I want to apply Girsanov's theorem to
$B_t = X_t - \int_0^t b(X_s)ds$
where $X_t$ is a Brownian motion and $b$ is Lipschitz. Therefore I have to show that
$Z_t = \exp(\int\limits_0^t b(X_s)dX_s - \frac{1}{2} \int\limits_0^t|b(X_s)|^2ds)$
is a martingale. In order to do so I want to apply Novikov's condition, which gives me the martingale property for $Z$, if
$\mathbb{E}[\exp(\frac{1}{2} \int\limits_0^t|b(X_s)|^2ds)] < +\infty$
Jensen and the Lipschitz property give me
$\mathbb{E}[\exp(\frac{1}{2} \int\limits_0^t|b(X_s)|^2ds)] \: \leq \: ...\leq \mathbb{E} [\int\limits_0^t\exp(\frac{C}{2}|X_s|^2)ds]$
But how can I continue to find the finiteness?