The reason I ask is because I am curious to whether Stan ever evaluates the integrand at t=0.
For the integral
\int^T_0 t^{\beta w_1}dt
I enter the lower and upper limits, but I have n samples of \beta and w_1. Does Stan sample t from \text{U}(0,T), and hence have n vectors (\beta, w_1, t), and then perform Monte Carlo integration?
So Stan (although it is saying 1D integration) is actually evaluating
\int_{W}\int_B\int^T_0 t^{\beta w_1}dtd\beta dw_1,
where B and W are the parameter spaces for \beta and W, respectively.
If so, then the probability that t=0 is drawn should be zero and the singularities should not be an issue. But, I am not sure what Stan is doing behind the scenes.