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Numerade Educator

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Problem 19 Hard Difficulty

Find the exact length of the curve.

$ y = \ln (1 - x^2) $ , $ 0 \le x \le \frac{1}{2} $

Answer

$-\frac{1}{2}+\ln 3$

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Video Transcript

it's Clara. So when you read here, So here, we're gonna find that exactly. We're gonna first start with the derivative. And we got negative two acts over one minus x square. We're gonna use this on plug it into our our Klink equation Square root of one plus negative to X over one minus x square square. When we factor this out we got four x square all over one minus two X square less x to the fourth when we're gonna make one bye. Using a common denominator two X square, it was four x because X to the fore. Excuse me one minus two X square plus X to the fourth. And when we add and simplify, this part becomes from zero to wouldn't have one plus x square over one minus x squared DX. Because we're taking this square root off the integral. We're gonna defy the top and bottom. Using long division in this equals from 0 to 1/2 negative one plus two over one minus x squared d x and we're going to use partial fractions. So a over one plus tax must be over one minus X or finding A and B and we get a to be one be to be one. So it becomes negative X plus from 0 to 1/2 one over one plus x plus one over one minus x the ex. When we just integrate this, we got negative one have plus Helton of three.