Primitive of Tangent Function/Secant Form
Theorem
- $\ds \int \tan x \rd x = \ln \size {\sec x} + C$
where $\sec x$ is defined.
Proof
| \(\ds \int \tan x \rd x\) | \(=\) | \(\ds -\ln \size {\cos x} + C\) | Primitive of $\tan x$: Cosine Form | |||||||||||
| \(\ds \) | \(=\) | \(\ds \ln \size {\frac 1 {\cos x} } + C\) | Logarithm of Reciprocal | |||||||||||
| \(\ds \) | \(=\) | \(\ds \ln \size {\sec x} + C\) | Secant is Reciprocal of Cosine |
$\blacksquare$
Also see
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Integration
- 1960: Margaret M. Gow: A Course in Pure Mathematics ... (previous) ... (next): Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xvii)}$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.13$
- 1972: Frank Ayres, Jr. and J.C. Ault: Theory and Problems of Differential and Integral Calculus (SI ed.) ... (previous) ... (next): Chapter $25$: Fundamental Integration Formulas: $10$.
- 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $5$. Integrals: Integrals of Special Functions: $5$
- 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: $16.13.$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $7$: Integrals
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $8$: Integrals