Question

Find the length of the curve \(y = 2\ln \left( {\sin \frac{1}{2}x} \right),\frac{\pi }{3} \le x \le \pi \)

Short answer

The length of the curve is \(2\ln (2 + \sqrt 3 )\)

Step-by-step solution

Applied the length of the curve formula

By using the formula for length of the curve

\(\begin{aligned}L &= \int_a^b {\sqrt {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} } dx\\b &= \pi \\a &= \frac{\pi}{3}\\y &= 2\ln \left( {\sin \frac{1}{2}x} \right)\\\frac{{dy}}{{dx}} &= - \ln \left( {{{\sin }^2}\left({\frac{x}{2}} \right)} \right)\end{aligned}\)

Solution of the length of the curve

\(\begin{aligned}L &= \int_a^b {\sqrt {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} } dx\\ &= \int_{\frac{\pi }{3}}^\pi {\sqrt {1 + {{\left( { - \ln \left( {{{\sin }^2}\left( {\frac{x}{2}} \right)} \right)} \right)}^2}} } dx\\L &= 2\ln (2 + \sqrt 3 )\end{aligned}\)