Answer
$$\kappa(t)= \frac{e^{t}}{\left(e^{2 t}+1\right)^{3 / 2}}.$$
Work Step by Step
Since $ r(t) = \lt 1,e^t, t\gt$, then $ r'(t) = \lt 0,e^t, 1\gt$ and hence $\|r'(t)\|=\sqrt{e^{2t}+1}$, $T(t)=\frac{r'(t)}{\|r'(t)\|}=\frac{ \lt 0,e^t, 1\gt}{\sqrt{e^{2t}+1}}$. Now, the curvature is given by
$$\kappa(t)=\frac{1}{\|r'(t)\|}\|\frac{dT}{dt}\|=\frac{e^{t}}{\left(e^{2 t}+1\right)^{3 / 2}}.$$
