While working on an integration problem I found that I will arrive at two different solutions depending on how I approach it.(adsbygoogle = window.adsbygoogle || []).push({});

I'm finding the arc length of y=ln(1-x^{2}) 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-x^{2})

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

4x^{2}/ ( x^{2}-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

4x^{2}/ ( 1-x^{2})^{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

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# Operations with negative sign

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