Question

find the laplace transform of t^2 sinat?

Answer 1

Given f(t)=sin(at)f(t)=sin⁡(at) I want to calculate the Laplace transform of f(t)f(t).I have determined by using integration by parts twice, that the answer should beF(s)=as2+a2F(s)=as2+a2Now I want to recalculate it by using just thatsin(at)=12i(eati−e−ati)sin⁡(at)=12i(eati−e−ati)So the integral becomes12i∫∞0e−st(eati−e−ati)=12i∫0∞e−st(eati−e−ati)=12i∫∞0et(ai−s)−e−t(ai+s)dt=12i∫0∞et(ai−s)−e−t(ai+s)dt=12i[1ai−set(ai−s)]∞0+12i[1ai+se−t(ai+s)]∞0