How to show that a language {w|ww^R in A} is regular, A being regular?
You might want to study If $L$ is a regular language then so is $\sqrt L=\{w\mid ww∈L\}$.

The solution is to simulate the DFA in parallel with itself, from both sides of the string. For each letter one step forward (for $w$) and one step backward (for $w^R$). If the forward state reading $w$ reaches the same state as the backward simulation, both parts joined form a computation on $ww^R$.