Can anyone show me the step by step process on how to solve this difficult integral by parts

I want to do a Laplace transform and I'm having difficulty solving this integral.

integral[0,infinity] (t*(e^(a-s)*t))

u'=e^(a-s)*t, v=t , denote integral by S. S(u'*vdt)=u*v-S(u*v'dt)
u=(1/(a-s)*e^(a-s)*t, v'=1 so S(u'*vdt)=u*v-S(u*v'dt)=(1/(a-s))*t*(e^(… S(1/(a-s))*e^[(a-s)*t] dt
=(1/(a-s))*t*(e^[(a-s)*t] - [1/(a-s)^2]*e^[(a-s)*t]
Sub limits and you get 1/(a-s)^2, I think. Note that as t --> infinity e^[(a-s)*t] ->0 as does t* e^[(a-s)*t]