# Primitive of x over Sine of a x

## Theorem

 $\ds \int \frac {x \rd x} {\sin a x}$ $=$ $\ds \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + C$ $\ds$ $=$ $\ds \frac 1 {a^2} \paren {a x + \frac {\paren {a x}^3} {18} + \frac {7 \paren {a x}^5} {1800} \cdots} + C$

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

## Proof

 $\ds \csc x$ $=$ $\ds \sum_{n \mathop = 0}^\infty \dfrac {B_{2 n} \paren {-1}^{n - 1} x^{2 n - 1} 2 \paren {2^{2 n - 1} - 1} } {\paren {2 n}!}$ Power Series Expansion for Cosecant Function $\ds \leadsto \ \$ $\ds \dfrac x {\sin a x}$ $=$ $\ds x \sum_{n \mathop = 0}^\infty \dfrac {B_{2 n} \paren {-1}^{n - 1} \paren {a x}^{2 n - 1} 2 \paren {2^{2 n - 1} - 1} } {\paren {2 n}!}$ Cosecant is Reciprocal of Sine $\ds$ $=$ $\ds \dfrac 1 a \sum_{n \mathop = 0}^\infty \dfrac {B_{2 n} \paren {-1}^{n - 1} \paren {a x}^{2 n} 2 \paren {2^{2 n - 1} - 1} } {\paren {2 n}!}$ $\ds \leadsto \ \$ $\ds \int \dfrac x {\sin a x} \rd x$ $=$ $\ds \dfrac 1 {a^2} \sum_{n \mathop = 0}^\infty \dfrac {B_{2 n} \paren {-1}^{n - 1} \paren {a x}^{2 n + 1} 2 \paren {2^{2 n - 1} - 1} } {\paren {2 n + 1}!}$ Primitive of Power, Linear Combination of Integrals

$\blacksquare$