Primitive of Cosecant Function

Theorem

Tangent Form

$\ds \int \csc x \rd x = \ln \size {\tan \frac x 2} + C$

where $\tan \dfrac x 2 \ne 0$.

Cosecant plus Cotangent Form

$\ds \int \csc x \rd x = -\ln \size {\csc x + \cot x} + C$

where $\csc x + \cot x \ne 0$.

Cosecant minus Cotangent Form

$\ds \int \csc x \rd x = \ln \size {\csc x - \cot x} + C$

where $\csc x - \cot x \ne 0$.

Also presented as

Some sources present this result as the primitive of the reciprocal of the sine function:

Also see

• Primitive of Sine Function
• Primitive of Cosine Function

• Primitive of Tangent Function
• Primitive of Cotangent Function

• Primitive of Secant Function

• Primitive of $\dfrac 1 {\sin a x}$

• Primitive of $\dfrac 1 {\cos a x}$

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: $\text {(xix)}$
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.16$
• 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $5$. Integrals: Integrals of Special Functions: $8$
• 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: $16.16.$
• 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