Primitive of Hyperbolic Cotangent of a x over x

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Theorem

\(\ds \int \frac {\coth a x \rd x} x\) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \frac {2^{2 k} B_{2 k} \paren {a x}^{2 k - 1} } {\paren {2 k}!} + C\)
\(\ds \) \(=\) \(\ds -\frac 1 {a x} + \frac {a x} 3 - \frac {\paren {a x}^3} {135} + \cdots + C\)

where $B_k$ denotes the $k$th Bernoulli number.


Proof

\(\ds \coth x\) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \frac {2^{2 k} B_{2 k} \, x^{2 k - 1} } {\paren {2 k}!}\) Power Series Expansion for Hyperbolic Cotangent Function
\(\ds \) \(=\) \(\ds \dfrac 1 x + \sum_{k \mathop = 1}^\infty \frac {2^{2 k} B_{2 k} \, x^{2 k - 1} } {\paren {2 k}!}\) extracting the first term, just in case
\(\ds \leadsto \ \ \) \(\ds \frac {\coth a x} x\) \(=\) \(\ds \dfrac 1 {a x^2} + \sum_{k \mathop = 1}^\infty \frac {2^{2 k} B_{2 k} \, \paren {a x}^{2 k - 1} } {x \paren {2 k}!}\)
\(\ds \) \(=\) \(\ds \dfrac 1 {a x^2} + \sum_{k \mathop = 1}^\infty \frac {2^{2 k} B_{2 k} a^{2 k - 1} x^{2 k - 2} } {\paren {2 k}!}\)
It is noted that the exponent of $x$ is always even, so there is no need to consider the special case where $x^{-1}$.
\(\ds \leadsto \ \ \) \(\ds \int \frac {\coth a x \rd x} x\) \(=\) \(\ds \int \paren {\dfrac 1 {a x^2} + \sum_{k \mathop = 1}^\infty \frac {2^{2 k} B_{2 k} a^{2 k - 1} x^{2 k - 2} } {\paren {2 k}!} } \rd x\)
\(\ds \) \(=\) \(\ds \int \dfrac 1 {a x^2} \rd x + \sum_{k \mathop = 1}^\infty \int \frac {2^{2 k} B_{2 k} a^{2 k - 1} x^{2 k - 2} } {\paren {2 k}!} \rd x\)
\(\ds \) \(=\) \(\ds -\dfrac 1 {a x} + \sum_{k \mathop = 1}^\infty \frac {2^{2 k} B_{2 k} a^{2 k - 1} x^{2 k - 1} } {\paren {2 k - 1} \paren {2 k}!} + C\) Primitive of Power
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \frac {2^{2 k} B_{2 k} \paren {a x}^{2 k - 1} } {\paren {2 k}!} + C\) bringing the first term back inside the summation

$\blacksquare$


Also see


Sources

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