How do I integrate a rational function with a square root in the denominator?

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Homework Help Overview

The discussion revolves around integrating a rational function with a square root in the denominator, specifically the integral of the form ∫(x/√(1-4x²))dx. Participants explore various methods to approach the problem without providing a complete solution.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster considers integration by parts but questions if there is a simpler method. Some participants suggest variable substitutions, such as u=4x² and u=1-4x², as potential approaches. There is also a discussion on how to identify when a substitution might be appropriate.

Discussion Status

Participants are actively engaging with the problem, offering different substitution methods and discussing the reasoning behind choosing substitutions. There is a sense of exploration regarding the techniques, but no consensus has been reached on a definitive method.

Contextual Notes

The original poster expresses uncertainty about the integration process due to the presence of the variable x in the numerator, indicating a potential challenge in applying standard integration techniques.

thomas49th
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[tex]\int \frac{x}{\sqrt(1-4x^{2})}dx[/tex]
If it was just a constant on top I could turn it into an arcsin, but there is a stupid x there, so not to sure. I though about intergrating by parts, but their must be an easier method than that?

Thanks
 
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Well, you could make the x above "disappear" by doing a change of variable u=4x².
 
Try the substitution u=1-4x^2.
 
cheers thanks got it!

btw, how do you spot a substitution? There is a rule(s) for it. When somthingis the product or differential of somthing else?

Thanks :)
 
When you see a multiple of the derivative of something else in the numerator, it's a dead giveaway for a substitution.
 

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