Primitive of Sine Integral Function

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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

$\blacksquare$