Laplace transform with 3 functions

Question

I'm supposed to evaluate: $L\{t^{2}e^{7t}\sinh(3t)\}$.

I know that this can be broken using $(-1)^n \frac{d^n}{ds^n}F(s)$ so I end up with $(-1)^2 \frac{d^2}{ds^2}L\{e^{7t}\sinh(3t)\}$. I'm not sure how to proceed from here, as there are now two functions inside of the Laplace transform. Do I use some sort of unit step function? I saw in my notes that $L\{e^{-as}F(s)\}=f(t-a)U(t-a)$ but I'm not sure how to use this, or if I'm supposed to use it as all.

Any help would be greatly appreciated, thank you.

Answer 1

First, use

$$\sinh{x} = \frac{1}{2} (e^x - e^{-x})$$

as well as

$$\int_0^{\infty} dt \: t^2 e^{-a t} = \frac{2}{a^3}$$

Note that

$$\int_0^{\infty} dt \: t^2 \, e^{7 t} \sinh{3 t} e^{-s t} = \frac12 \int_0^{\infty} dt \: t^2 e^{-(s-10) t} - \frac12 \int_0^{\infty} dt \: t^2 e^{-(s-4) t} $$