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Find the exact length of the curve.

$ y = \ln (\cos x) $ , $ 0 \le x \le \frac{\pi}{3} $

$\ln (2+\sqrt{3})$

Applications of Integration

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we want to find the ark length of y equals Ellen co sign X. So, um, on the interval, um, X is greater than or equal to zero, but less than or equal to, um, pie over three. Uh, first, we'll want to find the derivative of why so we can plug that into her formula, so that will end up being negative. Sign of ex who account for co sign co sign X the U substitution. So one of her co sign, We can just say this is C can't x um c can X times sign is gonna be tangent, so it'll end up being negative Tangent X. So when we plug into our equation over here, we can take zero as a pie over three as b one, plus, um, tangent squared X D x, um, one plus tangent squared can also be written as c can't squared. So this is now a radical, um radical See? Can't squared, which cancels out, obviously. So we'll go to the next page. Um, and now we just have our integral from zero to pi over three of C can't d x and this just equals Ellen C Can't x plus tangent X on pi over three. Come on. And when we apply the fundamental their calculus and write all of this out it and, uh being Ln c can't pi over three plus tangent high over three minus Ellen C can't zero plus tangent zero and this comes down to Ellen two plus Brad three minus Ellen one plus zero which ends up just being Ellen Too close Rad three.