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-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
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.