How to parameterize a curve by arc-length, if the curve's position at time t=0 does not coincide with the origin.

Question

$$ \mathbf x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} \sinh t \\ 0 \end{bmatrix}$$ $ \mathrm D $ is the derivative operator. $$ \mathrm D \mathbf x (t) = \begin{bmatrix} \cosh t \\ 0 \end{bmatrix} $$ $$ (\mathrm d s)^2 = \mathrm d t \mathrm D^\mathtt T \mathbf x (t) \mathrm D \mathbf x (t) \mathrm d t $$ $$ \int \mathrm d s = \int \cosh t \, \mathrm d t $$ $$ s (t) = \int_{q=0}^{q=t} \cosh q \, \mathrm d q = \sinh t - \sinh 0 = \sinh t $$ Proposition: $$ x_1 (t=0) = 0 \land x_2 (t=0) = 0 \implies \int \sqrt {\mathrm D^\mathtt T \mathbf x (0) \mathrm D \mathbf x (0)} \, \mathrm d t = 0 + c $$ Here is the question: Under what conditions is the proposition true? If the proposition is false, then I think the arc-length, $ \mathit s $, at time $\mathrm d t $ can be some quite large quantity; this situation would be difficult to interpret.