Context: using this from another thread: https://quant.stackexchange.com/questions/25660/integration-in-the-hull-white-sde
The relevant bit which i am trying to solve is the following part of the thread: $$\int_t^T e^{\int_0^u b(v) dv} (a(u) du + \sigma(u) dW_u) = \int_t^T a(u) e^{\int_0^u b(v) dv} du + \int_t^T \sigma(u) e^{\int_0^u b(v) dv} dW_u$$ of course with some changes. My version of the above SDE to solve is: $$\int_0^t e^{\lambda s} (\theta(s) ds + \sigma dW_s) = \int_0^t \theta(s) e^{\lambda s} ds + \int_0^t \sigma e^{\lambda s} dW_s$$ But seem to be stuck solving the first integral as it seems to loop into differentiations of $\theta(s)$ $$ \int_0^t \theta(s) e^{\lambda s} ds $$ Is it solvable just by parts or does it have a different approach which I might be missing. Any help would be appreciated.