Primitive of Hyperbolic Tangent of a x over x

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Theorem

\(\ds \int \frac {\tanh a x \rd x} x\) \(=\) \(\ds \sum_{k \mathop \ge 1} \frac {2^{2 k} \paren {2^{2 k} - 1} B_{2 k} \paren {a x}^{2 k - 1} } {\paren {2 k - 1} \paren {2 k}!} + C\)
\(\ds \) \(=\) \(\ds a x - \frac {\paren {a x}^3} 9 + \frac {2 \paren {a x}^5} {75} - \cdots + C\)

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


Proof

\(\ds \tanh x\) \(=\) \(\ds \sum_{k \mathop = 1}^\infty \frac {2^{2 k} \paren {2^{2 k} - 1} B_{2 k} \, x^{2 k - 1} } {\paren {2 k}!}\) Power Series Expansion for Hyperbolic Tangent Function
\(\ds \leadsto \ \ \) \(\ds \frac {\tanh a x} x\) \(=\) \(\ds \sum_{k \mathop = 1}^\infty \frac {2^{2 k} \paren {2^{2 k} - 1} B_{2 k} \paren {a x}^{2 k - 1} } {x \paren {2 k}!}\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^\infty \frac {2^{2 k} \paren {2^{2 k} - 1} B_{2 k} a^{2 k - 1} x^{2 k - 2} } {\paren {2 k}!}\)
\(\ds \leadsto \ \ \) \(\ds \int \frac {\tanh a x \rd x} x\) \(=\) \(\ds \int \paren {\sum_{k \mathop = 1}^\infty \frac {2^{2 k} \paren {2^{2 k} - 1} B_{2 k} a^{2 k - 1} x^{2 k - 2} } {\paren {2 k}!} } \rd x\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^\infty \int \frac {2^{2 k} \paren {2^{2 k} - 1} B_{2 k} a^{2 k - 1} x^{2 k - 2} } {\paren {2 k}!} \rd x\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^\infty \frac {2^{2 k} \paren {2^{2 k} - 1} B_{2 k} a^{2 k - 1} x^{2 k - 1} } {\paren {2 k - 1} \paren {2 k}!} + C\) Primitive of Power
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^\infty \frac {2^{2 k} \paren {2^{2 k} - 1} B_{2 k} \paren {a x}^{2 k - 1} } {\paren {2 k - 1} \paren {2 k}!} + C\)

$\blacksquare$


Also see


Sources