Compute the integral $\int_0^\pi \frac{\sin x}{x}dx$ . How to compute precisely the integral $$\int_0^\pi \frac{\sin x}{x}dx$$ analytically? It is well-known that $$\int_0^{+\infty}\frac{\sin x}{x}dx=\frac{\pi}{2}.$$ One way to compute the above integral using Taylor expansion for $\sin x$, we then obtain a power series value approximately 1.84. Thanks for any useful hints.
$$\color{#00f}{\large{\rm Si}\left(\pi\right)}$$


  $\displaystyle{\rm Si}\left(x\right)$ is the
  Sine Integral Function.
Your number is sometimes called the Gibbs constant and evaluates to

1.85193705198246617036105337015799136334580972898115
  49098047837818769818901663483585327103365029547577


with 100 digit accuracy. It is most likely not expressible in a simple way in terms of other "well-known" constants.