Integrating with Negative Signs: Effects on Solutions

[rizardon]

While working on an integration problem I found that I will arrive at two different solutions depending on how I approach it.

I'm finding the arc length of y=ln(1-x²) on the interval [0,0.5]

The formula for finding the arc length is ∫sqrt[1+[f'(x)]²]dx

So f'(x) = -2x / ( 1-x² )

Here I first simplify this to 2x / ( x² - 1 ) and squaring gives

4x² / ( x² -1 )²

Working from here I end up integrating from 0 to 0.5

∫ [1 + 1/(x-1) - 1/(x+1)] dx = 0.5 - ln3

On the other hand if I leave f'(x) as it is without simplifying, when I squared f'(x) I get

4x² / ( 1-x² )²

and end up integrating from 0 to 0.5

∫ [1 + 1/(1+x) + 1/(1-x)] dx = -0.5 - ln3

Should both have the same solution or is this simply a possible effect from squaring numbers?

Thank you

 

[arildno]

Note that when taking the square root of the perfect square in your denominator, you must use the ABSOLUTE value, |x^2-1| as your new denominator.