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:

\(\ds \int \dfrac {\d x} {\sin x}\)

\(=\)

\(\ds \ln \size {\tan \frac x 2} + C\)

\(\ds \int \dfrac {\d x} {\sin x}\)

\(=\)

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

\(\ds \int \dfrac {\d x} {\sin x}\)

\(=\)

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

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 $\dfrac 1 {\cos a x}$