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Numerade Educator

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Problem 37 Hard Difficulty

Find the arc length function for the curve $ y = \sin^{-1} x + \sqrt{1 - x^2} $ with starting point $ (0, 1) $.

Answer

$\sqrt{2}[2 \sqrt{1+t}]_{0}^{x}=2 \sqrt{2}(\sqrt{1+x}-1)$

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Video Transcript

he is clear. So when you read here, so we have f of acts is equal to inverse sine plus square root of one minus x square. So we take the derivative When we got one over square of one minus X square minus two x over to times square root of one minus x square. This becomes equal to one minus X over a square of one minus X square, which is equal to square root of one minus X over one plus nuts. So in our arguing formula, we know that there is the square root of one plus F, the derivative of F square. So one points here, I'll write it. This becomes equal to the square root of two over one plus X, which is equal to we just square with them individually. So we just have to plug it into our Klink function. So from zero to x square, Burt of two over square root of one plus t 18 we're gonna make you be equal to one plus t. So d t is equal to do you. This changes our limits of integration too. One one plus x. We're just adding one when we get square root of two integration 11 plus X one over square root of you, Do you? This becomes equal to square it of you to you to the 1/2 power from 1 to 1 plus acts, miss equals two square root of two runs a swear it of one plus x minus one