Primitive of Product of Cosecant and Cotangent

Theorem

$\ds \int \csc x \cot x \rd x = -\csc x + C$

where $C$ is an arbitrary constant.

Proof

From Derivative of Cosecant Function:

$\dfrac \d {\d x} \csc x = -\csc x \cot x$

The result follows from the definition of primitive.

$\blacksquare$

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
• 1960: Margaret M. Gow: A Course in Pure Mathematics ... (previous) ... (next): Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xvi)}$
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.24$
• 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: $17$.
• 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $5$. Integrals: Integrals of Special Functions: $12$
• 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: $16.24.$