My Stochastic calculus professor always used to say "When in doubt use Ito!"
So let $f(t,x) = t x $ and compute $\partial_t f(t,x) = x$, $\partial_x f(t,x) = x $ and $ \partial_{xx} f(t,x) = 0$
Now the Ito lemma says for $f$ twice differentiable with respect to $x$ and once differentiable with respect to $t$ then the following formula holds for any Ito process:
$f(t, X_t) = f(0,X_0) + \int_{0}^{t}\partial_s f(s,X_s) ds + \int_{0}^{t}\partial_x f(s,X_s) dX_s$ $+ \int_{0}^{t}\partial_{xx} f(s,X_s) d \left< X,X \right>_s $
So applying the above fact to the function $f(t,x) = tx$ gives:
$t X_t = 0 + \int_{0}^{t}X_s ds + \int_{0}^{t} s dX_s + 0$