Primitive of Square of Hyperbolic Cosine of a x over Hyperbolic Sine of a x
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Theorem
- $\ds \int \frac {\cosh^2 a x \rd x} {\sinh a x} = \frac {\cosh a x} a + \frac 1 a \ln \size {\tanh \frac {a x} 2} + C$
Proof
\(\ds \int \frac {\cosh^2 a x \rd x} {\sinh a x}\) | \(=\) | \(\ds \int \frac {\paren {\sinh^2 a x + 1} \rd x} {\sinh a x}\) | Difference of Squares of Hyperbolic Cosine and Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\sinh^2 a x \rd x} {\sinh a x} + \int \frac {\d x} {\sinh a x}\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \sinh a x \rd x + \int \csch a x \rd x\) | Definition 2 of Hyperbolic Cosecant | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\cosh a x} a + \int \csch a x \rd x\) | Primitive of $\sinh a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\cosh a x} a + \frac 1 a \ln \size {\tanh \frac {a x} 2} + C\) | 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 $\cosh a x$: $14.600$