Primitive of Reciprocal of Square of Hyperbolic Sine of a x

Theorem

$\ds \int \frac {\d x} {\sinh^2 a x} = -\frac {\coth a x} a + C$

Proof

\(\ds \int \frac {\d x} {\sinh^2 a x}\)

\(=\)

\(\ds \int \csch^2 a x \rd x\)

Definition 2 of Hyperbolic Cosecant

\(\ds \)

\(=\)

\(\ds -\frac {\coth a x} a + C\)

Primitive of $\csch^2 a x$

$\blacksquare$

Also see

• Primitive of $\dfrac 1 {\cosh^2 a x}$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sinh a x$: $14.549$