Let $\Omega$ be a bounded region in $R^3$ and for each $x\in\Omega $ define
$$I(x) = \int_\Omega \dfrac{1}{|x-y|}dy$$
I want to show that $I$ is differentiable by differentiating $I$ under the integral, but I am having a difficulty with finding a dominating function for the integrand. By using mean value theorem, I was able to get
$$(I(x+he_i)-I(x))/h = \int_\Omega \dfrac{x+k-y}{|x+ke_i-y|^3} dy$$ where $k$ is a real number with $|k|<h$, but how do I get a dominating function for this integrand independent of $k$?
A few other references that I looked up seems to simply say that:
$|\partial_{x_i} \dfrac{1}{|x-y|} | \leq \dfrac{1}{|x-y|^2}$ and since the function on the right hand side is integrable, we can differentiate under the integral, but I do not understand this argument. Could you please explain?