Primitive of Hyperbolic Cosecant of a x over x

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Theorem

\(\ds \int \frac {\csch a x \rd x} x\) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \dfrac {2 \paren {1 - 2^{2 k - 1} } B_{2 k} \paren {a x}^{2 k - 1} } {\paren {2 k - 1}\paren {2 k}!} + C\)
\(\ds \) \(=\) \(\ds -\frac 1 {a x} - \frac {a x} 6 + \frac {7 \paren {a x}^3} {1080} + \cdots + C\)

where $B_{2 k}$ denotes the $2 k$th Bernoulli number.


Proof

\(\ds \csch x\) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \dfrac {2 \paren {1 - 2^{2 k - 1} } B_{2 k} x^{2 k - 1} } {\paren {2 k}!}\) Power Series Expansion for Hyperbolic Cosecant Function
\(\ds \leadsto \ \ \) \(\ds \frac {\csch a x} x\) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \dfrac {2 \paren {1 - 2^{2 k - 1} } B_{2 k} \paren {a x}^{2 k - 1} } {x \paren {2 k}!}\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \dfrac {2 \paren {1 - 2^{2 k - 1} } B_{2 k} a^{2 k - 1} x^{2 k - 2} } {\paren {2 k}!}\)
\(\ds \leadsto \ \ \) \(\ds \int \frac {\csch a x \rd x} x\) \(=\) \(\ds \int \paren {\sum_{k \mathop = 0}^\infty \dfrac {2 \paren {1 - 2^{2 k - 1} } B_{2 k} a^{2 k - 1} x^{2 k - 2} } {\paren {2 k}!} } \rd x\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \int \dfrac {2 \paren {1 - 2^{2 k - 1} } B_{2 k} a^{2 k - 1} x^{2 k - 2} } {\paren {2 k}!} \rd x\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \dfrac {2 \paren {1 - 2^{2 k - 1} } B_{2 k} a^{2 k - 1} x^{2 k - 1} } {\paren {2 k - 1}\paren {2 k}!}\) Primitive of Power
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \dfrac {2 \paren {1 - 2^{2 k - 1} } B_{2 k} \paren {a x}^{2 k - 1} } {\paren {2 k - 1}\paren {2 k}!}\)

$\blacksquare$


Also see


Sources

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