# I need. to find $E\exp(luX_{t})$, where $u \in \mathbb R$

$$X_{t} = \int _{0}^{t}h(t)dW_{t}$$ where $$h(t)$$ is a determined function (not random) and $$W_{t}$$ is Wiener process.

My thoughts so far:

1)use Ito's formula and find X's probability density.

2)Try to do it by definition of Ito's integral maybe.

3)I know for sure that $$X_{t}$$ is a gauss process, so maybe there's something there, but still, I'd need to find density first.

Nothing came out so far. So any help is appreciated.

$$X_t$$ is normal with mean $$0$$ and variance $$\int_0^{t}h(s)^{2}ds$$. Hence $$Ee^{luX_t}=e^{l^{2}u^{2}\int_0^{t}h(s)^{2}ds}$$.