Help with Integration Problem: \int \frac{\ x^2}{1+x^2} dx

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Discussion Overview

The discussion revolves around the integration of the function \(\int \frac{x^2}{1+x^2} dx\). Participants explore various methods for solving the integral, including integration by parts, partial fractions, and long division, while sharing their experiences and suggestions.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses difficulty with integration by parts for the given integral.
  • Another suggests using partial fractions, but acknowledges that it may lead to complex numbers, complicating the method.
  • A different participant proposes using long division to rewrite the integral as \(1 - \frac{1}{x^2+1}\), which they claim is easier to integrate.
  • Some participants reiterate the need for long division due to the degree of the numerator being equal to that of the denominator, making it a non-proper fraction.
  • One participant mentions that integration by parts is not necessary and points to an alternative approach involving trigonometric inverses.
  • A participant expresses gratitude for the help and indicates understanding after the discussion.

Areas of Agreement / Disagreement

There is no clear consensus on the best method to solve the integral, as participants propose different approaches and express varying levels of confidence in their suggestions. Some methods are contested, particularly regarding the use of partial fractions.

Contextual Notes

Participants highlight that the integral's structure complicates the use of certain methods, such as partial fractions, due to the nature of the fraction being non-proper. There are also references to potential complexities arising from the use of complex numbers.

trap
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can anyone help me with this,

[tex]\int \frac{\ x^2}{1+x^2} dx[/tex]

thanks.

I've tried with integration by parts but does not work..
 
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Hmm... I don't quite remember how to use partial fractions for that... but I remember there was a way (at least with a similar problem). You could try

[tex]\int{\frac{x^2}{1+x^2}dx}=\int{\frac{x^2}{(x-i)(x+i)}dx}[/tex]

and use partial fractions from there (but it gets a bit "hairy," as my math teacher always says).
 
trap said:
can anyone help me with this,

[tex]\int \frac{\ x^2}{1+x^2} dx[/tex]

thanks.

I've tried with integration by parts but does not work..
Did you try partial fractions?

edit: nevermind read above :)
 
trap said:
can anyone help me with this,

[tex]\int \frac{\ x^2}{1+x^2} dx[/tex]

thanks.

I've tried with integration by parts but does not work..

Partial fractions work only for proper fractions. This is not a proper fraction in the sense that the degree of the numerator is the same as the degree of the denominator. So to fix this little problem you need to use long division after which you get.

[tex]1-\frac{1}{x^2+1}[/tex]

Which is now easy to integrate.

Regards
 
Last edited:
Use x^2/(1 + x^2) = (x^2 + 1 - 1)/(1 + x^2) = 1 - 1/(1 + x^2) and integrate that instead...
 
In this case partial fractions would lead indeed to a decomposition containing complex numbers.Therefore the method is not good.

Daniel.
 
I did this on my calculator and got a simple answer. I think you can do integration by parts. What have you tried?
 
Townsend said:
Partial fractions work only for proper fractions. This is not a proper fraction in the sense that the degree of the numerator is the same as the degree of the denominator. So to fix this little problem you need to use long division after which you get.

[tex]1-\frac{1}{x^2+1}[/tex]

Which is now easy to integrate.

Regards

Integration by parts isn't necessary. Townsend clearly explains the best way to do this.

[tex]\int\frac{x^2}{x^2+1} dx = \int\frac{x^2+1}{x^2+1}-\frac{1}{x^2+1}dx[/tex]

Hint for the second part: trig inverses.
 
thank you everyone for the help, i understand now :smile:
 

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