Primitive of Reciprocal of Square of Hyperbolic Sine of a x by Square of Hyperbolic Cosine of a x
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Theorem
- $\ds \int \frac {\d x} {\sinh^2 a x \cosh^2 a x} = \frac {-2 \coth 2 a x} a + C$
Proof
\(\ds \int \frac {\d x} {\sinh^2 a x \cosh^2 a x}\) | \(=\) | \(\ds \int \frac {\d x} {\paren {\sinh a x \cosh a x}^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\d x} {\paren {\dfrac {\sinh^2 2 a x} 2}^2}\) | Double Angle Formula for Hyperbolic Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \int \frac {\d x} {\sinh^2 2 a x}\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \int \csch^2 2 a x \rd x\) | Definition 2 of Hyperbolic Cosecant | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \frac {-\coth 2 a x} {2 a} + C\) | Primitive of $\csch^2 a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-2 \coth 2 a x} a + C\) | simplifying |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sinh a x $ and $\cosh a x$: $14.598$