TITLE: Expected Value of Brownian motion using ito isometry
QUESTION [7 upvotes]: Find 
$$
E\ \left[\left(\int_{0}^T e^{s+W_s}dW_s \right)^2\right],
$$ 
where $(W_s)$ is a Brownian  motion.
I tried to use Ito isometry to solve this question, but still not yet to find the right path.  Appreciate if you could shed the light on this question.  Thanks.

REPLY [6 votes]: By Ito isometry, this is $\mathsf{E} \intop_0^T e^{2(s+W_s)} ds$. Then use Fubini theorem to interchange $\intop$ and $\mathsf{E}$, and recall the exponential moments of a Gaussian distribution, or calculate them if you have never done it, or otherwise look here.