Q1) Let $(B_t)_t$ the Brownian motion. How can I compute $$\int_0^T e^{2B_s^4}ds\ \ ?$$
I'm sorry, I'm a novice in this kind of thing. Since I don't know $B_s$, I can't conclude.
Q2) An other question, if $f(X_s)$ for $f$ a continuous function and a continuous stochastic process $(X_s)_s$, do we always have $$\mathbb E\int_0^T f(X_s)ds=\int_0^T \mathbb E[f(X_s)]ds\ \ \ ?$$
I would say yes. Indeed, we have that $f$ continuous and $s\longmapsto X_s$ continuous. Therefore, $f(B_s)$ continuous, and thus integrable over $[0,T]$. Therefore, using fubini we can conclude. Is my justification correct ?