evaluate-the-given-definite-integrals-int_23-frac8leftx21rightleftx33-xright2-d-x-3

Question

Evaluate the given definite integrals.
$$\int_{2}^{3} \frac{8\left(x^{2}+1\right)}{\left(x^{3}+3 x\right)^{2}} d x$$

Dwijendra Rao · Accepted Answer

Hello Friends We have to evaluate their given definite integers. That is integration of book three head of X squared plus one upon Excuse Plus three X square of the X. Okay so this can it can be indicated with so now we will differentiate it with respect to X. So this will be three access well plus three because to do you upon the X. So we can I did Access couple is one of the x equals two. one of 1. three of new when actually cause you to the lower limit then lower limit of you will be To the power of 3-plus 3 into two. That is eight plus six. That is 14 When exposed to three. Then you will be into the power three Last three into 3. So that is the question nine plus 27 Plus nine. So that is a costume 36. Mhm. So integration will be 14 to 36 yet by three of the U upon U squared Get up on three of Indication of 14: 36. Your department is two of the U. C. Integration of this will be close to Hit up on three you two departments two plus one upon managed to plus one 14- 36. So this will because we ate up on three You two departments 7 upon -1 fought in 2 36 So does the U -1 half for it by three When upon 36 months one upon 14 so we can write it. So that&#x27;s when we questioned you. And 36 to the power of minus one- of 14 to the power of -1 multiplied wade upon three. So this will be a cost to 22.189. Mhm. Okay and they said the costume 0.116 four. So this is the answer. I hope you are not sure. Thank you.

Ian Maurer · Answer

So here we have, from negative to to to the end of a row of X plus three squared D X. I don&#x27;t know what the integral is of parentheses, and so what I would do is distribute their out, and then you can look at each piece separately. Any time you&#x27;re seeing things in parentheses squared, it&#x27;s a good idea to just distribute it out. And so we&#x27;ll end up getting X squared. There&#x27;s two different three exes, and then there&#x27;s a nine. And then you can anti do the anti derivative, each piece individually, so X cubed over three plus six x squared over two plus nine x. We can simplify six over to to make that into three x squared. Then we&#x27;re doing our F, A, B and F K. It&#x27;s work looking into I&#x27;m plugging in negative, too. So playing into would be two to the third over three. Remember, I simplify that to be a three x squared, so three times two squared nine times, too. So that&#x27;s my F B. And then I&#x27;m going to subtract half of a so negative two. So the third we were three. Those three times negative two squared plus nine times negative, too. So just doing a little bit. Arithmetic. 8/3 plus 12 plus 18 minus and it&#x27;s J basically the same. Answers. Negative. 8/3 plus 12 minus 18. Um, distribute the negative. Get rid of my brackets. 1/3 plus 12 most 18 Double negative. Eight. Fareed&#x27;s single Negative 12. Double negative. 18. I could see those 12. They&#x27;re canceling out. Then I have eight plus 8 16 3rd 18 plus 18 plus 36 to make them into a common denominator. I could multiply this by 3/3. So 16 3rd and then plus another 108 herds gives me Is that 1 24 dirt? Thank you.

Doruk Isik · Answer

in this film or Estivill eight. Integral three over X Plus three to deport power points, making it to two. Let&#x27;s start by reading my fingers and check the less negative, too. We have three times explosive, three to the negative four power and we know that freezes the constant so we could just take it outside the entire left left if I find anything. So we know that the entire derivative pull this portable perform explosive three to the negative three divided by negative three and told me to start at this end. And two, um so, Nate, freeze Well, can slow and we know that this will then be negative. One five Negative. Third, Probert, My 13 negative third power. And then that&#x27;ll be negative. One times one over 135 like this one and we find and the answer to be 1.4

