Mastering Integrals: Solving Tricky Equations with x and Quotient Rule

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Homework Help Overview

The discussion revolves around integrating rational functions, specifically focusing on the integrals of the expressions x/(x^2+x-6) and 4/(x^2+4x+4). Participants explore various methods for solving these integrals, including the quotient rule and partial fractions.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the use of the quotient rule, question its applicability to integrals, and suggest using partial fractions instead. There are attempts to factor the denominators and apply substitution methods. Some participants express uncertainty about their solutions and seek clarification on the correct approach.

Discussion Status

The discussion is active, with participants providing feedback on each other's attempts. Some guidance has been offered regarding the use of partial fractions and substitution methods. There is a recognition of errors in initial attempts, and participants are encouraged to refine their approaches.

Contextual Notes

There is mention of homework constraints, including the requirement to avoid certain methods and the need for clarity in presenting work. Participants are also navigating through different interpretations of the integral solutions.

zombieguy
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Homework Statement



How do I do these 2?

a) x/(x^2+x-6)

b) 4/(x^2+4x+4)

Homework Equations





The Attempt at a Solution



a) Tried quotient rule it gave me -x^2 - 6/(x^2+x+6)^2, I don't think it's right

b) Tried substitution (because there's another similar question between these two on the paper and that was the way to go for it) Don't think my answers worth mentioning
 
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There's no 'quotient rule' for integrals! Try partial fractions.
 
Factor first the denominator then use the partial fractions.
 
Ok, Thanks

I've now got:

a) 3/5ln(x+3)+2/5ln(x-2)

b) 2/(x+2)ln(x+2)^2

Is that right?
 
Part b is not correct. After factoring you have
[tex]\int \frac{4}{(x + 2)^2}~dx[/tex]

You can do this with an ordinary substitution. Your answer should NOT have a log in it!
 
If you mean (3/5)ln(x+3)+(2/5)ln(x-2), (use more parentheses!) then the first one looks ok. I don't like the looks of the second one. If you had shown your work, I might have been able to tell you where you went wrong.
 
Last edited:
OK, I've tried b) again but this time by using:

u.v-(integral of)(v.du/dx)

u=4
du/dx=0 (That gets rid of the integral part)

v=-(x+2)^-1
dv/dx= (x+2)^-2

so: 4.-(x+2)^-1

It would be nice if you could show me how to do it your way (using substitution) aswel
 
Let u=(x+2). du=dx. Then 4/(x+2)^2 dx=4/u^2 du=4*u^(-2) du. Use the power law integral u^n=u^(n+1)/(n+1).
 
Thank you all for your help
 

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