Primitive of Square of Hyperbolic Secant Function

Theorem

$\ds \int \sech^2 x \rd x = \tanh x + C$

where $C$ is an arbitrary constant.


Proof

From Derivative of Hyperbolic Tangent:

$\map {\dfrac \d {\d x} } {\tanh x} = \sech^2 x$

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: $10$.
  • 1960: Margaret M. Gow: A Course in Pure Mathematics ... (previous) ... (next): Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xxix)}$
  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Indefinite Integrals: General Rules of Integration: $14.31$
  • 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $5$. Integrals: Integrals of Special Functions: $21$
  • 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 16$: Indefinite Integrals: General Rules of Integration: $16.31.$
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