The first thought I had to solve this problem was using the integral,
$$ \int_1^\infty \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\: \:dt $$ Once you solve for the derivatives you get this expression and do a lot of simplifying via maxima:
$$ \int_1^\infty \sqrt{\frac{1}{t^4}+ \left( \frac{2t\cos t\sin t + (1-t^2)\cos(t)^2 - 1}{t^4} \right) } \, dt $$
adding and taking the square root(sorry if my latex is bad this is my first time).
However one cannot integrate this so I am stuck. I have thought about converting this to cartesian coordinate, however the graph looks radically different. The function is $x\cdot sin(\frac{1}{x})$ which instinct tell me is also not integratable. Is there any way to solve this problem please help I have not found a solution anywhere. Could converting to polar and then taking the integral help. This is from calculus eight edition. I have found out that is path length is divergent can you prove that?
Consider the following sequence of $t$-values: $$t = \frac{\pi}{2} , \,\,\,\, 3 \cdot \frac{\pi}{2}, \,\,\,\, 5 \cdot \frac{\pi}{2}, \,\,\,\, 7 \frac{\pi}{2},\,\,\,\, \ldots $$ The corresponding $y$-values are $$y = \frac{2}{\pi}, \,\,\,\, - \frac{1}{3} \cdot \frac{2}{\pi}, \,\,\,\, \frac{1}{5} \frac{2}{\pi}, \,\,\,\, - \frac{1}{7} \frac{2}{\pi}, \,\,\,\,\ldots $$ So as your graph zig-zags up and down from one of these $y$-values to the next, its length must be no less than than $2/\pi$ times $$1 + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \cdots = +\infty $$