Primitive of Hyperbolic Secant Function/Arcsine Form
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Theorem
- $\ds \int \sech x \rd x = \map \arcsin {\tanh x} + C$
Proof
Let:
| \(\ds \int \sech x \rd x\) | \(=\) | \(\ds \int \frac {\sech^2 x} {\sech x} \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int \frac {\sech^2 x} {\sqrt {1 - \tanh^2 x} } \rd x\) | Sum of Squares of Hyperbolic Secant and Tangent |
Let:
| \(\ds u\) | \(=\) | \(\ds \tanh x\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds u'\) | \(=\) | \(\ds \sech^2 x\) | Derivative of Hyperbolic Tangent |
Then:
| \(\ds \int \sech x \rd x\) | \(=\) | \(\ds \int \frac {\d u} {\sqrt {1 - u^2} }\) | Integration by Substitution | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \arcsin u + C\) | Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$: Arcsine Form | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \arcsin {\tanh x} + C\) | Definition of $u$ |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.29$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Appendix $2$: Table of derivatives and integrals of common functions: Hyperbolic functions