Primitive of Reciprocal of Square of Hyperbolic Sine of a x by Hyperbolic Cosine of a x
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Theorem
- $\ds \int \frac {\d x} {\sinh^2 a x \cosh a x} = -\frac 1 a \map \arctan {\sinh a x} - \frac {\csch a x} a + C$
Proof
\(\ds \int \frac {\d x} {\sinh^2 a x \cosh a x}\) | \(=\) | \(\ds \int \frac {\paren {\cosh^2 a x - \sinh^2 a x} \rd x} {\sinh^2 a x \cosh a x}\) | Difference of Squares of Hyperbolic Cosine and Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\cosh^2 a x \rd x} {\sinh^2 a x \cosh a x} - \int \frac {\sinh^2 a x \rd x} {\sinh^2 a x \cosh a x}\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\cosh a x \rd x} {\sinh^2 a x} - \int \frac {\d x} {\cosh a x}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\cosh a x \rd x} {\sinh^2 a x} - \int \sech a x \rd x\) | Definition 2 of Hyperbolic Secant | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\coth a x \rd x} {\sinh a x} - \int \sech a x \rd x\) | Definition 2 of Hyperbolic Cotangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \csch a x \coth a x \rd x - \int \sech a x \rd x\) | Definition 2 of Hyperbolic Cosecant | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-\csch a x} a - \int \sech a x \rd x\) | Primitive of $\csch^n a x \coth a x$ for $n = 1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-\csch a x} a - \frac 1 a \map \arctan {\sinh a x}\) | Primitive of $\sech a x$ |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sinh a x $ and $\cosh a x$: $14.596$