Primitive of Reciprocal of Hyperbolic Sine of a x by Square of Hyperbolic Cosine of a x
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Theorem
- $\ds \int \frac {\d x} {\sinh a x \cosh^2 a x} = \frac 1 a \ln \size {\tanh \frac {a x} 2} + \frac {\sech a x} a + C$
Proof
\(\ds \int \frac {\d x} {\sinh^2 a x \cosh a x}\) | \(=\) | \(\ds \int \frac {\left({\cosh^2 a x - \sinh^2 a x}\right) \rd x} {\sinh a x \cosh^2 a x}\) | Difference of Squares of Hyperbolic Cosine and Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\cosh^2 a x \rd x} {\sinh a x \cosh^2 a x} - \int \frac {\sinh^2 a x \rd x} {\sinh a x \cosh^2 a x}\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\d x} {\sinh a x} - \int \frac {\sinh a x \rd x} {\cosh^2 a x}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \csch a x \rd x - \int \frac {\sinh a x \rd x} {\cosh^2 a x}\) | Definition 2 of Hyperbolic Cosecant | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \csch a x \rd x - \int \frac {\tanh a x \rd x} {\cosh a x}\) | Definition 2 of Hyperbolic Tangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \csch a x \rd x - \int \sech a x \tanh a x \rd x\) | Definition 2 of Hyperbolic Secant | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \csch a x \rd x - \frac {-\sech a x} a + C\) | Primitive of $\sech^n a x \tanh a x$ for $n = 1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \ln \left\vert {\tanh \frac {a x} 2} \right\vert + \frac {\sech a x} a + C\) | Primitive of $\csch a x$ |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sinh a x $ and $\cosh a x$: $14.597$