Primitive of Square of Hyperbolic Sine Function
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Theorem
- $\ds \int \sinh^2 x \rd x = \frac {\sinh 2 x} 4 - \frac x 2 + C$
where $C$ is an arbitrary constant.
Corollary
- $\ds \int \sinh^2 x \rd x = \frac {\sinh x \cosh x - x} 2 + C$
where $C$ is an arbitrary constant.
Proof
\(\ds \int \sinh^2 x \rd x\) | \(=\) | \(\ds \int \paren {\frac {\cosh 2 x - 1} 2} \rd x\) | Square of Hyperbolic Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \paren {\frac {\cosh 2 x} 2} \rd x - \int \frac 1 2 \rd x\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \paren {\frac {\cosh 2 x} 2} \rd x - \frac x 2 + C\) | Primitive of Constant | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {\frac {\sinh 2 x} 2} - \frac x 2 + C\) | Primitive of Function of Constant Multiple and Primitive of Hyperbolic Cosine Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sinh 2 x} 4 - \frac x 2 + C\) | rearranging |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.35$