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
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\tanh a x$: $14.612$