integrate-each-of-the-given-functions-int-frac12-x2-4-x-24-x3-x-d-x-3

Question

Integrate each of the given functions.
$$\int \frac{12 x^{2}-4 x-2}{4 x^{3}-x} d x$$

Vikash Ranjan · Accepted Answer

in the problem we have indication. 12 X. Quiet -4 X -2 upon four XQ -X. Index. Mhm. No this can&#x27;t be represented us. 12 X Squire -4 X -2. Phone X into two x plus one and 2. two X -1. Or this is A upon X. Class be upon. Do you explain us 1? Let&#x27;s see upon two X- Man. No. To find the village of abc. So this is Develop Squire -4 X -2 equals into two x plus one In the two X -1 plus bx into two x -1 plus six. Into do explainable. No the value of A is equal to two B equals three and C is equal to -1. Therefore the integration become Diggle two upon X plus three upon two x plus one. My husband upon two x plus one. The X There four days twice a landmark. X Plus 3.2. Ellen Mod two weeks plus one minus half. Ellen Mod two X -1 Plus C. Or further after taking half as common. This is for L. N more X Plus three. Ln mud two x plus 1 -1 Ellen Mod two X -1. Let&#x27;s see now here this is equal to half and then extra. The power fall plus Ln two weeks plus one. to the power three minus Ln two x -1 plus the this is equal to half into Ellen mod export food into two X -1. This is plus one to the power three upon two X -1. Let&#x27;s see. So for all we have answered us half Ellen actually, Power fall into two X plus one to the Power three divided by To explain this one plus C. So this is the overall answer.

Tom Greenwood · Answer

trying to do this integral. We, uh, first half Teoh dx on their the ah factor the denominator. So first we can factor out in X, and that makes that X squared minus four, which will then factor down because that&#x27;s the difference of two squares. So our decomposition to simplify this big fraction into a bunch of smaller, simpler fractions. The X cubed minus four x Well, we already know X is a factor, So there will be something over X and then the X squared minus four factors into an X minus two with a value up in the numerator and an X Plus two with Don&#x27;t value up in the numerator. So this big fraction can be separated into the three smaller fractions over X be over X minus two and see over X plus two all combined together. Question is where a B and C so we multiply by the common denominator which will get rid of all our fractions. So we get the numerator X squared plus 12 X minus four equals multiplying. Each of these the denominators going to reduce and you&#x27;re just left with the numerator times. The other two parts, so eight times The X minus two and X plus two is eight times the X squared minus four. So that&#x27;s a X squared minus four A. B times The X and the X Plus two will give us be X squared plus to be Times X and then C times the X and the X minus two is going to give us plus c X squared minus two. See X. Now we can equate our terms and the coefficients. The X squared has to equal our X squared. So one is gonna equal Hey plus B posse and then 12 X is going to equal our ex terms over here of the to be minus two C and our constant of the minus four has to equal are constant over here. Which means that we have negative four equals negative for hay, which tells us that a has two equal one. And then we can substitute that back in here and figure out being see and we end up with be equal street and C equals negative three. So this fraction and it&#x27;s interval can be rewritten as one over x plus three over X minus two minus three over x plus two DX. And now we can integrate all three of those individually pulling the constant out in front of the integral, using the one over you Do you rule? So we end up with the natural log of the absolute value of X plus three times the natural log of the absolute value of X minus two minus three times the natural look of the absolute value of X plus two. And then we will have our policy, the arbitrary constant. Yeah, and now it can use our walls of algorithms to combine all this into one algorithmic expression algorithm of the apse with value of X Times X minus two to the third power, the adding, multiplying those together and the constant of three becoming the exponents and the subtracting becomes division so divided by the X Plus two raised to the third power. Same thing, the constant becomes the exponents and we&#x27;ll under absolute value and plus our

Ernest Castorena · Answer

they want to get the integral, uh, exquisite First while that minus 40. Over. Excuse mine. That&#x27;s for it yet. And what if we do partial fraction decomposition on this rational function? We&#x27;ll end up with one over X minus three over X plus two plus three over X minus two. So we&#x27;re actually taking the interval. Uh, but see, one of the wrecks my street over expressed clustering over X man. It&#x27;s too. Yes. Uh, I will give us natural. Look, uh, let&#x27;s see. You can collect the money. Thanks. Times X minus two with metre log properties. Divide by explodes to give also with better luck properties. And I need that, and I&#x27;m a trade constancy.

