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Use residues to find the integrals in Exercises.$$ext { P.V. } int_{-infty}^{-} frac{x^{3} sin 2 x d x}{x^{4}+4}$$...

Question

Use residues to find the integrals in Exercises.$$ext { P.V. } int_{-infty}^{-} frac{x^{3} sin 2 x d x}{x^{4}+4}$$

Use residues to find the integrals in Exercises. $$ ext { P.V. } int_{-infty}^{-} frac{x^{3} sin 2 x d x}{x^{4}+4} $$



Answers

Evaluating a Definite Integral In Exercises $41-48,$ use a table of integrals to evaluate the definite integral.
$$\int_{-\pi / 2}^{\pi / 2} \frac{\cos x}{1+\sin ^{2} x} d x$$

All right, so let's integrate this. Ah eight signed before power x dx from your pie. Um, person you want to do is we noticed what kind of power this is. You haven't even power of sign. So, um, seems like the best way to move this is using power reduction formulas, since we can't pull out a a singular sign of X and use triggered And these, uh, of of squares. So let's go ahead and try to produce this. We know we can power reduce sine squared of X into one minus co signed two x over two. Let's go ahead and try that. So that's turned us into eight times sine square. It's weird. Yeah, DX And then we can basically pullout date and turn this into one minus co sign Square deck are sorry. Coastline two x over to It's where DX, and we'll notice that this two square on the bottom becomes for so we have 8/4 front or simply just to times here to integrate with Europe. I of one minus co sign two x squared the eggs. Now all we have to do is expand this. So if we expand this we end up with following one minus two Co sign two X plus Co signed two x squared all the X and then was simply do another power reduction formula for this coastline square to give us no growth too. Two times is your interval visitor to pie of one on this to co sign two X and we know co sine squared of Let's say you is one plus co sign to you over to. So in this case we end up with, um um let's say we have a 1/2 and then a plus co sign four x over two, all dx and then we can start integrating. So we have basically a one that would have, which is that Three halves and three hives. Integrated is just three has X, uh, negative to cosign two x integrated is, um simply sign of two X and then coast on forex of two is generation bring to to side of forex Allah Great. And then we integrate this from Judah pie uh, notice from Europe. I signed a zero on both of these angles, so multiplying it by a integer two or four does not change value. So these go to zero in both end points and we're simply left with a two times three halves pie for a three pie as our

It is a problem. Eight by 35 problems for 64 from a 640.3 in which we have on indefinite anjiro sine X sign Q Becks, you're going great Course, Ianto for X t X. So Oh, so I swifts in many times before a interrupt trick in there like this usually requires u substitution. So which one would be better? You equal society circus acts? Well, let's see Phil at you ecos co sign X Then we been in the Nam That we only have. This is only this's just equal to our u two forth so that makes that seems to make things simpler. So let's go ahead with it. And so what happens with this? Then we will have a deal. It cost negative. Sign off ex DX. So that's good. Which means we're taking one power from this three powers of sign. So what? So we have what's left us. We have signed Squared X, but luckily this is simply one minus. You squared. So this turns our into into negative Inderal. Yeah. One lies. You squared. Do you You want bring you to fourth? Now this is a Now this is a rational internal. Khun do So this is where if you split the two parts you have whenever you two forth to you plus one over, you squared you and we can do both of them. You mean so the first term gears one over you Q divided by three. Um so it's it's stupid this way. So if if if this bothers who should write this as you too negative for power. And this is you two NATO second power. And if you remember the formula and tender it off you too. And do you when Anis An integer That's not in a beta one Then we have you and plus one for us during Grandal's. So with this in mind, we're going to have, um here's my name. So with this might have the negative sign and we have any call Senator for here. So we're going to have you too. Negative three divided by their four plus one, which is negative. Three and then plus you Have you You too. Negative two plus one with just one. They're wanted by you to news just native to persona which is needed. What? Plus See you, Khun taking clearly like that but we weigh 200. You represent our original interval in terms of X. So this would be our co sign. Too negative. Three. Power X divided by three minus co sign. Oh, while this is a vanity ation because this could be confused with the universe. Course I act. So let's just use one of record sine X here, Class C.

In this question were to Seoul interrogation zero toe by by four x cost two x bx Okay, so we will do this integral by integration by parts. So you because toe X that will give us the U equals toe the X and rest of the part that is Devi because toe cause two x bx that will give us we equals two we equals toe sign two X divided by two Okay, now education off. Zero to pi by four x cost two x dx will be you not we that is you is x dot obvious signed two x by two minus interrogation off V that is signed two X door to divide. But to do that is DX and the limit will be zero toe by by four. Okay, and now it will be x by two sine x and by NUS and education off signed to expect you will be minus minus plus because who X divided by four. Okay. And it will be 02 by by four. Okay, Now we will put the values by by four first and then zero and it will be like bye bye four and two that it's by by eight. Sign by by two Bless one by four dot Costs Bye bye to okay and minus it will be actually zero than zero by to sign zero and plus cause zero divided by four. Okay, so there are four times +123 and four. So first time will be pie by eight and signed by Beto is one that it is same plus one by four caused by by two so caused by by +20 Then it will be zero minus third time will be multiplied. Sign zero That's zero and minus plus minus. And cause zero will be one that it will be won by four. Okay, on it will be by by eight, minus one by four. Okay, so in short, we can say it will be bye bye to divided by eight. And when we put the value of pie in this, it will be 0.143 And that will be your final answer. Thank you.

Hail in this problem, we are given integral delimit 0 to 5 it into signed to the power for texting here and is even so we used the formula. Science, where X is equal is true one minus cause I into makes divided. By then we get eight into integral with 11 0 to buy one minus co sign to it they wanted. By two we'll square into the which is equals true, eight of one poor, integral with the limit. Zero to buy but minus co sign two weeks or square into the We're gonna be dick two into integral with the limit. Zero. To buy one place school science quite way minus two into co sign twigs into the it would be equals two. Go into and ignore the delimit. Zero Go by the X plus two into integral with the limit. Zero to buy one place co sign for it. They rented by in duty Yes, minus were into integral with the limit. Zero to buy. Who signed two weeks into the It's indignation of dysfunction is do indo okay and then amid zero to buy less express one by four in to sign for its that the limit. Zero to buy my news for and do science to wakes divided by two with the limit zero opera by By substituting the upper and lower limits we get do in tow by minus you less by, plus fund upon poor signed for by my new zero minus one upon four in blue Sign We're into zero minus do in tow trying to buy my new signs You don't, It would be equals Cool, right? Best buy it is equals to three. By therefore there were you dainty girl the elements Do you go by eight into signed to the problem for a 60 x is the worst. Please. This is the final answer.


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