The context of this question is variational inference (vriational Bayes) assuming factoring posterior distributions, which is then also known as mean field theory. However the argument is purely algebraically.
Let $q(\theta)$ be a function where $\theta$ a parameter vector. Assume $$q(\theta)=\prod_{i=1}^D q_i(\theta_i),$$ where $\int q_i(\theta_i) d \theta_i=1$. Then another function is given by $$\int q(\theta) \ln q(\theta) d\theta$$ which, under the assumption made above, apparently factorizes to
$$\sum_{i=1}^D \int q_i(\theta_i) \ln q_i(\theta_i)d\theta_i.$$
I do not understand why. I can see that
$$\prod_{i=1}^D \int q_i(\theta_i) \sum_{j=1}^D \ln q_j(\theta_j) d\theta_i$$
but I do not know how to get to the result from here.