Algebraic manipulation for easier integration

[stmbs02]

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

 

[rock.freak667]

I don't think what you did made it simpler, but here is what I'd do


$$\int_0 ^{1} \frac{x^2}{x^2+1}dx = \int_0 ^{1} \frac{x^2+1-1}{x^2+1}dx = \int_0 ^{1} \left( 1- \frac{1}{x^2+1} \right)dx$$

 

[Hurkyl]

1/(1/x^2 +1)

which is equal to x^2/2.

 

[stmbs02]

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?

 

[Hurkyl]

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:
$$\frac{1}{\frac{1}{q}} = q$$

but you're working with
$$\frac{1}{\frac{1}{q} + r}$$

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)

 

[stmbs02]

Thanks for the help. rock.freak667, I think what you did makes a lot more sense than what I did.