Primitive of Reciprocal of Hyperbolic Sine of a x

Theorem

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

Proof

\(\ds \int \frac {\d x} {\sinh a x}\)

\(=\)

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

Definition 2 of Hyperbolic Cosecant

\(\ds \)

\(=\)

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

Primitive of $\csch a x$

$\blacksquare$

Also see

• Primitive of $\dfrac 1 {\cosh a x}$

• Primitive of $\dfrac 1 {\tanh a x}$

• Primitive of $\dfrac 1 {\coth a x}$

• Primitive of $\dfrac 1 {\sech a x}$

• Primitive of $\dfrac 1 {\csch a x}$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sinh a x$: $14.545$