evaluate-the-given-definite-integrals-int_-12-frac8-x-2left2-x2-x1right3-d-x-3

Question

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

Dwijendra Rao · Accepted Answer

Hello. We have to value their given definite integral. That is given Money is 1-2 eight x managed to upon to access square minus X plus one. To the power of three the X. So we will substitute to prescribe you Well we can it can be integrated with their but the substitution of the excess carbon is X plus one as it goes to you. So now we will do defense here differentiation of this with respect to X. So this will be four X -1. Yeah because too do you upon the X. four X -1 of the X. Because who do you so we can add it And then x equals to one is 1. Then you will be consumed two and two minus one squared minus of minus one plus one. So there&#x27;s a little bit too Plus one Plus 1. That is the cost of four. And then actually closer to You will be two into two square minus of two plus one. So this will be he had managed to that is six plus one, that is seven. So these are the new limits so you can buy it 4 to 7. Okay. Goes off four X -1 DX so we can like it&#x27;s a cause to be you upon Here to the power of three. Okay, so this will be cause to integration of this will be goes off you believe our ministry plus one upon monastery plus one 4 to 7. So this will be considered to myself, You two departments to upon managed to four per seven. So this will be a cost to- of 75 I -2- of 4 to the power monastery. So this will be a question 7 to the power of manage to- of 4 to the power of manage to so that we request to minus of 33 upon 784. So this will be a question 33 upon 784 So this is the has it And it will be close 0.04-1. Okay, so this is the answer. Mhm. I hope you understood things.

Kurt Kleinberg · Answer

So here here we have another. Definitely your question. We have the in a role of X squared minus two X plus three quantity from negative one toe one So on and I derive. And so we&#x27;re going to get 1/3 X cubed, minus X squared plus three X all of that evaluated from negative 1 to 1. So that&#x27;s gonna be negative. Probably 1/3 times. Well, one cute minus one squared plus 3 10 1 minus 1/3 times negative. One cubed minus negative. One squared. Plus three times. Negative one. The big thing here is he&#x27;s got What about the negatives and the powers. So here we have up 1/3 minus one plus three. Now, when do couple things at once we&#x27;re gonna get a Well, I&#x27;ll leave the premises minus 20. Get a negative 1/3. We&#x27;re gonna get a minus one. His native one squared is positive one, but the negative in front makes it, uh, negative and then minus three. Scenario left. Basically with 1/3 minus one plus three plus 1/3 plus one and plus three. So the once cancel, you&#x27;re left with six, six and 2/3 a little better. You left with six and 2/3. Or, in the words, 23rd that your final answer

Kurt Kleinberg · Answer

Hi. Here we have another definite Inderal. It&#x27;s the in a row from native to two of the quantity execute minus two X plus three. So as we and I derive, get 1/4 x to the fourth minus X squared plus three X ray waited from negative to to to it&#x27;s this becomes put premises there one force to the fourth minus two squared plus three times to minus parentheses. 1/4 negative to quantity of the fourth minus native two squared plus three times. Need to now keep in mind how a negative numbers behaved with even powers. This will end up being the same as this because to the fourth is the same thing as negative. Fourth, for that matter. Negative two squared and negative quantity. Navy Square to the same thing. So you&#x27;re gonna have Is this canceled? This Cancel this. Cancel this. Cancel until you&#x27;re left with essentially six minus negative. 61 of the words 12

