Primitive of Cosine Integral Function

Theorem

$\ds \int \map \Ci x \rd x = x \map \Ci x + \sin x + C$

where:

$\Ci$ denotes the cosine integral function

$x$ is a strictly positive real number.

Proof

By Derivative of Cosine Integral Function, we have:

$\ds \frac \d {\d x} \paren {\map \Ci x} = -\frac {\cos x} x$

So:

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

\(=\)

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

\(\ds \)

\(=\)

\(\ds x \map \Ci x - \int \paren {-x \frac {\cos x} x} \rd x\)

Integration by Parts

\(\ds \)

\(=\)

\(\ds x \map \Ci x + \int \cos x \rd x\)

\(\ds \)

\(=\)

\(\ds x \map \Ci x + \sin x + C\)

Primitive of Cosine Function

$\blacksquare$