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2. [5 pts] Find a first integral of the Euler-Lagrange equation that makes the following integral stationary (no need to simplify answer):(22 y 23/2 dx '1+y2...

Question

2. [5 pts] Find a first integral of the Euler-Lagrange equation that makes the following integral stationary (no need to simplify answer):(22 y 23/2 dx '1+y2

2. [5 pts] Find a first integral of the Euler-Lagrange equation that makes the following integral stationary (no need to simplify answer): (22 y 23/2 dx '1+y2



Answers

Calculate the integrals. $$\int \frac{y^{2}}{25+y^{2}} d y$$

We want to evaluate this indefinite integral here now in the top left corner. We are told in the chapter that if we integrate won over you, that should give us the natural look of the absolute value of you. Plus a constant see. So let's make are integral here. Look like that they released, try to and then integrate. Well, we might say, Let's let you equal to why squared minus 25. Then to get our differential, we take the derivative with respect to why we're going to get do you is equal to, or at least for a differential to you. Is it too, do I d y? And we'll look that that's exactly what we have in the numerator there. So it's going to be do you over you now. Interesting. That should give us natural log of you at least the absolute value of you lost our constancy. Then we go ahead, look in what he was, and that's gonna give the natural log of the absolute value of Y squared minus 25 plus C or our anti derivative

Hello there. So for the following exercise we need to integrate this function. So the first thing is always to try to separate this as some. So we have the integral sum of two terms or obstruction of the terms or more terms. The DEA integrate this in a different way. So we are going to have this uh is equal to equal to the integral of y minus one over Y. Dy okay. And it is simple to interact. There's well corresponds to the integral of y minus dy minus the integral of one over wide wine and then we can integrate this. So the integral of why is just wide square over two and the integral of one over Y is just the algorithm, the natural logarithms of why? Classy of course.

Welcome to this lesson in this lesson. Both of the definite integral. Using the integration by parts matter. So integration by part simply haven't two functions that can be expressed as u and DV. And that is he called to do the, uh, the evaluation of I am be the minus the integral from A to B over, do you? Okay, so the best part about this is to identify what is your you and what you see our debate. So here you is a function that is easily differentiable or when differentiated, it goes to zero quickly. Okay, so here will pick that. That's why. So that the you would be called to dy. Okay, Then again, we'll pick the deal. Very as okay. The hyperbolic sign of y y. So that here, if we took the integral e f y is because true they have a bolic costs of y. Okay, so none of that you have, uh, got that. We have the, um we have the way we can divide. Uh, we can find the we'll see why ass i you us? Why, then? May is close. Cool. Okay, so we value that at 02 Will come back to that later. Then I've been becomes then not do you becomes t y. Okay, so here. Yeah, Why then they hyperbolic costs. Now, if we integrate that, we have negative. A parabolic sign of why, Okay, Then you value it to the at this point. Yeah. Now the hyperbolic side why is is given us eat the power y minus e to the negative. Why all over to than the hyperbolic costs? Why is also giving us eat the power plus speed to the power negative y on to. So you will place those mhm then the whole of the differential. Uh, the whole of the integral. Yeah, now becomes, Why then needs the power. Yeah. Yeah, right. Yeah. So that is 02 Then we have this photos for the side. The hyperbolic sine. Yeah. Yeah. Okay, so now this gives us Yeah. If you put the zero out there because of this, it becomes zero. And that is gone. So would have only one by four black. The second part we have right this through. Okay. Mhm. And we have one minus one all over to, uh that is minus one. Okay, so that goes to zero. Well, yeah. I love this. This causes all that. So we have E Yeah, yeah, yeah. Okay. Oh, yeah, Yeah. Mhm. Okay. Yeah, yeah, yeah. Okay, so here. Wow, we have two e Thank you. Last to hope that. Oh, yeah. Not bad. Oh. Oh, Okay. So we're just multiplying to buy this so that we can let all of them sit on a bath. So this becomes very eat about you, then minus e to the power. Negative, too. All over. So this is that and of the lesson. And this is the answer for that definite integral. Yeah. Okay, thanks to a time, The end of the lesson.

So if we want to find the anti derivatives off this function here, So we really don't have any kind of tools right now too. Do this one directly. But if we first expand this square so let's go ahead and do that off on this side, we might be able to get something that we know how to work with. So this here will expand to sew for Why squared minus or why plus one And this excusing those sum of squares formula. So if we go ahead and rewrite this inside using that so for why squared minus or why plus one d y? And now I can use the linearity and scaler property of control to rewrite this. So first I'll just go ahead and we fight the expression for why we need a little space before why squared minus or why Plus and so remember one I can rewrite days times x to the zero for I'm sorry not expert times Why two zero power since we're interfering with respect to buy on DH So this will be one times why to zero And now it can go on in place all my integration symbols so control D Why into girl? Why then go do it now I can use the power rule for each of these. So get for why squared And then I want to add one to the power and then divide this by that power. So three and I add on some integration constant which will call c one than minus or why so why by itself is really wide to the first powers will be one plus one over the new power too. Plus another constant of integration that's called wants me to And then it will be so one times anything is just itself Sole dropped one there and they'LL be Why? To zero wass on color plus one over our new power which is one plus some integration constant c three. So first notice that we have three constants here so we can addle the constance up to give us a single constant geologist called constant C. And now let's go ahead and clean the rest of this up a little bit. So first part becomes four thirds fly cubed minus so four divided by two will give me to sew to why squared and then the last term just becomes Why? So let's go ahead and move this over a little bit, the constant and then then our anti derivative for this problem Is this right here?


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