Probably my question is related to this question, but this question does not provide the answer to my question.
Let $f:X\times Y \rightarrow \mathbb{R}$. Suppose I am interested in integration of $f(x,y)$ over $x\in X$. But problem is that I know $x$ and $y$ are related through variable $z$. For example, $x=g(z)$ and $y=h(z)$. In this case, usually I need to take the relationship between $x$ and $y$ in consideration when I integrate. But is there any way to integrate $f(x,y)$ over $x$ ignoring relationship between $x$ and $y$ just like partial derivative?
Integrating the function $f$ over one of it's inputs is just as valid as differentiating with respect to one of it's inputs. However $f$ and the function $f(g(z), h(z))$ are very different functions. In fact, they are 2 completely different functions. $f$ is the way it's defined. $f(g(z), h(z))$ is a composition of functions and therefore it's a totally different function from $f$. It's a single variable function of $z$ (over a totally different domain than $f$). You can either
$$ \int f(x,y) dx$$
Or equivalently integrate over the parameterized path $x = g(z)$
$$\int f(g(z), y) g'(z) dz $$
Or integrate the completely different composition function $J(z) := f(g(z), h(z))$
$$\int f(g(z), h(z))dz = \int J(z) dz $$