Primitive of Hyperbolic Tangent Function
Theorem
- $\ds \int \tanh x \rd x = \map \ln {\cosh x} + C$
Proof 1
| \(\ds \int \tanh x \rd x\) | \(=\) | \(\ds \int \frac {\sinh x} {\cosh x} \rd x\) | Definition of Hyperbolic Tangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \frac {\paren {\cosh x}'} {\cosh x} \rd x\) | Derivative of Hyperbolic Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \ln \size {\cosh x} + C\) | Primitive of Function under its Derivative | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \ln {\cosh x} + C\) | Graph of Hyperbolic Cosine Function: $\cosh x > 0$ for all $x$ |
$\blacksquare$
Proof 2
| \(\ds \int \tanh x \rd x\) | \(=\) | \(\ds -i \int \tan i x \rd x\) | Hyperbolic Tangent in terms of Tangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\int \tan i x \rd \paren {i x}\) | Primitive of Function of Constant Multiple | |||||||||||
| \(\ds \) | \(=\) | \(\ds \ln \cmod {\cos i x} + C\) | Primitive of $\tan x$: Cosine Form | |||||||||||
| \(\ds \) | \(=\) | \(\ds \ln \cmod {\cosh x} + C\) | Definition of Hyperbolic Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \ln {\cosh x} + C\) | Graph of Hyperbolic Cosine Function: $\cosh x > 0$ for all $x$ |
$\blacksquare$
Also see
Sources
- 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 8$. Change of Variable
- 1960: Margaret M. Gow: A Course in Pure Mathematics ... (previous) ... (next): Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xxv)}$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.27$
- 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $5$. Integrals: Integrals of Special Functions: $17$
- 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
- 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
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals
- 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: $16.27.$