write-out-the-form-of-the-partial-fractions-similar-to-that-shown-in-eq-i-of-example-2-that-would-be

Question

Write out the form of the partial fractions, similar to that shown in Eq. (I) of Example 2, that would be used to perform the indicated integrations. Do not evaluate the constants.
$$\int \frac{3 x+2}{x^{2}+x} d x$$

Yaw Asomani · Accepted Answer

at this one. Tu minus X right here and then X squared those eggs E x. Okay, we confirm dinner The X out off the denominator, so we just have this one left. Okay? None of this canoe decompose into partial fractions. So let us do the decomposition first, and then come back later. This is gonna be a your ex plus B over X plus one, right? And then we is gonna equate the numerator a X plus one close. Be X right now, If you let eggs of what is happening for lead exposure, this one goes to zero. So we eliminate be okay for, like, zero here. Then we&#x27;re just gonna have to if zero here, we just have one. So a is equal to two right now. We let eggs be negative one. What is happening? Let X is negative. One negative. One plus one is zero. So effectively eliminate a and then negative one times B is negative. Negative B. Right. And I put negative one here you have to minus negative one, which is positive. Three, therefore be is also negative three. Okay, So any is positive to NBC&#x27;s negative three. So here our integral known becomes that. This is This is positive to right. So I&#x27;m gonna put positive two over X, and then this is negative. Three is gonna be negative. Three over X close. What? Okay, on on the X right here. And then we&#x27;re going forward, so I&#x27;m just gonna make it turn by term integration. So two of her ex miners three over one plus X plus one. This is the X. This is the X o. K. So what is the integral of this one is just twice the natural log affects. And what is the integral of this one? It&#x27;s just three times the natural Log off, X close one. Okay, Now I want to combine the two. So I&#x27;m just gonna bring this one to the top. It&#x27;s gonna be natural. Log X squared and bring Just wanted to talk. It&#x27;s gonna be natural. Log X plus one. Que can cubic power. So it is this cubic power. Now I can combine them. This is gonna be natural love X squared over X plus one. Oh, cubic. Power. And then we&#x27;ll see. Ok, so, uh, that is that is, uh, one way you can do it so those becomes the answer to the question. Natural log eggs swayed right. Natural log X squared over experts one or cubic policy S. C.

Odera Egbuonu · Answer

So we want to find the partial fraction decomposition of X plus three over X squared plus two multiplied by two ex post one. So our denominator in this case will be broken into two parts. We have X X squared plus two and two X plus one. So each denominator in this case is already in its most simplified state For the numerator over the X squared plus two term. We want to make sure that the numerator is one power less than the tended denominator. So our numerator is going to be a X plus B. Now notice that the X term in the numerator is to the first power, while the denominator X squared plus two is to the second power. Now for the numerator over the two x plus one term. We know that X and the denominator is to the first power and so are numerator. For the two x plus one term, we&#x27;ll just have to be to the zero power. So that just leaves us with a constant of C. And so this will be our partial fraction decomposition

Amy Jiang · Answer

gets another integral of two over exposed to squared. I&#x27;m suing his ex yet. Okay, so we&#x27;re gonna have to take for use partial fraction to composition. To separate this, we get to over experts juice word 206. This is equal to a over. Experts still be over repeating. See, over its human is X Can&#x27;t No. Let&#x27;s multiply bar or else you&#x27;II get Sue if you go to a exposed to to metaphysics plus B which is two minutes x and see which is explicit Students wait. Okay, listen then to two is equal to or a minus a squirt. X word was to be minus B X. That&#x27;s the X squared Such force e x. Let&#x27;s foresee now it&#x27;s combined like Tim&#x27;s X squared is negative Papacy plus X, which is B B, but or C and a constant terms, which is for a for us to be with you. Okay, so we have that for a us to be that&#x27;s worth E. If he could do to you could have a plus sees You could schedule and we have it for C minus B. Is todo okay? Yeah. What you do here? Be okay. So based on this equation, we know that sees a So let&#x27;s plug in every sea value for a he gets We&#x27;re actually looks like an A for Caesar. We foresee missing or C And then we have a four. See here. So that&#x27;s Betsy. It was to be Is he good to you? It could multiply this by two. Yet it&#x27;s C, but it&#x27;s to be identical to don&#x27;t do we have 16 c is equal to so C is equal to two over 16 which is one over eight now itself for you know, that&#x27;s cuz go to a So actually 80 would see what, over a priest of this when you were here. We have four c. So for one over eight, is it, uh, might I speak? That&#x27;s equal to B. What? B is equal to 1/2. Okay, now that we have all our values, you can rewrite this as in a girl of A which was won over eight. So one over eight explicit, too. Plus B which was 1/2. So that&#x27;s one have experts to squared, and I believe we have won over eight again for C won over eight to minus X Okay, now it&#x27;s six in around this. He gets one over eight ln of exposed to minus went over to exports to and then minus. We&#x27;re actually plus well. And or one of her aides. You never ate Oh, and, uh, to minus X. Oh, you know, we get a negative here because it&#x27;s two minutes, except it&#x27;s negative over eights. Well, in a scene when his ex

