Primitive of x by Hyperbolic Tangent of a x
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Theorem
- $\ds \int x \tanh a x \rd x = \frac 1 {a^2} \paren {\frac {\paren {a x}^3} 3 - \frac {\paren {a x}^5} {15} + \frac {2 \paren {a x}^7} {105} + \cdots + \frac { 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$
where $B_{2 n}$ denotes the $2 n$th Bernoulli number.
Proof
\(\ds \int x \tanh a x \rd x\) | \(=\) | \(\ds \frac 1 {a^2} \int \theta \tanh \theta \rd \theta\) | Substitution of $a x \to \theta$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac{2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \, \theta^{2 n - 1} } {\paren {2 n}!} \rd \theta\) | Power Series Expansion for Hyperbolic Tangent Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {2^{2 n} \paren {2^{2 n} - 1} B_{2 n} } {\paren {2 n}!} \int \theta^{2 n} \rd \theta\) | Fubini's Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + C\) | Substituting back $\theta \to ax$ |
$\blacksquare$
Also see
- Primitive of $x \sinh a x$
- Primitive of $x \cosh a x$
- Primitive of $x \coth a x$
- Primitive of $x \sech a x$
- Primitive of $x \csch a x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\tanh a x$: $14.610$