# Laplace Transform of Sine Integral Function/Proof 2

## Theorem

$\laptrans {\map \Si t} = \dfrac 1 s \, \arctan \dfrac 1 s$

where:

$\laptrans f$ denotes the Laplace transform of the function $f$
$\Si$ denotes the sine integral function

## Proof

 $\displaystyle \laptrans {\dfrac {\sin t} t}$ $=$ $\displaystyle \arctan \dfrac 1 s$ Laplace Transform of $\dfrac {\sin t} t$ $\displaystyle \leadsto \ \$ $\displaystyle \laptrans {\int_{u \mathop \to 0}^{u \mathop = t} \frac {\sin u} u \rd u}$ $=$ $\displaystyle \dfrac 1 s \arctan \dfrac 1 s$ Laplace Transform of Integral $\displaystyle \leadsto \ \$ $\displaystyle \laptrans {\map \Si t}$ $=$ $\displaystyle \dfrac 1 s \arctan \dfrac 1 s$ Definition of Sine Integral

$\blacksquare$