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Algebraic manipulation for easier integration

  • Thread starter stmbs02
  • Start date
  • #1
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Homework Statement



I am trying to integrate x^2/(1+x^2 ) from 0 to 1.

The Attempt at a Solution



We recently worked on trig substitutions in class, but, rather than substituting x for tan(theta) I think there may be an easier way via algebraic manipulation. If I divide both numerator and denominator by the highest power of X in the denominator (x^2), then I get back out

1/(1/x^2 +1)

which is equal to x^2/2.

Now, I can easily integrate 1/2 x^2 without trig substitution.

My main question is: Is this a viable method for simplifying the integral, or must I go through trig substitution?
Thanks
 

Answers and Replies

  • #2
rock.freak667
Homework Helper
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I don't think what you did made it simpler, but here is what I'd do


[tex]\int_0 ^{1} \frac{x^2}{x^2+1}dx = \int_0 ^{1} \frac{x^2+1-1}{x^2+1}dx = \int_0 ^{1} \left( 1- \frac{1}{x^2+1} \right)dx[/tex]
 
  • #3
Hurkyl
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1/(1/x^2 +1)

which is equal to x^2/2.
:confused:
 
  • #4
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:confused:
Doesn't it? Algebra is not exactly my strong suit. 1/(1/x^2) is x^2 (the denominator of the denominator is the numerator), right?
 
  • #5
Hurkyl
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1/(1/x^2) is x^2 (the denominator of the denominator is the numerator), right?
This is true, but not relevant to the problem -- there's a "+1" in there, and you can't just ignore it.

This is true:
[tex]\frac{1}{\frac{1}{q}} = q[/tex]

but you're working with
[tex]\frac{1}{\frac{1}{q} + r}[/tex]

which is not of the form of the left hand side.


Incidentally, if you're ever unsure about an algebraic identity, you should do a sanity check -- try plugging in some numbers. If your algebraic manipulation is valid, then plugging in numbers should give equal results.

(Though beware -- getting equal results doesn't prove your manipulation is valid)
 
  • #6
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Thanks for the help. rock.freak667, I think what you did makes a lot more sense than what I did.
 

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