Could you please find the Laplace transform?
Find Laplace transform of  sinωt i.e X(s)

2 .replace s by (s+a) in Laplace transform of sinωt ,you got X(s+a)
3.differentiate X(s+a)   by  n  times with respect to s and that's you answer
Method for n times differentiation-
1.partial fraction X(s+a) ,here poles will be of complex form
2.and then differentiate both partial fractions by n times and after that combine then , hopefully you'll get your answer