Laplace Transform of Exponential Integral Function

Theorem
Let $\Ei: \R_{>0} \to \R$ denote the exponential integral function:


 * $\map \Ei x = \ds \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {e^{-t} } t \rd t$

Then:
 * $\laptrans {\map \Ei t} = \dfrac {\map \ln {s + 1} } s$

where $\laptrans f$ denotes the Laplace transform of the function $f$