Limit at Infinity of Sine Integral Function

Theorem
Let $\Si: \R \to \R$ denote the sine integral function.

Then $\Si$ has a (finite) limit at infinity:


 * $\displaystyle \lim_{x \mathop \to +\infty} \Si \paren x = \frac \pi 2$

Corollary

 * $\displaystyle \lim_{x \mathop \to -\infty} \Si \paren x = -\frac \pi 2$

Proof
The limit:


 * $\displaystyle \lim_{x \mathop \to +\infty} \Si \paren x = \lim_{x \mathop \to +\infty} \int_{t \mathop \to 0}^{t \mathop = x} \frac {\sin t} t \rd t$

is the Dirichlet Integral.