let $\displaystyle \kappa$ denote the curvature of a closed spacecurve $\displaystyle \alpha$. Find $\displaystyle \oint_{\alpha} \frac{1}{\kappa} \ d\kappa$

Looking at this, I would imagine that this would either be $\displaystyle 0$ since $\displaystyle \alpha$ is closed or $\displaystyle T$, the tangent Vector, but I'm not to sure on how to achieve the desired result.

My other idea would have been to write it down as follows:

$\displaystyle \oint_{\alpha} \frac{\|\alpha'\|^3}{\|\alpha' \times \alpha''\|} \cdot \|\alpha'\| \ dt$

But these attempts don't seem to lead to anything.