RSA Accumulators with auxiliary information or known trapdoor information
For the inclusion proof you might need the trapdoor or the accumulated set. The exclusion proof can be generated without knowing the trapdoor. See this related question and answer.

How can one generate inclusion proofs if knew the trapdoor, i.e. the factorization of $N=p\cdot q$?

Let's $A$ denote the accumulator's value, $x$ be an item to-be-accumulated and $N$ the modulus whose factorization is known.

Generating a valid membership proof for an item $x$ essentially means taking $x$-th root of $A$ $\mod N$. Since you know the factors of $N$ you can just compute $x$-th root $\mod p$ and then $\mod q$ and afterwards compute the $x$-th root $\mod N$ by applying the Chinese-remainder theorem for moduli $p,q$.

Note, that computing efficiently arbitrary roots mod a prime is simple. See, for instance, here.