Integration of Rational Functions with Partial Fractions

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Homework Help Overview

The discussion revolves around the integration of the rational function 1/(x^2-5x+6) using techniques such as partial fractions. Participants are exploring various methods to approach the integration problem.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation, Assumption checking

Approaches and Questions Raised

  • Participants discuss the possibility of factoring the denominator and the use of partial fractions. There are attempts to integrate the function using substitution and integration by parts, with some questioning the effectiveness of these methods. Others express uncertainty about the correctness of their values for A and B in the partial fraction decomposition.

Discussion Status

The discussion is active, with participants sharing their attempts and reasoning. Some guidance has been offered regarding the use of partial fractions, and there is a recognition of the need to verify the values of A and B. Multiple approaches are being considered, but no consensus has been reached on the best method to proceed.

Contextual Notes

Participants are working under the constraints of homework rules, which may limit the types of methods they can use or the information they can assume. There is an ongoing exploration of assumptions related to the integration process.

alexis36
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Homework Statement


Integration of 1/(x^2-5x+6)


Homework Equations





The Attempt at a Solution


I know i cannot do ln|x^2-5x+6|
I've tried some form of substitution or intergration by parts, and they don't work.
Should I factor the bottom?
 
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Have you tried partial fractions, as the title suggests? To be brief, you can rewrite \frac{1}{x^2-5x+6} as \frac{a}{x-2}+\frac{b}{x-3} where you must find a and b.
 
Last edited:
Okay.. I've solved the question up to a certain point. (I've decided that it's do-able without partial practions..)
however; now I am stuck on an integration:
i might find the integral of
1/(x^2-5x+6)
and now I am not sure if i use substituion, integration by parts, or bring the bottom up to the top of the fraction for this integration?
 
alexis36 said:
I've solved the question up to a certain point. (I've decided that it's do-able without partial practions..)
Ok, show us then. Maybe we can help you from there.
 
So partial fractions is the way to go I think..
I have
A/(x-3) + B/(x-2) = )A(x-2)+B(x-3))/(x-2)(x-3)
Ive solved for A and B and I got 1 for each of them.
Now I need to go and integrate 1/(x-2) and 1/(x-3) right? and set that equal to (t+c) and then solve the autonomous equation as i would any?
 
Now I need to go and integrate 1/(x-2) and 1/(x-3) right?
Yes, you would integrate it from here.
 
A=1 and B=1 doesn't work. Think about the signs again.
 

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