Question

show that the length of the curve y= log Sec x between the point x=0 and x=60 is log(2+√ 3)

Answer 1

Answer on Question #37620, Math, Differential Calculus | Equations

The length of the curve if it is given in form $y = f(x)$ from point $x_0$ to $x_1$ is

$$S = \int_ {x _ {0}} ^ {x _ {1}} \sqrt {1 + \left(f _ {x}\right) ^ {2}} d x. \text{ For } f (x) = \ln \sec x = \ln \frac {1}{\cos x}.$$ $$S = \int_ {0} ^ {\frac {\pi}{3}} \sqrt {1 + \frac {\sin^ {2} x}{\cos^ {2} x}} d x = \int_ {0} ^ {\frac {\pi}{3}} \sqrt {1 + t g ^ {2} x} d x = \int_ {0} ^ {\frac {\pi}{3}} \frac {d x}{\cos x} = - \ln (\cos \frac {x}{2} - \sin \frac {x}{2}) + \ln (\cos \frac {x}{2} + \sin \frac {x}{2})) _ {0} ^ {\frac {\pi}{3}} = - \ln (\frac {\sqrt {3} - 1}{2}) + \ln (\frac {\sqrt {3} + 1}{2}) = \ln \left(\frac {\sqrt {3} + 1}{\sqrt {3} - 1}\right) = \ln (\sqrt {3} + 2).$$