Tom Greenwood · Answer

okay To evaluate this in a girl, we have to first do the partial fraction decomposition of our seven X minus two over the three X squared minus two x for the first thing you notice is the way the denominator factors bacon factor in X out of both of those. And so one of the factors is X. The other factor is three X minus two. So those would be the denominators in our partial fraction decomposition. So we&#x27;ll have a over X plus B over the three X minus two. Now, I would multiply everything by the common denominator so that we can set up our system of equations to find a MB so multiplying gets rid of the fraction on the left hand side, we just have seven x minus two more Upon the right hand side, each factor will reduce out. So we have the numerator times the opposing factor in the other denominator. So eight times a three x minus two his three a nex minus two A. And then be times the X is possibly X. And now we can equate our terms with each other and equipped to coefficients. So seven is gonna equal three a plus B from X terms and then the negative too as two equal negative to a from the constant terms. So a has to be one and then that means that b has to be for so now we can rewrite are integral as the partial fraction decomposition of one over X plus for over three x minus two d x. That being a girl being evaluated from wonder too. So we can separate this into two separate and girls and the 1st 1 is going to be pretty straightforward. You have one over X the X So you won over you, Do you? That&#x27;s that perfect form. Don&#x27;t have to do anything there. So that would be the natural algorithm, the absolute value of X. Second, though, I&#x27;m gonna pull that four out. We got four times the integral of one over three X minus two d x. So I&#x27;m not sure how you were talking with this You do you form and how to account for things. But if you is three X minus two, that means that do you his three d x. So I was taught that I needed three times dx so I&#x27;m gonna multiply that by three. And to compensate for multiplying in here by three. I&#x27;m gonna multiply outside about 1/3. So that way, my three and 1/3 would reduce with each other and equal one. And I&#x27;m not changing the value of the so that gives me plus 4/3 times. And now I do have my one over you, Do you? So 4/3 times the natural order of them of the three x minus two. And this is all getting evaluated from 1 to 2. So that means I have to put those in for X. So I&#x27;m going to have they natural algorithm of to plus 4/3 times The natural algorithm of three times two is six six minus two is for there&#x27;s plugging in my to minus the natural order of them off one plus 4/3 times the natural algorithm off three times one minus two is one again, and the algorithm of one is just zero. So this all just becomes zero. So we have the hunger of them of to natural algorithm of to plus 4/3 times and natural Loker them off. Four. There&#x27;s our

Nathalie Luna · Answer

All right. So in this problem, whereas to evaluate this integral, it&#x27;s a definite integral from X equal to minus one up to one of this function effects. And we&#x27;re integrating with respect to X. So before we start on this, I want to review the Powerball, uh, immigrants. And so when we are integrating a power of X with respect to x thie intercooled B x, No, the the power we&#x27;re going to at one and divide this will be look, um, over a constant that will be the same ass, this new power. So So this quirks, whenever this end is different from one. In any other case, um, we can play this route. So we&#x27;re gonna with that in mind, we&#x27;re going to begin integrating this. So we have three terms and we&#x27;re going to We&#x27;re going to play the cereal to all three of them. Um, in case it&#x27;s not a hundred percent clear. This the sequel to my existence by itself means the explaining this one. And when we&#x27;re talking about a constant, this will be we keep the the coefficients, and then this eggs will have. It&#x27;s just one race to the first power And if we see a constant, this is X rays, too. Zero. All right, so with that in mind, we&#x27;re going to use the parable the first first term here. This is this expressionist equal to for this one. Powers too. So once we integrate integrate, this will be three and dividing. It&#x27;s like the same number. So no for this one. Um, Constance, just stay as they are. So we leave it here and now this X we add one to the power making it two. And we divide by this new power over to you Foster center for this last true, it&#x27;s a constant. But here, um once we integrate, I mean to integrate. We&#x27;re going to think of it as X to the serum. So that means the conch, the coefficient stays here. It&#x27;s not going anywhere. And now X is to build first power, which is just excellent, deleted like that. And we&#x27;re dividing by this new power. That&#x27;s one. All right, so who this is once it&#x27;s integrated and the limits are X equal to minus one up to one I wreck. So let&#x27;s waken simplify this a little bit. Great. This too cancels with this chin, and we don&#x27;t really need to write this. All right, So now what we&#x27;re going to do is we&#x27;re going to apply the fundamental theory. So this expression is equal to whatever&#x27;s in parentheses is equal to a means not equal when x with eggs taking this value minus this expression with X taking the lower. Right. So let&#x27;s start. So if we have, I&#x27;m gonna color. Could this so here. Minus one and positive one. So first is when X equals it will, too One so we can start. So substitute substituting. Um, one place too. Third power over. Over three minus one squared Plus three times. What? So that is his expression when exit Cool one. And we&#x27;re going to subtract the same expression with eggs equal to minus. So we start with minus one rates. D&#x27;Oh, um I&#x27;m sorry. Cubed over thanks over his three. Linus. My God. Squared plus three times minus one Parady Soon and we have all these constant. Uh, we can plug this into the calculator o r doing my hound A calculator iss useful when either you&#x27;re in a rush or you don&#x27;t care about having the exact number. Um any kids. We&#x27;re going to simplify this a little bit. So this one here one race to no matter which power that is always going to be one over three. Once cleared again, It&#x27;s one and three times going this way. I&#x27;m sorry. I&#x27;m sorry. Okay, So for this one, so here are it gets a little tricky with the plus minus its very BC to miss a sign. So, um, for this part, this is positive. Um, let&#x27;s take care of the parentheses first, and then we&#x27;LL deal with this one. I always find it&#x27;s best to do an extra step if it means that you&#x27;re doing You&#x27;re an actor. You&#x27;re being working so minus one times three. Since the power is odd, that means the result will be negative. And here, um, so one minus one squared. Since the power is even, that means is this positive? But we have this one here, silly have minus one and laugh. It&#x27;s minus time&#x27;s passage. That&#x27;s minus all right. You know, we can&#x27;t add in this kind of distributing this minus sign over each of these terms. So since this one is negative, like all three of them are negative So with this one day, all of them become positive. Plus one plus three. Awesome. So now for this next print again, huh? Could use a calculator at this point. But if you want to figure out the exact value Um so we have to good two fractions. Both of them have the numerator three. So I want all the terms here and to be din America with toe haven&#x27;t the numerator of three. So to do that, where it was going to add it in here. Um, but since we do it on the bottom of this number, have to do it on the top. That&#x27;s now instead of one one times three same on each term That has C you numerator different then. Great. Hey, so you think if this number as being that number over one so are multiplying by three here we&#x27;LL multiply by three on top already so won&#x27;t suite at now. We can add all of them together, So it&#x27;s going to be so the but the numerator. They&#x27;re all the same. You don&#x27;t have those, you just transfer them and I have one minus three plus nine plus one plus, uh, three. And this is like so all that a together it&#x27;s twenty. And so this is what the integral evaluates to

