Primitive of x by Cosecant of a x

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Theorem

$\displaystyle \int x \csc a x \rd x = \frac 1 {a^2} \paren {a x + \frac {\paren {a x}^3} {18} + \frac {7 \paren {a x}^5} {1800} + \cdots + \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_n \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$

where $B_{2 n}$ is the $2 n$th Bernoulli number.


Proof

\(\displaystyle \int x \csch a x \rd x\) \(=\) \(\displaystyle \frac 1 {a^2} \int \theta \csch \theta \rd \theta\) Substitution of $a x \to \theta$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \, \theta^{2 n - 1} } {\paren {2 n}!} \rd \theta\) Power Series Expansion for Cosecant Function
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} } {\paren {2 n}!} \int \theta^{2 n} \rd \theta\) Fubini's Theorem
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + C\) Substituting back $\theta \to ax$

$\blacksquare$


Also see


Sources