p(X, Y) is joint distribution
p(X/Y) is conditional distribution
p(X) is marginal distribution (Y is marginalized out).
You can not get conditional distribution from joint distribution with just by integration. There is no such relationship.
There are just two rules for probability. Sum rule and product rules. And then there is Bayes theorem.
We might want to look at a table like below and calculate joint and conditional distribution and marginalized out one of the variable. 
Further reading :