Primitive of Reciprocal of Power of Hyperbolic Sine of a x
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Theorem
- $\ds \int \frac {\d x} {\sinh^n a x} = \frac {-\cosh a x} {a \paren {n - 1} \sinh^{n - 1} a x} - \frac {n - 2} {n - 1} \int \frac {\d x} {\sinh^{n - 2} a x}$
for $n \ne 1$.
Proof
\(\ds \int \frac {\d x} {\sinh^n a x}\) | \(=\) | \(\ds \int \csch^n a x \rd x\) | Definition 2 of Hyperbolic Cosecant | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-\csch^{n - 2} a x \coth a x} {a \paren {n - 1} } - \frac {n - 2} {n - 1} \int \csch^{n - 2} a x \rd x\) | Primitive of $\csch^n a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-\coth a x} {a \paren {n - 1} \sinh^{n - 2} a x} - \frac {n - 2} {n - 1} \int \frac {\d x} {\sinh^{n - 2} a x}\) | Definition 2 of Hyperbolic Cosecant | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-\cosh a x} {a \sinh a x \left({n - 1}\right) \sinh^{n - 2} a x} - \frac {n - 2} {n - 1} \int \frac {\d x} {\sinh^{n - 2} a x}\) | Definition 2 of Hyperbolic Cotangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-\cosh a x} {a \paren {n - 1} \sinh^{n - 1} a x} - \frac {n - 2} {n - 1} \int \frac {\d x} {\sinh^{n - 2} a x}\) | simplifying |
We note that when $n = 1$, $\dfrac {n - 2} {n - 1}$ is undefined, rendering this derivation invalid.
$\blacksquare$
Also see
- Primitive of $\dfrac 1 {\sinh a x}$ for the case where $n = -1$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sinh a x$: $14.560$