Primitive of Square of Hyperbolic Sine of a x by Square of Hyperbolic Cosine of a x
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Theorem
- $\ds \int \sinh^2 a x \cosh^2 a x \rd x = \frac {\sinh 4 a x} {32 a} - \frac x 8 + C$
Proof
\(\ds \int \sinh^2 a x \cosh^2 a x \rd x\) | \(=\) | \(\ds \int \paren {\sinh a x \cosh a x}^2 \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \paren {\frac {\sinh 2 a x} 2}^2 \rd x\) | Double Angle Formula for Hyperbolic Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 4 \int \sinh^2 2 a x \rd x\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 4 \paren {\frac {\sinh 2 \paren {2 a x} } {4 \paren {2 a} } - \frac x 2} + C\) | Primitive of $\sinh^2 a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sinh 4 a x} {32 a} - \frac x 8 + C\) | simplifying |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sinh a x $ and $\cosh a x$: $14.594$