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Given Js(x)dr = 4 jpgtr) 6h(r)]ur" IeolFind [ntr)dx _[enco; cwalualc Er" 2hr)jr.|6) By using appropriale subslitulion solve the integralKx dx | 2r...

Question

Given Js(x)dr = 4 jpgtr) 6h(r)]ur" IeolFind [ntr)dx _[enco; cwalualc Er" 2hr)jr.|6) By using appropriale subslitulion solve the integralKx dx | 2r

Given Js(x)dr = 4 jpgtr) 6h(r)]ur" Ieol Find [ntr)dx _ [enco; cwalualc Er" 2hr)jr.| 6) By using appropriale subslitulion solve the integral Kx dx | 2r



Answers

Use a change of variables to find the following indefinite integrals. Check your work by differentiation. $$\int x^{3}\left(x^{4}+16\right)^{6} d x$$

For a problem. 31 were asked to u substitution to find this integral. So we're gonna let you equal whatever is in the denominator. So for us to be to, the six are and then we go ahead and take the derivative of both sides with respect to our So for is a constant. So the only derivative we have to check is he to this six are which would be six. Eat to the six are D R. And as you can see, we have six. Either the six are right up there, so it works out perfectly. Now we can rewrite this as won over you, since this is the same and we must fight by do you? This gives us Ellen absolute value of you. Let's see. And you cannot have a you in your final equations. So we have to substitute the original you back into the equation. Would be or waas B to the six are cosi, and this would be I know equation

The first thing we should do here is such a cz factor that the not leader as much as we can. So here will get X squared minus four x squared plus four. And then we should check to see if he's fatter. So if we look at the first one here, this is just X plus two X minus two about the second one. Well, this one factor. So you look at the discriminatory here, B squared minus four a. C. It's a negative number here for this problem. For this x squared plus four. So that means it does not factor. So have to write it like that. And then we could go straight to the partial fraction so constant for the first two and then because we have irreducible contract IQ on the bottom and green, we have to put a linear up top. So we have CX plus de. So here, let me write this. We have to find a B C ity. Of course. So is And I got him on over thirty two. Be positive on over thirty two, and then we have that c zero and dia's minus one over eight, So I just pulled off the minus there. And then we have X square plus four. Now, the first two hundred girls, those air easier. And then we have one over thirty two l. A cops. Thirty two. They're not me. That's sloppy There. And then here we have X minus two. And then for this last inner girl here a little more difficult. In the first two, you could do a train from here, but sixty toothy data. And so when we integrate, this will have the one over eight with the minus from disturb right here and then after interbreeding. That's ten inverse of X over to and then divide by two again, all coming from the tricks up over here. Let me just go to the next page and write that out. So combining the log rhythms. So here, just combining log using the law of properties and then combining the the two that sixteen. And that's your final answer

All right. So we're looking at this interval of two are cube minus three over R squared plus four d r. And the very first thing I would do I would make my students rewrite. This innit rule with rational exponents makes the problem a little bit easier. Eso I expect my students to know that the r to the second power and the denominators the same mazar to the native second power. So then when you do the anti derivative, you add one to your experiment. You multiply by the reciprocal well, two divided by four produces 21 half, Uh, because the constant does go along for the ride. That's why I said two divided by four. Same thing with the next one. Add one to your experiment when they give to plus one is negative one. When I take this three that goes along for the ride and divided by a negative one The new exponents it changes to plus three andan This one the anti drove to this four are and don't forget about plus c Now some teachers will let you leave your answer like this. Other teachers might say if we start with a the negative exponents the denominator. Go ahead and put it back into the denominator. But what they want you to do is double check that. The derivative of this is the same as the original. Well, if you double check Well, if you bring this forward in front four times one half is to subtract one from your expense. Three, uh, negative. One times three is negative three and are to the native second power. The drift of a four are is four or four up there in the derivative of CIA zero. That's why we need that value, which is equivalent to what I circled down here in green is your correct answer.

In the given question we have an integration and the integration is next square Through 1 -4 x Cube X. Okay let I represented by by trying to solve this interrogation. I am making a substitution that is one minus poor. Excuse me. It was to W If they substitute this. No it's delivered to put awaits me -12 access where the X equals two data group means access. Whether the X can be replaced by minus data blew up on to it. Okay now we make the substitution in the integration than our integration changes into are the place of access where in the X. We have to put Minour of do the blow by 12 and that the place so this 1 -4. Excuse me. You have to put w. So it changes into or out of the blue. Now we can say that this integration finally can be Britain is okay -1 Bird. Well into W. To the power lines are way too into it. But in the given question it is being said that your integration is it was too okay. W to the power and did okay. If you compare both this integration by comparison we can say that here. Kay is it was too minus one by well and you're and and also we can say that and he's also here one way to and we have to also show the value of W. The substitution we had taken. That is 1 -4 experience. OK so these are all the values here in the question that R. K and N W.


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