Tom Greenwood · Answer

okay, If we&#x27;re gonna break this down, we have to first realise, weaken, factor out an X squared. So the decomposition is going to be a two over X squared times X plus one. And we have this repeated factor of the X. So that means we have a fraction off excellent first and a fraction over X to the second here. So have to have that repeated group giving us the in this case, two separate fractions since everything&#x27;s linear just a and A B from a numerator. And then we need a fraction that was added on here that had the denominator off exports one. So now we multiply everything by the common denominator. And that means that a lot of stuff is gonna divide out with each of these. So we end up with two equals. One of the exes divides out with the first traction. So we&#x27;ve got a times X Times X plus one. It will give us a X squared plus a X. Both of the exes divide out. So we have B times the X plus one. So that would give us B X plus B, and then the X plus one divides out and we have C Times, he explained. Aired. So there&#x27;s that term and now we can set the terms on each side to each other are square terms. There&#x27;s no square term on the left hand side, and we have a X squared plus a C X squared on the right. So zero, he close a posse. We have no X terms on the left hand side. What? We have a couple of ex terms there in this case, a plus B and then our constant of two pastie who are constant over on the other side, which is just the B. So we know that b equals two and from because two we can figure out that a equals negative, too, from the middle equation there. And if a equals negative to the top equation tells us that C has to equal to just like he does. So now we can rewrite are integral as the integral of negative two over X plus two over X squared plus two over X plus one, because that&#x27;s all being multiplied times our DX, and now it&#x27;s integrated to these individually, letting you equal what&#x27;s in the denominator. And then the D you would just equal the D X that will have with each one. The twos can get pulled out in front of the integral. That&#x27;s constant. So we have negative, too Times the natural log of X. For this middle fraction, we&#x27;re going to use the power ruble that&#x27;s actually next to the negative second power. So we add one to the exponents and divide by the new exponents That gives us a minus two X to the negative first power. And then we&#x27;re back to using our natural log integral rule for the last fraction. And so we have plus two times and natural Aga the X plus One. Now we can do a little rewriting of this. We can combine our algorithms together so we have both of these twos becoming exponents, and that the negative means were divide by X. So we have the natural log the X plus one divided by the X squaring that and then minus two over the X negative exponents moving it into the denominator, plus our constant c Oh, sorry about that. Now, harm on the touch

Ernest Castorena · Answer

in a row of their actions. You all their excuse, those expert. Yeah, but no. This is going to be a repeated it, uh, zero. So she&#x27;s not a man when you saw for your form, but we&#x27;re gonna get over X plus one minus two over X plus two over export. So are integral, actually. Simplifies. It&#x27;s a two over X plus one minus two over its most two over X squared yet? Well, that&#x27;s equal. Teoh. See, I don&#x27;t think that you had a very the natural log of the absolute value of exports. One over at. And then let&#x27;s see, we&#x27;ll have a minus one over X still under the tea, and then we can add an arbitrary constants. E we have.

