How does one integrate a quotient?

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    Integrate quotient
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Discussion Overview

The discussion revolves around techniques for integrating quotients, specifically through the example of the integral \(\int\frac{3x + x^{2}}{5x - 1} dx\). Participants explore various methods for approaching this type of integral, including polynomial long division and substitution.

Discussion Character

  • Homework-related, Mathematical reasoning

Main Points Raised

  • One participant asks how to integrate a specific quotient and seeks assistance.
  • Another participant questions whether the example is related to homework.
  • A participant clarifies that the example is not from homework but is intended to aid future understanding.
  • One proposed method involves using polynomial long division to express the integrand as a polynomial plus a proper rational expression.
  • Another participant suggests a change of variable, specifically letting \(t=5x-1\), to rewrite the integral without \(x\) or \(dx\).

Areas of Agreement / Disagreement

Participants present different methods for integrating the quotient, indicating that multiple approaches exist without a consensus on a single preferred method.

Contextual Notes

Some methods may depend on familiarity with polynomial long division or substitution techniques, which could affect the applicability of the proposed approaches.

James2
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How do you integrate quotients? Let's practice with one like... \int\frac{3x + x^{2}}{5x - 1} dx

Thanks for the help!
 
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James2 said:
How do you integrate quotients? Let's practice with one like... \int\frac{3x + x^{2}}{5x - 1} dx
Is this homework?
 
No, I just pulled that one out of thin air. This will help me with homework in the future, but right now, that is just an example.
 
I would use polynomial long division to write the integrand as a polynomial plus a remainder term that is a proper rational expression. If you don't know this technique, do a search on wikipedia for "polynomial long division."
 
Another method : change of variable
t=5x-1
Rewrite the integral with t and dt (witout x nor dx in it)
 

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