Evaluation of Integral using Laplace Transform

Theorem
Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of the real function $f$.

Then:
 * $\ds \int_0^{\to \infty} \map f t \rd t = \map F 0$

assuming the integral is convergent.

Proof
By definition of Laplace transform:


 * $\ds \int_0^{\to \infty} e^{-s t} \map f t \rd t = \map F s$

The result follows by taking the limit as $s \to 0$.