Primitive of Hyperbolic Cosecant Function/Inverse Hyperbolic Cotangent of Hyperbolic Cosine Form/Proof 2
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Theorem
- $\ds \int \csch x \rd x = -2 \map {\coth^{-1} } {e^x} + C$
Proof
Let:
| \(\ds u\) | \(=\) | \(\ds \cosh x\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds u'\) | \(=\) | \(\ds \sinh x\) | Derivative of Hyperbolic Cosine |
Then:
| \(\ds \int \csch x \rd x\) | \(=\) | \(\ds \int \frac 1 {\sinh x} \rd x\) | Definition of Hyperbolic Cosecant | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \frac {\sinh x} {\sinh^2 x} \rd x\) | multiplying top and bottom by $\sinh x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \frac {\sinh x} {\cosh^2 x - 1} \rd x\) | Difference of Squares of Hyperbolic Cosine and Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \frac {\rd u} {u^2 - 1}\) | Integration by Substitution | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\coth^{-1} u + C\) | Primitive of Reciprocal of $\dfrac 1 {x^2 - a^2}$: $\coth^{-1}$ form | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\map {\coth^{-1} } {\cosh x} + C\) | Definition of $u$ |
$\blacksquare$