Primitive of Power of Hyperbolic Cotangent of a x by Square of Hyperbolic Cosecant of a x
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Theorem
- $\ds \int \coth^n a x \csch^2 a x \rd x = \frac {-\coth^{n + 1} a x} {\paren {n + 1} a} + C$
Proof
\(\ds z\) | \(=\) | \(\ds \coth a x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d z} {\d x}\) | \(=\) | \(\ds -a \csch^2 a x\) | Derivative of $\coth a x$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \coth^n a x \csch^2 a x \rd x\) | \(=\) | \(\ds \int \frac {-1} a z^n \rd z\) | Integration by Substitution | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-1} a \frac {z^{n + 1} } {n + 1}\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-\coth^{n + 1} a x} {\paren {n + 1} a} + C\) | substituting for $z$ and simplifying |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\coth a x$: $14.618$