Patrick Vaughn · Answer

and this problem. We want to use partial fractions to derive the integration Formula one over a squared minus X squared D X is equal to one over x u a l in absolute value of a plus X over a minus X complicity constant. Okay, so first up, we want to factor the denominator. So it&#x27;s difference of squares. So we get a minus x times a plus size. Next, uh, we want to set up our form of partial fraction decomposition. So we get one over a minus X times a plus X on that it&#x27;s gonna be Teoh is equal to some concept over a my sex. You know, Let&#x27;s just make this X minus egg in a negative year Expo Say, just like it&#x27;s easier for me to look at like this. So we have a over x minus. So capital a less excuse us if you being a over X minus a plus c over expose so that, uh uh, we can use the heavy side covered method since the factors or both linear and non repeated. So what are the roots? So X equals Azour in this annex equals. So then B is gonna be negative. Warm over. Do so I&#x27;m gonna have a close A which is gonna be negative. One over to A and C is going to be negative. One over. Negative. A minus eight. Just gonna equal one over to a right so we can rewrite the interval. So the integral becomes, um, negative one over x. You a times one. Over. Um, X plus. Explain it. Say sorry. Plus one over its you eight times one of her x plus a DX. Good. Good. Uh, when we immigrate this we&#x27;re gonna get one over to a Ellen absolute by you of X minus a. Okay. Plus one over its you a Helen Absolute value of X plus a. Now we can combine the l in terms, so we&#x27;re gonna get Ln absolute value. Uh, so we get a one over to a in front, so we get, uh, X plus A in the numerator and X minus a in the dominator. Was there constant? They might say, Hey, that&#x27;s not this cause the denominators 80 minus x not experience eight, But because the absolute value, we can just back her out negative if we want to you. And it doesn&#x27;t affect the absolute value targets. Right. That as negative acts plus a or whatever and that completes from

Odera Egbuonu · Answer

So we want to find the partial fraction decomposition of negative too over X squared, multiplied by X minus one. So in this part, we have on X squared term and then we have a nex minus one term. So because we have the X squared term, we have to break this into two terms and X to the first power and also an X to the second power. This is this term is only to the first power. And so we can leave it as it is. And so are partial fraction. Decomposition becomes a over X plus be over X squared, plus see over X minus one, and that is our partial fraction decomposition.

Problem

Write out the form of the partial fractions, simi…

Problem 3 Easy Difficulty

Write out the form of the partial fractions, similar to that shown in Eq. (I) of Example 2, that would be used to perform the indicated integrations. Do not evaluate the constants.
$$\int \frac{3 x+2}{x^{2}+x} d x$$

Answer

$\frac{A}{x}+\frac{B}{x+1}$

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Grace H.

Catherine R.

Anna Marie V.

Samuel H.

Precalculus Review - Intro

Functions on the Real Line - Overview

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$\frac{A}{x}+\frac{B}{x+1}$

Basic Technical Mathematics with Calculus

Grace H.

Catherine R.

Anna Marie V.

Samuel H.

Precalculus Review - Intro

Functions on the Real Line - Overview

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

Grace H.

Catherine R.

Anna Marie V.

Samuel H.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus