Primitive of Cosecant of a x over x

Theorem

$\ds \int \frac {\csc a x} x \rd x = \frac {-1} {a x} + \frac {a x} 6 + \frac {7 \paren {a x}^3} {1080} + \cdots + \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n - 1} \paren {2 n}!} + \cdots + C$

where $B_n$ is the $n$th Bernoulli number.

Proof

$\ds \int \frac {\csc a x} x \rd x$

$=$

$\ds \int \frac 1 x \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n}!} \rd x$

Power Series Expansion for Cosecant Function

$\ds $

$=$

$\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} a^{2 n - 1} } {\paren {2 n}!} \int x^{2 n - 2} \rd x$

Primitive of Constant Multiple of Function

$\ds $

$=$

$\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} a^{2 n - 1} } {\paren {2 n}!} \paren {\frac {x^{2 n - 1} } {2 n - 1} } + C$

Primitive of Power

$\ds $

$=$

$\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n - 1} \paren {2 n}!} + C$

$\blacksquare$