Let $\rho(x,t)$ denote a function of space, $x$ and time, $t$, and let $\hat{\rho}(x,s)$ denote its Laplace transform wrt time.
Assume I need to calculate the Laplace transform of the integral $$ \int_0^\pi \left( \int_x^1 \rho(\xi,t) d\xi\right) dx . $$ Is it true that: $$ \mathcal{L}\left[{\int_0^\pi \left( \int_x^1 \rho(\xi,t) d\xi\right) dx}\right]= \int_0^\pi \left( \int_x^1 \hat{\rho}(\xi,t) d\xi\right) dx $$ where $\mathcal{L}[f]=\hat{f}$?