Consider a filtered probability space $(\Omega,F_{t\geq 0},P)$ and two independent stochastic processes, $X$ and $Y$ under the measure $P$.
$$\int_0^L E_P(X(t)Y(t))dt$$
By independence,
$$\int_0^L E_P(X(t))E_p(Y(t))dt$$
$$\int_0^L (\int X(t)dP)(\int Y(t)dP)dt$$
While it is illicit to do the following, is there ANY case under which the following is true? $$\int_0^L (\int X(t)dP)dt\int_0^L(\int Y(t)dP)dt$$
Let $f(t) = \int X(t)dP = E[X(t)]$ and $g(t) = \int Y(t) dP = E[Y(t)]$. Your statement is now
$$\int_0^L f(t) g(t) dt = \left( \int_0^L f(t) dt \right) \left( \int_0^L g(t) dt \right).$$
Do you have any reason to believe this is true in general? Try with some simple examples.