Primitive of x by Hyperbolic Cotangent of a x

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Theorem

$\ds \int x \coth a x \rd x = \frac 1 {a^2} \paren {a x + \frac {\paren {a x}^3} 9 - \frac {\paren {a x}^5} {225} + \cdots + \frac {2^{2 n} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$

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


Proof

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

$\blacksquare$


Also see


Sources

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