How Can I Evaluate This Integral of a Rational Function?

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Discussion Overview

The discussion revolves around evaluating integrals of rational and radical functions, specifically focusing on the integral \(\int\frac{dx}{x\sqrt{2-x-x^2}}\) and a related integral \(\int\frac{\sqrt{2-x-x^2}}{x^2}dx\). Participants explore various substitution methods and express concerns about the complexity of rewriting results in terms of the original variable.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant suggests using substitutions \(x+\frac{1}{2}=\frac{3}{2}\sin t\) and \(u=\tan\frac{t}{2}\) to solve the integral \(\int\frac{dx}{x\sqrt{2-x-x^2}}\), expressing uncertainty about the final expression in terms of \(x\).
  • Another participant agrees that the suggested substitution is correct but notes that their resulting expression differs slightly, indicating potential variations in the approach.
  • Several participants express that while the method is valid, they find the process tedious and seek alternative methods or substitutions to simplify the integration.
  • One participant points out that the integrals discussed are not rational functions as stated in the thread title, suggesting that "radical functions" may have been the intended term.
  • Another participant elaborates on the substitution process, providing a detailed transformation of variables, but acknowledges that rewriting the result in terms of \(x\) is not overly complicated.

Areas of Agreement / Disagreement

Participants generally agree on the validity of the substitution methods proposed, but there is no consensus on the ease or complexity of rewriting the final answers in terms of \(x\). Additionally, there is a disagreement regarding the classification of the functions involved, with some participants noting the inaccuracy in referring to them as rational functions.

Contextual Notes

Participants express varying levels of comfort with the complexity of the integration process and the transformations involved. There is also a lack of clarity on the definitions being used, particularly regarding the classification of the functions as rational or radical.

y_lindsay
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i'm trapped with a problem: \int\frac{dx}{x\sqrt{2-x-x^2}}.

i think this problem could be solved by subtitutions: \ x+\frac{1}{2}=\frac{3}{2}sint and \ u=tan\frac{t}{2}.
and finally we would get an expression in \ u: \frac{\sqrt{2}}{4} log\left|\frac{2\sqrt{2}+u-3}{2\sqrt{2}-u+3}\right|
(am i right so far?)

however i find it difficult and tedious to write the result in x and get the final answer.

does anyone know how to evaluate this integral in an alternative way?

Thanks a lot.
 
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another question is how to evaluate the integral \int\frac{\sqrt{2-x-x^2}}{x^2}dx.
i used the method of integration by parts, anyone knows some smarter way to do it?
 
y_lindsay said:
i'm trapped with a problem: \int\frac{dx}{x\sqrt{2-x-x^2}}.

i think this problem could be solved by subtitutions: \ x+\frac{1}{2}=\frac{3}{2}sint and \ u=tan\frac{t}{2}.

yes this is the correct substitution. my answer is slight different from yours, but it could be just me not doing it carefully, or they are actually the same but written in slightly different form. Anyway method is correct.
 
thanks mjsd.

i know that the method itself is correct, but it just seems a little tedious, especially when we need to write the final answer in variable x.

is there any other substitution we could use to attack this integral? or any alternative methods rather than the routine process to integrate rational function?
 
y_lindsay said:
thanks mjsd.

i know that the method itself is correct, but it just seems a little tedious, especially when we need to write the final answer in variable x.

is there any other substitution we could use to attack this integral? or any alternative methods rather than the routine process to integrate rational function?

when I have time I may try to do this again, but I do not believe it is overly complicated when you put it in terms of x. (well, that depends on your definition of "complicated"), you probably just got to know some tricks to simplify it.
 
By the way, you do understand that these are not "rational functions", as you said in your title, don't you? Was that just a typo for "radical functions"?
 
It may be a tiny bit 'tedious' to write again in terms of x, but not difficult, and no where near as tedious as have doing that integral in the first place..

x+\frac{1}{2}=\frac{3}{2}\sin t so \sin t = (x+\frac{1}{2})(\frac{2}{3}) = \frac{2x+1}{3} hence, t= \arcsin \left(\frac{2x+1}{3}\right)

So

u = \tan \left( \frac{\arcsin \left(\frac{2x+1}{3}\right)}{2} \right).
 

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