Let $\gamma$ be a smooth unit-speed curve $[0,L] \to\mathbb{R}^{3}$, and let $X \colon [0, L] \to \mathbb{R}^{3}$ be a normal vector field along $\gamma$, i.e., a vector field along $\gamma$ such that $\langle X, \gamma' \rangle =0$.
Given a positively oriented orthonormal frame $(\gamma', V, W)$ along $\gamma$, one can write $$ X = X^{1}V + X^{2}W $$ for some functions $X^{1}$ and $X^{2}$.
I wonder if the integral
$$ \int_{\gamma}X = \int_{0}^{L} X^{1} \, dt + \int_{0}^{L} X^{2} \, dt $$ is well-defined, meaning that its value does not depend on the choice of vector fields $V, W$ composing the frame $(\gamma', V, W)$.
I suspect that the answer is no. But can one say anything interesting if, for instance, one finds that the integral computed with respect to a particular frame vanishes?
EDIT. As pointed out by lisyarus in the comments, the answer to both question in negative. Would anything change if we assumed the vector fields $V,W$ be closed, i.e., $V(0) = V(L)$ and $W(0) = W(L)$?