I will begin with the description to the problem.
Let X have a geometric dist. with success prob. P, where P follows a beta(a = 2, b = 1). Derive the unconditional distribution of X and use it to find P(X< 2).
So I use,
P(X=x) = integratal over all y of P(X|p=p) *P(P=p)
= inegratal[p*(1-p)^(x-1) * 2p dp]
= inegratal[2p^2 *(1-p)^(x-1) dp]
Which you can use the Beta Kernel(a = 3, b = x) to solve, however after solving for the gamma functions I get
which gives me values greater than 1, so can't be a prob function. I was wondering where I was going wrong in trying to find P(X=x).