A How you so in this case, because we're eating a former because seek it of you people to one over co se you so this formula is directly applicable. But it just looks different because we have a coastline instead of seeking. But it is the proper former. And if you can see formula here Yeah, and he was three. It's the exponents of our coastline here, and we have a is equal to one because we don't have any to appreciation for the opening of one in front of our very you. So the formula looks like this. We have one over in minus one times. Sign A That was you. Over close. Eye to the power anyway, this one. What a time to you. All right. Plus in minus two. Over N writes one horns. General, you over. Cool. Sign hour and minus two. Eight times you. So this is what our formula, it looks like you just need to plug in our numbers for and okay, which we already said three in 81. So wasn't these numbers in are people? Does it mean to one over three, minus one. It was a sign. You over call sign squared you because advice one is three lines. 1 to 2. Worse. Three minus 2/3, minus one. There's extra for you over. Who say you It's here. We have three ways to just one. So we have coastline of you. So this whole side here is completely dealt with. You don't need to be waiting for the However, Listen to go over here recently to evaluate we know this into page because one of the co sign let yourself first stage people to see you can't. And I know the integral of c can you, from our cables These people too. The national life rhythm. I she can't new bus. Yeah, you was seen so we can point this, uh, formula for each will. Right here that we've been in the final answer so before we know, simplify this part right here. Three months from its just, too. So we have 1/2 also coach squares and get on here so we can just pull it up to the new writer Seacat Square. So her solution that becomes 1/2 I was sign you. I was seeking Swear you also, uh this we're here is one or two. So we have us 1/2. That's the natural life for him, can't you? Plus Kim, you plus c So we're just about done or need to do now is just re substitute from you back t so over here for a very start. It said that use it puts you either power t minus one. So you play that back in here, Open a new page through this, so we know how 1/2 times sign of e our T minus one times seeking square. Uh, e our t minus one plus the national lottery. Uh, see again, you did have power team like this one us and with the power t minus one and then plus C because it's a indefinitely. That's the final solution to the question. So just to summarize, we had former interval where we couldn't defy our production from those. But just one simple substitution of to it's a either must be operated on Bersih Well, as Chancellor. Oh, the other for political. We didn't need to do by parts that love is to use this direction from keeping in mind this event of you hear that she can just pull over coastline. That was that We're going to solve the Did you go by turning our. It's really been solved into a much simpler one has coincided with no one seeking one. Instead, she can't cube and keep. We are writing our final solution to the question.


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