$\textbf{Problem}:$ Find the hyperbolic length of the path $t \mapsto t+i$ for $t \in [0,x]$.
Of course I immediateley considered the length formula: \begin{equation*} L(\gamma) = \int_0^x ||\gamma'(t)||dt \end{equation*}
Where $\gamma$ is the path in question.
My question is, how do I explicitly compute $\gamma'(t)$ here? I attempted it and got $\frac{x}{y^2}$ for the integral but this doesn't seem right. Any input is appreciated.