Primitive of Sine Integral Function

Theorem

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

where $\Si$ denotes the sine integral function.

Proof

By Derivative of Sine Integral Function, we have:

$\map {\dfrac \d {\d x} } {\map \Si x} = \dfrac {\sin x} x$

So:

\(\ds \int \map \Si x \rd x\)

\(=\)

\(\ds \int 1 \times \map \Si x \rd x\)

\(\ds \)

\(=\)

\(\ds x \map \Si x - \int x \frac {\sin x} x \rd x\)

Integration by Parts

\(\ds \)

\(=\)

\(\ds x \map \Si x - \int \sin x \rd x\)

\(\ds \)

\(=\)

\(\ds x \map \Si x + \cos x + C\)

Primitive of Sine Function

$\blacksquare$