Primitive of Cotangent Function
Theorem
- $\ds \int \cot x \rd x = \ln \size {\sin x} + C$
where $\sin x \ne 0$.
Proof
| \(\ds \int \cot x \rd x\) | \(=\) | \(\ds \int \frac {\cos x} {\sin x} \rd x\) | Definition of Real Cotangent Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \frac {\paren {\sin x}'} {\sin x} \rd x\) | Derivative of Sine Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \ln \size {\sin x} + C\) | Primitive of Function under its Derivative |
$\blacksquare$
Also see
Sources
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- 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: $\text {(xviii)}$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.14$
- 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: $11$.
- 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $5$. Integrals: Integrals of Special Functions: $6$
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- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals
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- 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: $16.14.$
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- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $8$: Integrals