Primitive of Hyperbolic Cosecant of a x

Theorem

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

Proof

\(\ds \int \csch x \rd x\)

\(=\)

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

Primitive of $\csch x$

\(\ds \leadsto \ \ \)

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

\(=\)

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

Primitive of Function of Constant Multiple

$\blacksquare$

Also see

• Primitive of $\sinh a x$
• Primitive of $\cosh a x$
• Primitive of $\tanh a x$
• Primitive of $\coth a x$
• Primitive of $\sech a x$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\csch a x$: $14.636$
• 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $130$.
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 17$: Tables of Special Indefinite Integrals: $(32)$ Integrals Involving $\csch a x$: $17.33.1.$