Primitive of Cosecant of a x/Tangent Form

Theorem

$\ds \int \csc a x \rd x = \frac 1 a \ln \size {\tan \frac {a x} 2} + C$

where $\tan \dfrac {a x} 2 \ne 0$.

Proof

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

\(=\)

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

Primitive of $\csc x$: Tangent Form

\(\ds \leadsto \ \ \)

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

\(=\)

\(\ds \frac 1 a \ln \size {\tan \frac {a x} 2} + C\)

Primitive of Function of Constant Multiple

$\blacksquare$

Also see

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

Sources

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