I'm not sure why you are saying that you are integrating over all of y but in a Bayesian scenario (which is what you are discussing) then P(X=x|P=p) = Integrate Out p P(X=x|P=p).
Anyway if you have identified a Beta distribution then you should just integrate it out for X <= 2 to get the answer.
But in terms of the actual marginal distribution we integrate out the joint distribution and the joint distribution is given as P(X = x, Y = y) = P(X=x|Y=y)*P(Y=y) so integrating out or summing out Y gives the marginal for X (which is what you have attempted), but you haven't given a final distribution in terms of x.
If your prior is a Beta distribution and Your Likelihood is a Geometric then you need to multiply both PDF's together and integrate out over the domain: You have not done this.
What is the distribution of a geometric distribution? What is the distribution of a Beta? What is the product? What are the parameters of each? What is the result when you integrate out p?