Can you please explain which formula I have to use to integrate the exponential of Brownian motion w.r.t to time: $$\int_0^t e^{uB_s} ds$$
where $B_t$ is Brownian motion and $u$ is a real positive number.
Using stochastic product rule with $d(te^{uB_t})$ I have: $$ = te^{uB_t} - \frac{u^2}{2} \int_0^t s e^{uB_s}ds- u\int_0^t se^{uB_s} dB_s$$
Is this correct?