# Primitive of Sine Integral Function

## Theorem

$\displaystyle \int \map \Si x \rd x = x \map \Si x + \cos x + C$

where $\Si$ denotes the sine integral function.

## Proof

By Derivative of Sine Integral Function, we have:

$\displaystyle \frac \d {\d x} \paren {\map \Si x} = \frac {\sin x} x$

So:

 $\displaystyle \int \map \Si x \rd x$ $=$ $\displaystyle \int 1 \times \map \Si x \rd x$ $\displaystyle$ $=$ $\displaystyle x \map \Si x - \int x \frac {\sin x} x \rd x$ Integration by Parts $\displaystyle$ $=$ $\displaystyle x \map \Si x - \int \sin x \rd x$ $\displaystyle$ $=$ $\displaystyle x \map \Si x + \cos x + C$ Primitive of Sine Function

$\blacksquare$