Laplace Transform Theorems
Thus, linearity of the Laplace transform follows immediately from the linearity of integration.
Proof: This follows immediately from integration by parts:
since by assumption.
Corollary: Integration Theorem
Thus, successive time derivatives correspond to successively higher powers of , and successive integrals with respect to time correspond to successively higher powers of .
Laplace Analysis of Linear Systems
Relation to the z Transform