what is the intuitive understanding when you integral for $x$ in pdf $f(y|x)$
as we all know that $f(y|x)$=$f(x,y)/f(x)$, and also if you integral for $x$ in joint pdf $f(x,y)$ then you get marginal pdf of $f(y)$
So back to my question, what is the meaning when you integral $x$ in $f(y|x)$?
If the question is what is the meaning of the function $g$ defined by $g(y)=\displaystyle\int f(y\mid x)\mathrm dx$, the answer is: None, except in special cases such as $X$ being uniform.
When $X$ is uniform on some set $U$ with finite measure $1/c$, then $f=c\mathbf 1_U$ hence $$ \int_U f(y\mid x)\mathrm dx=c^{-1}\int_U f(y\mid x)f(x)\mathrm dx=c^{-1}\int_U f(x,y)\mathrm dx=c^{-1}f_Y(y), $$ that is, $g$ is proportional to $f_Y$. Otherwise, everything can happen...