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Solve the given problems by integration.To integrate $\int \frac{x^{2} d x}{(x-2)(x+3)},$ explain why $A$ and $B$ cannot be$$\text { found if we let } \frac{x^{2}}{(x-2)(x+3)}=\frac{A}{x-2}+\frac{B}{x+3}$$
Calculus 1 / AB
Chapter 28
Methods of Integration
Section 9
Integration by Partial Fractions: Nonrepeated Linear Factors
Integrals
Missouri State University
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Baylor University
University of Michigan - Ann Arbor
Lectures
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So we have the integral off two X plus one over X squared plus one DX, and ah, based on no question a we need to fill in the blanks by splitting the inte grand into two fractions. Okay, so this is also equal to so based on part A If you want to split it into two, that'll be two X over X squared plus one plus in to go off one over X squared, plus one. So this is the answer for part A. Here for part B, you need to evaluate the to and two girls. Ah, so let's do that. So for part A, that's integrate 1st 2 x over X squared, plus one. So let's take you as X square plus one. So I'll be using substitution here, do you? Over DX is two x, which means that ah DX is do you over to X so moving on from there. That's two X over you times do you over to x two x into X Cancel. Now we have into girl off one over you, Do you? So the answer to that is Ln of you, Ellen of you can just be written as Ellen off next squared, plus one next squared plus one. So there's your answer to your first part of your for this part here. Now we need to solve for the second part here. So it's labeled it says one for two. You have integral off one over X squared plus one. This is a standard integral. So it'll be arc tanned X So that would be your answer for the second part. So finally your integral would be Ln off x squared plus one plus arc 10 hex plus c. So this is your final answer for the integral?
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