Primitive of Tangent Function/Cosine Form

Theorem

$\ds \int \tan x \rd x = -\ln \size {\cos x} + C$

where $\cos x \ne 0$.

Proof

\(\ds \int \tan x \rd x\)

\(=\)

\(\ds \int \frac {\sin x} {\cos x} \rd x\)

Definition of Real Tangent Function

\(\ds \)

\(=\)

\(\ds -\int \frac {-\sin x} {\cos x} \rd x\)

Primitive of Constant Multiple of Function

\(\ds \)

\(=\)

\(\ds -\int \frac {\paren {\cos x}'} {\cos x} \rd x\)

Derivative of Cosine Function

\(\ds \)

\(=\)

\(\ds -\ln \size {\cos x} + C\)

Primitive of Function under its Derivative

$\blacksquare$

Also see

• Primitive of $\tan x$: Secant Form

Sources

• 1945: A. Geary, H.V. Lowry and H.A. Hayden: Advanced Mathematics for Technical Students, Part I ... (previous) ... (next): Chapter $\text {III}$: Integration: Integration
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.13$
• 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $5$. Integrals: Integrals of Special Functions: $5$
• 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: Trigonometric 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
• 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