Ian Maurer · Answer

get the in euro from negative one toe, one of X squared, minus two X plus three the ex. The nice thing is that you can split this up and imagine it&#x27;s really three different separate intervals X squared to eggs and three so X, where he becomes X to the third over. Three, two acts becomes too and squared over two and three becomes three X. That&#x27;s the opposite of the derivative process. Instead of doing the derivative, you do the anti derivative Add one to the power and divide. This is gonna be from negative 1 to 1. Before I go on. I&#x27;m just gonna cancel these twos and hope to make it a little easier. So first I plug in the one for everything. One cubed over three minus one square because we simplified it. Plus three times one that&#x27;s plugging in the F B. They were talking about in this section and then you minus FAA. So that means I&#x27;m plugging a negative one. I got one cubed over three minus negative. One squared. Make sure using those parentheses like that. Yeah, I&#x27;m gonna bring that up here. Uh, getsem answers here. One cubed over three is 1/3 minus one squared plus three and then over here we have negative. Warm. Third, minus one minus three. Okay, um, I&#x27;m gonna distribute that negative just so we can see if anything&#x27;s gonna cancel out. So 1/3 minus one plus three. Double negative one fared. Double negative. One double negative. Three. So minus one. Cancers with a plus one, three, and three makes six. So we have six and 2/3.

Problem

Evaluate the given definite integrals. $$\int_{2}…

Problem 29 Easy Difficulty

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

Answer

$\frac{33}{784}$

Related Courses

Basic Technical Mathematics with Calculus

Related Topics

Discussion

Top Calculus 1 / AB Educators

Grace H.

Anna Marie V.

Kristen K.

Michael J.

Calculus 1 / AB Courses

Precalculus Review - Intro

Functions on the Real Line - Overview

Recommended Videos

Watch More Solved Questions in Chapter 25

Video Transcript

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Grace H.

Anna Marie V.

Kristen K.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

$\frac{33}{784}$

Basic Technical Mathematics with Calculus

Grace H.

Anna Marie V.

Kristen K.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

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

Grace H.

Anna Marie V.

Kristen K.

Michael J.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus