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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.$$\int \frac{3 x+2}{x^{2}+x} d x$$

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

Calculus 1 / AB

Chapter 28

Methods of Integration

Section 9

Integration by Partial Fractions: Nonrepeated Linear Factors

Integrals

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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'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's negative three. So here our integral known becomes that. This is This is positive to right. So I'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're going forward, so I'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's just three times the natural Log off, X close one. Okay, Now I want to combine the two. So I'm just gonna bring this one to the top. It's gonna be natural. Log X squared and bring Just wanted to talk. It'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'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.

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