Question

I've 2 questions:

let be $W_s$ a standard Brownian motion:

• using Ito's formula show that $\left( W_t,\int_0^t W_sds \right)$ has a normal distribution;
• and calculate $ E\left[e^{W_t}e^{\int_0^t W_sds} \right] .$

For the first part, i know that $W_t$ and $\int_0^t W_sds$ have normal distribution with mean and variance respectively $(0,t)$ and $(0, t^3/3)$, but i need help with Ito's formula.

For the second part i've tried to solve $E\left[e^{W_t}e^{\int_0^t W_sds} \right]= \iint e^{W_t}e^{\int_0^t W_sds} \;\phi \left( W_t,\int_0^t W_sds \right)\: dW_t \int_0^t W_sds$...

Is these the only way?

P.S. sorry for my poor english