Primitive of Cosecant of a x/Cosecant minus Cotangent Form

Theorem

$\ds \int \csc a x \rd x = \frac 1 a \ln \size {\csc a x - \cot a x} + C$

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

Proof

\(\ds \int \csc x \rd x\)

\(=\)

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

Primitive of $\csc x$: Cosecant minus Cotangent Form

\(\ds \leadsto \ \ \)

\(\ds \int \csc a x \rd x\)

\(=\)

\(\ds \frac 1 a \ln \size {\csc a x - \cot a x} + C\)

Primitive of Function of Constant Multiple

$\blacksquare$

Also see

• Primitive of $\sec a x$: Secant plus Tangent Form

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\csc a x$: $14.461$