Primitive of Square of Hyperbolic Sine of a x over Hyperbolic Cosine of a x
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Theorem
- $\ds \int \frac {\sinh^2 a x \rd x} {\cosh a x} = \frac {\sinh a x} a - \frac 1 a \map \arctan {\sinh a x} + C$
Proof
| \(\ds \int \frac {\sinh^2 a x \rd x} {\cosh a x}\) | \(=\) | \(\ds \int \frac {\paren {\cosh^2 a x - 1} \rd x} {\cosh a x}\) | Difference of Squares of Hyperbolic Cosine and Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \frac {\paren {\cosh^2 a x} \rd x} {\cosh a x} - \int \frac {\d x} {\cosh a x}\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \cosh a x \rd x - \int \sech a x \rd x\) | Definition 2 of Hyperbolic Secant | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sinh a x} a - \int \sech a x \rd x\) | Primitive of $\cosh a x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sinh a x} a - \frac 1 a \map \arctan {\sinh a x} + C\) | 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 $\cosh a x$: $14.599$