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.
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.