Primitive of Square of Hyperbolic Sine Function/Corollary

Corollary to Primitive of Square of Hyperbolic Sine Function

$\ds \int \sinh^2 x \rd x = \frac {\sinh x \cosh x - x} 2 + C$

where $C$ is an arbitrary constant.

Proof

\(\ds \int \sinh^2 x \rd x\)

\(=\)

\(\ds \frac {\sinh 2 x} 4 - \frac x 2 + C\)

Primitive of Square of Hyperbolic Sine Function

\(\ds \)

\(=\)

\(\ds \frac {2 \sinh x \cosh x} 4 - \frac x 2 + C\)

Double Angle Formula for Hyperbolic Sine

\(\ds \)

\(=\)

\(\ds \frac {\sinh x \cosh x - x} 2 + C\)

rearranging

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.35$
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 16$: Indefinite Integrals: General Rules of Integration: $16.35.$