Nathalie Luna · Answer

So in this problem asked Valley. Same to grow up from my hands to two three at this pulling on you with respect to X. So we have three turns and disintegrate Interval. And, uh, since it&#x27;s a point of view, let&#x27;s meet up. Each term is part of eggs. So we want to buy D Pearl so powerful when it has this planet we&#x27;re integrating apart X. With respect to X, the integral will be faithful to Exley just at one to the power. And then we divide by this new power. Plus in the constant in this case, we&#x27;re dealing with a definite into crop. So you don&#x27;t need to worry about this contest suit. First up is so here. Uh, for this, uh, let&#x27;s do it. Terminate him. Systems were either adding or subtracting. We can into great turn by turn. So open parentheses have hard and long term. So for X equal to three, we&#x27;re going to add one here, which will make get four and then by by the steam that gives me power. No, All Constance, this constant states the same. Just we just keep it there and this power is one. So this scene poorest gonna be. We&#x27;re going to act one to it. That&#x27;s two. And then we divide by this new power. That&#x27;s two right. So in this last one, it&#x27;s not. If you don&#x27;t see it at first, you can think of three as Spain. The big time sex race to the Syrah. No, right, Jason. Us. Anything to Ciro. It&#x27;s one. And we can say that three is equal to three personal self. Uh, that&#x27;s really think of the constant. Still, when we integrate it, I&#x27;m going to keep the three because it&#x27;s a constant no X. We&#x27;re gonna add one to the power since it isthe Ciro just one. Which means we don&#x27;t really have to write out want us a power. And we did buy a new problem just that&#x27;s equal to itself. So we&#x27;re gonna write it down anyway. Kiss. We&#x27;re dealing with fractions, and that&#x27;s going to make it easier. So weii said that this was a definite integral, so we don&#x27;t really We don&#x27;t put a don&#x27;t add a constant. Instead, we say that this is evaluated from X equal to minus two, two, three. So So now we can can simplify this a little bit. That&#x27;s not just it. So I guess now we can go ahead and applied the fundamental fear of calculus. That means we&#x27;re going to evaluate this expression. Yeah, at X equals three and then subtract the same expression evaluated X equal to minus two. So just to keep that in mind, X pressure getting to x equal and then X equals minus two. Great. This whole thing is he up to? Could be race you to the power divided by four minus two times. Great place to, uh, Square. Great. You. He&#x27;s cleared. Richard. That&#x27;s State Time X in this case, is three over one not going to write that right now. Oh, so then we subtract the same expression, evaluated X equal to minus two. So we have I meant to Tim&#x27;s for him. So he raised to the fourth car over minus two times X. That is now minus two. Squared over two, three kinds. Lines, too. Yeah. All right. So, um, we can start on simplifying olive office, and so that&#x27;s looking. I can&#x27;t This too. Uh, these two cancel each other out. Same with these two. And, uh, that start on your line, so three. Uh, cute. This anyone? That&#x27;s a four. This Teo three squared that isthe night. And this three times three. It&#x27;s also nice. And now we have, um my nips don&#x27;t lose. This practice is we&#x27;re not going to distribute this monies just yet. Yeah, men&#x27;s too. Race you to the fourth power. That is sixteen over four. And here we have minus two squared, minus two attempts. Minus two is for we have this negative sign here. So minus four. And then this last term positive in mind multiplied by minus minus six. All right, so here we have the&#x27;s cancel out. It&#x27;s here. And the sixteen over four. That&#x27;s equal to four. So since we have four minus part, there&#x27;s also cancel. That cancels out. And what are we, elect with just two terms. This a four over getting on over four. So and this ninety six. But since this isthe and prophecies and we have this man&#x27;s here, there&#x27;s this plastics No, to be ableto at. These two things seem to have the same the same generators. So I&#x27;m going Teo, ad in here. This can&#x27;t be instead of six. We can write this six over one and I&#x27;m going. Teo Motive Pipe broke top and bottom before, so that gets me. Yeah. Any one of the workers is this Trump doesn&#x27;t change at all. No. Uh, twenty four over four? No. Now, since that seemed a numerator, this stasis ink, This isn&#x27;t changed. In a top, we have eighty one. Yes, twenty four. And that is equal to one hundred fine over four. So thiss iss What? This integral alarm agrees, too.

Problem

Evaluate the given definite integrals. $$\int_{0}…

Problem 30 Easy Difficulty

Evaluate the given definite integrals.
$$\int_{2}^{3} \frac{8\left(x^{2}+1\right)}{\left(x^{3}+3 x\right)^{2}} d x$$

Answer

$\frac{22}{189}$

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Grace H.

Kayleah T.

Caleb E.

Kristen K.

Precalculus Review - Intro

Functions on the Real Line - Overview

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$\frac{22}{189}$

Basic Technical Mathematics with Calculus

Grace H.

Kayleah T.

Caleb E.

Kristen K.

Precalculus Review - Intro

Functions on the Real Line - Overview

Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Grace H.

Kayleah T.

Caleb E.

Kristen K.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus