Question

The laplace transform of sin3 2t is

Answer 1

Using

sinh(x)=ex−e−x2

then

sinh3(3t)=[e3t−e−3t2]3

=e9t−e3t+e−3t−e−9t8

Now, eat, where t is time does not have a Laplace transform so sinh3(3t) will not have one either. However, if you meant to ask what is the Laplace transform of sinh3(3t)u(t) where u(t) is the unit step function, the problem is quite different.

Since the Laplace transform of x(t)=e−atu(t) is given by X(s)=L{x(t)}=1s+a and the Laplace transform is linear, we can write

L{sinh3(3t)u(t)}=L{e9t−e3t+e−3t−e−9t8u(t)}

=18[1s−9−1s−3+1s+3−1s+9]