For suitable families of locally infinitely divisible Markov processes $\{ξ^{ε}_t\}_{0\leq t\leq T}$ with frequent small jumps depending on a small parameter $ε>0,$ precise asymptotics for large deviations of integral forms $\mathbb{E}^ε[\exp\{ε^{-1}F(ξ^ε)\}]$ are proved for smooth functionals $F.$ The main ingredient of the proof in this paper is a recent result regarding the asymptotic expansions of the expectations $\mathbb{E}^ε[G(ξ^ε)\}]$ for smooth $G.$ Several connections between these large deviation asymptotics and partial integro-differential equations are included as well.