integration of Laplace transform with respect to parameter

We use the curved from the Laplace-transformed functions to the corresponding initial functions.

If

$$f(t,x)\curvearrowleft F(s,x),$$

then one can integrate both functions with respect to the parametre $x$ between the same which may be also infinite provided that the integrals converge:

${\displaystyle {\int }_a^b}f(t,x)𝑑x\curvearrowleft {\displaystyle {\int }_a^b}F(s,x)𝑑x$(1)

(1) may be written as

$ℒ\{{\displaystyle {\int }_a^b}f(t,x)𝑑x\}={\displaystyle {\int }_a^b}ℒ\{f(t,x)\}𝑑x.$(2)

Proof.  Using the definition of the Laplace transform, we can write

$${\int }_a^{b}f(t,x)𝑑x\curvearrowleft {\int }_0^{\mathrm{\infty }}\left(e^{-st}{\int }_a^{b}f(s,x)𝑑x\right)𝑑t={\int }_0^{\mathrm{\infty }}\left({\int }_a^b{e}^{-st}f(s,x)𝑑x\right)𝑑t.$$

We change the of integration in the last double integral and use again the definition, obtaining

$${\int }_a^{b}f(t,x)𝑑x\curvearrowleft {\int }_a^b\left({\int }_0^{\mathrm{\infty }}{e}^{-st}f(s,x)𝑑t\right)𝑑x={\int }_a^{b}F(s,t)𝑑x,$$

Q.E.D.