Primitive of Cosecant of a x/Cosecant plus 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

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

\(=\)

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

Primitive of $\csc x$: Cosecant plus 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: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $89$.