This question is partially inspired by Qiaochu Yuan's answer to "Will moving differentiation from inside, to outside an integral, change the result?".
Essentially, I would like to know, if we have:
$$\sum_{x=a}^b{ \int{ f(x)dx } } \quad\text{ or }\quad \sum_x{ \int{ f(x)dx } }$$
If we then perform a substitution $\int{ f(x)dx } \to \int{ g(y) dy}$, when can we carry the substitution over to the summation? In other words, when does
$$\sum_{x=a}^b{ \int{ f(x)dx } } = \sum_{y=a_2}^{b_2}{ \int{ g(y)dy } }$$
I'm hoping that someone can cover as much as possible.
I'm wondering how we would derive the new constants of summation as well.
The question doesn't really make sense. $x$ is the "dummy variable" in the integration, so $\int_c^d f(x)\; dx$ is not a function of $x$. Thus $$ \sum_{x=a}^b \int_c^d f(x)\; dx = (b-a+1) \int_c^d f(x)\; dx$$
EDIT: For the new version of the problem, let $F(x) = \int f(x)\; dx$ and $S = \sum_{x=a}^b F(x) = F(a) + F(a+1) + \ldots + F(b)$ where $a$ and $b$ are integers, $a < b$. A change of variables is a substitution $x = j(y)$, so that $F(x) = F(j(y)) = G(y)$ where $G = F \circ j$. Thus $F' = f$ and $G'(y)= g(y) = j'(y) f(j(y))$. For $S$ we have $S = G(y_a) + \ldots + G(y_b)$ where $j(y_x) = x$. In general there's no reason for these $y_x$ to be consecutive integers. Of course there are special cases such as $j(x) = x + constant$.