Laplace Transform of Sine of t over t

Theorem
Let $\sin$ denote the real sine function.

Let $\laptrans f$ denote the Laplace transform of a real function $f$.

Then:


 * $\laptrans {\dfrac {\sin t} t} = \arctan \dfrac 1 s$

Proof
From Limit of $\dfrac {\sin x} x$ at Zero:


 * $\ds \lim_{x \mathop \to 0} \frac {\sin x} x = 1$

From Laplace Transform of Sine:


 * $(1): \quad \laptrans {\sin t} = \dfrac 1 {s^2 + 1}$

From Laplace Transform of Integral:


 * $(2): \quad \ds \laptrans {\dfrac {\map f t} t} = \int_s^{\to \infty} \map F u \rd u$

Hence:

Also see

 * Laplace Transform of Sine Integral Function