How can I integrate this using partial fractions?

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Discussion Overview

The discussion revolves around the integration of a rational function using partial fractions. Participants explore methods for factoring the denominator and simplifying the integral, while also addressing challenges in the process.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses uncertainty about how to proceed with the integral and seeks guidance.
  • Another suggests using partial fractions and refers to textbook resources for assistance.
  • A participant initially doubts the factorability of the denominator but later acknowledges it can be factored.
  • Some participants propose using substitution with \( u \) and its derivative \( du \) to simplify the integration process.
  • There is confusion regarding the factoring process, with one participant stating they are struggling to understand a previous suggestion.
  • Another participant mentions using Kramer's rule to find a root of the denominator and suggests that the integral is tedious but not particularly difficult.
  • One participant confirms a specific factorization of the denominator and encourages the use of partial fractions thereafter.

Areas of Agreement / Disagreement

Participants generally agree on the approach of using partial fractions, but there are differing opinions on the factorability of the denominator and the complexity of the integral. The discussion remains unresolved regarding the best method to proceed.

Contextual Notes

Some participants express uncertainty about the factoring process and the applicability of certain methods, indicating potential limitations in their understanding or the clarity of previous posts.

Adventurehero
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\int\frac{3x^4+4x^3+16x^2+20x+9}{x^5+2x^4+6x^3+12x^2+9x+18}dx

I tried using du/u for Lnu but I couldn't get it into that form. I don't have any idea what to do next, really. Can someone point me in the right direction?
 
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You should try expanding it via partial fractions. There should be a section in your textbook about this.
 
I would have sworn that denominator couldn't be factored, but it can!
 
take u = (x^5+2x^4+6x^3+12x^2+9x+18)
du = 5x^4+8x^3+18x^2+24x+9
then take the numerator & cancel it out with du.
 
What?

I'm having trouble factoring it, and I'm not sure what Nachore means.
 
Use Kramer on the denominator, you will find that -2 is a root.
The remaining part can be factorized easily.
Then use partial fractions to split the integral into more classical ones.

This is not a particularly difficult integral, although a very tedious one to solve.
You might say that this makes it a difficult one after all.

Edit: It is not as tedious as I thought at first, the solution is ln(x+2)+ln(x^2+3)-2/(x^2+3)+C
 
Last edited:
Nachore said:
take u = (x^5+2x^4+6x^3+12x^2+9x+18)
du = (5x^4+8x^3+18x^2+24x+9)dx
then take the numerator & cancel it out with du.
Yes, that's true. but since the numerator is nothing like that, it doesn't help at all!
 
I get (x+2)(x^2+3)^2 as the bottom, factored out. Is this correct? I'm really not good at factoring.
 
Yes, that's correct. Now use partial fractions.
 

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