I am trying to evaluate the following integral: \begin{equation} \int_{M} \frac{(g')^2}{ g^{5/2}} - \frac{(g'')}{g^{3/2}} \ dx \end{equation} where $(M,g)$ is a one - dimensional closed and compact Riemannian manifold with metric g. So g' = \frac{dg}{dx} and likewise g'' = \frac{d^2g}{dx^2} (locally).
I suspect the integral to be zero but I am not sure how to proceed. In particular I'm not even sure whether the integral above makes sense .. I am totally new to Riemannian Geometry so in case the above is ill - defined please say so. If anyone could help that would be great, many thanks!