\(\text{From the given statement, the function is }\ f(t)=6+\sin(3t)\)

\(\text{Step 2}\)

\(\text{To find the Laplace transform of the function as follows.}\)

\(L(f(t))=L(6+\sin(3t))\)

\(=L(6)+L(\sin(3t))\)

\(\text{Known fact: }\)

\(L(1)=\frac{1}{s}\)

\(L(\sin(\omega t))=\frac{\omega}{s^2+\omega^2}\)

\(\text{Therefore, }\)

\(L(6)+L(\sin(3t))=6L(1)+L(\sin(3t))\)

\(=6\bigg(\frac{1}{s}\bigg)+\frac{3}{s^2+3^2}\)

\(=\frac{6}{s}+\frac{3}{s^2+9}\)

\(=\frac{6s^2+3s+54}{s^3+9s}\)

\(\text{Thus, the Laplace transform of the function is } \frac{6s^2+3s+54}{s^3+9s}\)