Question

laplace of sin(cube t)

Answer 1

.

Answer 2

Do you mean:
\[\sin^3t\\
\sin(\sqrt[3]t)\\
\sin t^3\]

Answer 3

Each possibility is piecewise continuous over \([0,\infty)\) and of exponential order, so the transform exists, but they might not be so easy to find.

The first one is the simplest.
\[\begin{align*}\sin^3t&=\sin t\sin^2t\\\\
&=\sin t\left(\frac{1-\cos2t}{2}\right)\\\\
&=\frac{1}{2}\sin t-\frac{1}{2}\sin t\cos2t\\\\
&=\frac{1}{2}\sin t-\frac{1}{2}\left(\frac{\sin3t-\sin t}{2}\right)\\\\
&=\frac{1}{4}\sin t-\frac{1}{4}\sin3t
\end{align*}\]
Presumably, you know the transform of \(\sin at\), correct?