= Solution
Apply the <Itô formula> to $M_t=e^{X_t}$. Since $d[X]_t=\sigma(X_t)^2dt$ and $b=-\sigma^2/2$, the two <drift> terms cancel:
$$
dM_t=e^{X_t}dX_t+\frac12e^{X_t}d[X]_t
=\frac{e^{X_t}}{1+X_t^2}\,dB_t.
$$
The integrand is continuous and hence locally square integrable after stopping at bounded levels of $X$. Thus $M$ is a positive <continuous local martingale>, with $M_0=1$ and <quadratic variation>
$$
\boxed{M_t=1+\int_0^t\frac{e^{X_s}}{1+X_s^2}\,dB_s,\qquad
[M]_t=\langle M\rangle_t=\int_0^t\frac{e^{2X_s}}{(1+X_s^2)^2}\,ds.}
$$
The cancellation is the reason $e^x$ is the relevant <scale function> for this diffusion.
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