Derivative of Sine Integral Function
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Theorem
- $\dfrac \d {\d x} \paren {\map \Si x} = \dfrac {\sin x} x$
where $\Si$ denotes the sine integral function.
Proof
We have, by the definition of the sine integral function:
- $\ds \map \Si x = \int_0^x \frac {\sin t} t \rd t$
By Fundamental Theorem of Calculus (First Part): Corollary, we have:
- $\ds \frac \d {\d x} \paren {\map \Si x} = \frac {\sin x} x$
$\blacksquare$