Primitive of Hyperbolic Cotangent of a x

Theorem

$\ds \int \coth a x \rd x = \frac {\ln \size {\sinh a x} } a + C$

Proof

\(\ds \int \coth x \rd x\)

\(=\)

\(\ds \ln \size {\sinh x} + C\)

Primitive of $\coth x$

\(\ds \leadsto \ \ \)

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

\(=\)

\(\ds \frac 1 a \paren {\ln \size {\sinh a x} } + C\)

Primitive of Function of Constant Multiple

\(\ds \)

\(=\)

\(\ds \frac {\ln \size {\sinh a x} } a + C\)

simplifying

$\blacksquare$

Also see

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

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\coth a x$: $14.615$
• 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $124$.