Primitive of Product of Hyperbolic Secant and Tangent
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Theorem
- $\ds \int \sech x \tanh x \rd x = -\sech x + C$
where $C$ is an arbitrary constant.
Proof
From Derivative of Hyperbolic Secant:
- $\dfrac \d {\d x} \sech x = -\sech x \tanh x$
The result follows from the definition of primitive.
$\blacksquare$
Sources
- 1960: Margaret M. Gow: A Course in Pure Mathematics ... (previous) ... (next): Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xxxi)}$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.37$
- 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $5$. Integrals: Integrals of Special Functions: $23$