Sid Wan · Answer

Okay. Finds an integral off That&#x27;s over full. Tend Actual full square the ah, that&#x27;s C If we take at our four square you to be that he was Explorer over sixteen D&#x27;You equals axe over Ate the X so we can run this a cz This guy is to see you, Tad. You too. Do you on your hands? You Because ah, well two in front collective national off cause I you we&#x27;LL see if you plugging you to be x square over sixteens that give to natural log because I asked Square over sixteen c, yeah.

Amy Jiang · Answer

Okay, so we want to use use up here. So why don&#x27;t we let are you equal are, um, larger polynomial, that is, Let&#x27;s let it equal X squared plus two x. But it&#x27;s what you know, if we take the derivative of that, you get to acts plus two and Indy X, So notice that we have two x puts to here and a DX so that&#x27;s equal to do you. So now let&#x27;s replace what we have. So we have do you over u to the power of four. Okay. So look to your right that as a negative exported so that you to the part of negative four. Do you? And I want to take her in a girl So he gets, um You took out negative three over negative three. Let&#x27;s see. And now, um, what&#x27;s your place? Are you with our X? So we have X squared plus two minus four to the part of negative three over the next three. Your name, plus C

Bobby Barnes · Answer

So if we want to find the anti derivative of this expression here, first thing we might want to look for is what most complicated portion of this function is that might so lead to something that if we take the derivative little kind of cancel out. So mine first thought is that X squared plus three is the most difficult thing here. And if I take the derivative of this, I end up with two eggs, which kind of looks like negative for ex. So you may say, Well, let you equal X squared plus running. Taking the derivative of this with respect to X gives you ideas Max physical to two hooks and then solving for what? Yeah, So now solving for d X, I want to multiply over by t X divided by six and one that will do over to ATS is equal to DX. So now let&#x27;s go ahead and do our substitution. Sze so negative or axe Well oh, so in the denominator will have are you and then DX becomes you over to so first noticed how thanks here and x there, counselor and the negative for over to give negative to remove the negative too. And I end up with just the integral of one of you. Did you so negative to now? Remember that he anti derivative of one over you is the natural log of the absolute hell you of you watch some constant C. Now we can go ahead and take what we&#x27;ve been partners for you, which is X squared plus one and plugged him and doing that, we have negative to natural law of the absolute value of X squared plus three kwasi and then X squared Plus three is always better than zero. So this here isn&#x27;t an issue, So I could go ahead and drop the absolute values and get natural log of exploited plus me pussy This year will be our indefinite.

Problem

Integrate each of the given functions. $$\int_{1}…

Problem 17 Easy Difficulty

Integrate each of the given functions.
$$\int \frac{12 x^{2}-4 x-2}{4 x^{3}-x} d x$$

Answer

$2 \ln |x|+\frac{3}{2} \ln |2 x+1|+\frac{1}{2} \ln |2 x-1|+C$

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Grace H.

Heather Z.

Caleb E.

Samuel H.

Precalculus Review - Intro

Functions on the Real Line - Overview

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$2 \ln |x|+\frac{3}{2} \ln |2 x+1|+\frac{1}{2} \ln |2 x-1|+C$

Basic Technical Mathematics with Calculus

Grace H.

Heather Z.

Caleb E.

Samuel H.

Precalculus Review - Intro

Functions on the Real Line - Overview

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

Grace H.

Heather Z.

Caleb E.

Samuel H.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus