Original problem: A point X is randomly chosen from the interval (0,1). Suppose X=x is observed. Then a coin with P(Heads) = x is tossed independently n times. Let Y be the number of heads in n tosses. Find the unconditional distribution of Y.
I've setup the problem below using total probability and a uniform pdf of x (a confirmation others agree my formulation is correct would be nice), but not sure how to integrate this: