giving the dynamics of a GBM: $dX_t=\alpha X_tdt+\sigma X_tdW_t$, I know that the solution $X_t=X_0 e^{(\alpha-\frac{1}{2}\sigma^2)t+\sigma W_t}$ is given by ito applied to $\ln(X_t)$. My question is: Where is the mistake if I compute the solution in this other way:
$\frac{dX_t}{X_t}=\alpha dt +\sigma dW_t$
$\int_0^t\frac{dX_t}{X_t}=\int_0^t\alpha dt +\int_0^t\sigma dW_t$
$[\ln(X_t)]_0^t=\alpha t+\sigma W_t$
$\ln(\frac{X_t}{X_0})=\alpha t+\sigma W_t$
$X_t=X_0 e^{\alpha t+\sigma W_t}$
For an Ito integral
$$\int_0^t \frac{dX_t}{X_t} \neq [\ln(X_t)]_0^t.$$
You need to apply Ito's lemma here.