EDIT: To make what I am asking more clear. I've simplified it and have a more direct question.
Let's say I am writing out an expression, and I want to write: \int_0^xF'(y)\,dy
However, for reasons outside the scope of this question, I don't want to use the notation F'(y) to represent the first derivative of $F(y)$ with respect to $y$. So, as far as I'm aware (correct me if there is another alternative) I must write:
$\int_0^x\frac{dF(y)}{dy} \,dy$
Which, I believe is equivalent to the first formula, but uses different notation. However, in the second form, immediately it looks like the $dy$s should cancel out. But that leaves only:
$\int_0^x dF(y)$
My question is: is this third formula in an acceptable and meaningful form? It looks weird to me, but maybe that's just because I'm inexperienced. Would this make sense to other people? Is it the best way for me to express it? Any help in understanding would be super appreciated!!!
Thank you!!
ORIGINAL QUESTION TEXT FOR REFERENCE:
Let's say I want to integrate something like this: \int_0^x F_1(x)F_2(y)F_3'(y)\,dy Where $F_1$, $F_2$, and $F_3$ are just placeholders for functions.
Since F_3'(y) = \frac{dF_3(y)}{dy} then this could also be written $\int_0^x F_1(x)F_2(y)\frac{dF_3(y)}{dy}\,dy$ Then, wouldn't the $dy$s cancel out, leaving: $\int_0^x F_1(x)F_2(y)dF_3(y)$
Is that correct, and how would one interpret that expression, or solve it in terms of $x$?