Primitive of Square of Hyperbolic Sine of a x
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Theorem
- $\ds \int \sinh^2 a x \rd x = \dfrac {\sinh a x \cosh a x} {2 a} - \frac x 2 + C$
Corollary
- $\ds \int \sinh^2 a x \rd x = \frac {\sinh 2 a x} {4 a} - \frac x 2 + C$
Proof
\(\ds \int \sinh^2 x \rd x\) | \(=\) | \(\ds \frac {\sinh x \cosh x - x} 2 + C\) | Primitive of $\sinh^2 x$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \sinh^2 a x \rd x\) | \(=\) | \(\ds \frac 1 a \paren {\frac {\sinh a x \cosh a x - a x} 2} + C\) | Primitive of Function of Constant Multiple | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sinh a x \cosh a x} {2 a} - \frac x 2 + C\) | simplifying |
$\blacksquare$
Also see
- Primitive of $\cosh^2 a x$
- Primitive of $\tanh^2 a x$
- Primitive of $\coth^2 a x$
- Primitive of $\sech^2 a x$
- Primitive of $\csch^2 a x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sinh a x$: $14.547$