I am currently trying to integrate the stochastic integral $\int_0^te^{\theta s}dW(s)$. My approach is to basically apply ito's lemma. If $ F(X(t), t) =e^{\theta t} X(t)$ where $dX(t)=dW(t)\,\,with\,\,\mu=0,\,\,\sigma=1$,
Then $$ dF(X(t),t)=\theta e^{\theta t} X(t)\,dt \,+\,e^{\theta t}\,dW(t) $$
And we can then integrate both the LHS and the RHS, my result was
$$
e^{\theta t} X(t)=\int_0^t \theta e^{\theta s} X(s)\,ds \,+\,\int_0^t e^{\theta s}\,dW(s)
$$
However if we rearrange them
$$ \int_0^t e^{\theta s}\,dW(s)=e^{\theta t}X(t)\,-\,\theta\int_0^t e^{\theta s} X(s)\,ds $$
In this case, the RHS seem to cancel out and equate to zero, I am just curious at which step did I go wrong?