Primitive of Square of Hyperbolic Cosecant Function
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Theorem
- $\ds \int \csch^2 x \rd x = -\coth x + C$
where $C$ is an arbitrary constant.
Proof
From Derivative of Hyperbolic Cotangent:
- $\map {\dfrac \d {\d x} } {\coth x} = -\csch^2 x$
The result follows from the definition of primitive.
$\blacksquare$
Sources
- 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $11$.
- 1960: Margaret M. Gow: A Course in Pure Mathematics ... (previous) ... (next): Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xxx)}$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.32$
- 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $5$. Integrals: Integrals of Special Functions: $22$