Primitive of x by Hyperbolic Cosecant of a x

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Theorem

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

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

$\blacksquare$


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