Jump-Diffusion Process: How to calculate the expectation of integral of S(t)

Question

Having a jump-diffuion process $S(t)$ and the transition density $f_{dS(t)}(x)$. How can I calculate the Expectation of the integral of $S(t)$ between two instants $t_0$ and $t_1$? $S(t_0)$ is considered to be known. Can we say that the Expectation of Integral $S(t)$ is equal to the Integral of the Expectation of $S(t)$? Note : I need to find the Expectation of the cumulative quantities of $S(t)$ between $t_0$ and $t_1$ knowing the value $S(t_0)$. Do not confuse with the expectation of the Integral of $dS(t)$ which is the expectation of $S(t)$ at time $t$. Thank you.