# What is the arclength of the polar curve f(theta) = 2thetasin(5theta)-thetacot2theta  over theta in [pi/12,pi/2] ?

Jan 17, 2018

$\infty$
For this solution, I am assuming that your $f$ is actually referring to the radial component.
If we travel an infinitely small angle, the distance will be f($\theta$) d$\theta$. Therefore, we can integrate that function across that whole range to get the value, i.e. ${\int}_{\frac{\pi}{12}}^{\frac{\pi}{2}} f \left(\theta\right) d \theta$
However, since cotangent goes to infinity very quickly at $\frac{\pi}{2}$, this integral does not converge to a finite value, hence the length is infinite.