What can we say about distribution of
,where $N(\mu,\sigma)$ is independent normally distributed with mean $\mu(s)$ and variance $\sigma^2(s)$, $T$ and $t$ are finite?
I think the object you really want to be integrating is a Gaussian process with some specified mean and covariance function. And you'll probably want the covariance function to have some regularity to it (e.g. continuous) in order to guarantee that the integral exists. In this case the result of the integral will be normally distributed, so you only need to compute its mean and variance. This is straightforward using Fubini's theorem. See also my answer here.