Primitive of Inverse Hyperbolic Tangent of x over a over x
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Theorem
| \(\ds \int \dfrac 1 x \artanh \dfrac x a \rd x\) | \(=\) | \(\ds \sum_{k \mathop \ge 0} \frac 1 {\paren {2 k + 1}^2} \paren {\frac x a}^{2 k + 1}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac x a + \frac {\paren {x / a}^3} {3^2} + \frac {\paren {x / a}^5} {5^2} + \frac {\paren {x / a}^7} {7^2} + \cdots\) |
Proof
| \(\ds \artanh \dfrac x a\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac 1 {2 n + 1} \paren {\frac x a}^{2 n + 1}\) | Power Series Expansion for Real Area Hyperbolic Tangent | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac 1 x \artanh \dfrac x a\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac 1 {2 n + 1} \frac 1 {a^{2 n + 1} } x^{2 n}\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \int \frac 1 x \artanh \dfrac x a \rd x\) | \(=\) | \(\ds \int \paren {\sum_{n \mathop = 0}^\infty \frac 1 {2 n + 1} \frac 1 {a^{2 n + 1} } x^{2 n} } \rd x\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \int \paren {\frac 1 {2 n + 1} \frac 1 {a^{2 n + 1} } x^{2 n} \rd x}\) | Fubini's Theorem | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac 1 {2 n + 1} \frac 1 {a^{2 n + 1} } \frac {x^{2 n + 1} } {2 n + 1} + C\) | Primitive of Power | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 0} \frac 1 {\paren {2 k + 1}^2} \paren {\frac x a}^{2 k + 1} + C\) | rearranging |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.659$