Primitive of Power of Hyperbolic Cotangent of a x
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Theorem
- $\ds \int \coth^n a x \rd x = \frac {-\coth^{n - 1} a x} {a \paren {n - 1} } + \int \coth^{n - 2} a x \rd x + C$
Proof
| \(\ds \int \coth^n a x \rd x\) | \(=\) | \(\ds \int \coth^{n - 2} a x \coth^2 a x \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int \coth^{n - 2} a x \paren {1 + \csch^2 a x} \rd x\) | Difference of Squares of Hyperbolic Cotangent and Cosecant | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \coth^{n - 2} a x \csch^2 a x \rd x + \int \coth^{n - 2} \rd x\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {-\coth^{n - 1} a x} {a \paren {n - 1} } + \int \coth^{n - 2} a x \rd x + C\) | Primitive of $\coth^n a x \csch^2 a x$ |
$\blacksquare$
Also see
- Primitive of $\sinh^n a x$
- Primitive of $\cosh^n a x$
- Primitive of $\tanh^n a x$
- Primitive of $\sech^n a x$
- Primitive of $\csch^n a x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\coth a x$: $14.625$
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $128$.
- 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 17$: Tables of Special Indefinite Integrals: $(31)$ Integrals Involving $\coth a x$: $17.31.8.$