TITLE: What is the integral of function $f(x) = (\sin x)/x$
QUESTION [7 upvotes]: I want to know if the function sinx/x is integrable and if it is, then what's its integral?
My high school book says its a non-integrable function while WolframAlpha says its integral is Si(x) + constant 
Please shed some light on this topic and explain from basic level like what is Si(x), who used it for the first time etc.

REPLY [4 votes]: In general, a function $f:\mathbb{R} \longrightarrow \mathbb{R}$ is integrable if it is bounded and the set of discontinuities (i.e. $x=0$ in this case) have measure zero.  Intuitively, this more or less amounts to the function being defined except at reasonably few exceptional points (i.e. a finite number of points as in this case is fine), so the function is integrable since it is clearly also bounded.  As the others have mentioned and linked to, the integral is nontrivial and has a special function defined to be its antiderivative called $Si(x)$.