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Numerade Educator

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Problem 15 Medium Difficulty

Find the exact length of the curve.

$ y = \ln (\sec x) $ , $ 0 \le x \le \frac{\pi}{4} $

Answer

$\ln (\sqrt{2}+1)$

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Video Transcript

Hey, guys, it's clear. So when you read here so we have y is equal to l end of seeking and access between zero and pi over four included. So when we take the derivative, we get a belt on ah, seeking just equal to one over. Seek it times seek it Times tangent. This becomes just engine No using the r. Klink formula. So we got zero high over four square root of one plus tangent square the ex and this is equal to from zero to pi over four square of secret square which is equal to from zero to pi over four second 50 x and we keep simplifying through the pi over four seeking limes Seek it less 10 gin over a seton plus engine the ex This is equal to zero high over four sequence square plus seek it times tension over seeking close tangent DX We see that the numerator is a derivative of the denominator. So this is equal to well and seek it plus 10 gents, absolute value 102 pi over four. We'll continue up here and this becomes L and seeking clover floor plus tangent pi over four minus felt end of absolute values. Speaking of zero minus 10 gin of zero, which is equal to album a skirt to plus one.