An equation (borrowed from statistics QLE) is
$$ \frac{\partial q(\mu,y)} {\partial\mu}= \dfrac{y-\mu}{a_i \phi V(\mu)}$$
Apparently this is equivalent to
$$q(\mu,y)=\int_0^\mu \frac{y-u}{a_i \phi V(u)}\,du + \textrm{a function of }y.$$
I'm not 100% sure how they got this equation. It seems like they computed
$$\int_0^\mu \frac{\partial q(u,y)} {\partial u}\,du = q(\mu,y) + \textrm{a function of }y$$
from the LHS. Instinctively, I try to evaluate this integral by cancelling the differentials of $u$, but I'm not sure that is allowed because one is a $d$ and one is a $\partial$, and even then the integral would turn into (I believe) $\int_0^\mu \partial q=\mu$, which does not give the equivalent statement. Can someone help clear up my confusion? Thanks!