Primitive of Hyperbolic Cosecant Function/Inverse Hyperbolic Cotangent of Hyperbolic Cosine Form
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Theorem
- $\ds \int \csch x \rd x = -\map {\coth^{-1} } {\cosh x} + C$
Proof 1
Let:
\(\ds u\) | \(=\) | \(\ds \cosh x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac {\d u} {\d x}\) | \(=\) | \(\ds \sinh x\) | Derivative of Hyperbolic Cosine |
Then:
\(\ds \int \csch x \rd x\) | \(=\) | \(\ds \int \dfrac {\d x} {\sinh x}\) | Definition 2 of Hyperbolic Cosecant | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \dfrac 1 {\sinh x} \dfrac {\d u} {\sinh x}\) | Integration by Substitution | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \dfrac {\d u} {\sinh^2 x}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \dfrac {\d u} {\cosh^2 x - 1}\) | Difference of Squares of Hyperbolic Cosine and Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \dfrac {\d u} {u^2 - 1}\) | Definition of $u$ | |||||||||||
\(\ds \) | \(=\) | \(\ds -\coth^{-1} u + C\) | Primitive of $\dfrac 1 {x^2 - a^2}$: $\coth^{-1}$ form | |||||||||||
\(\ds \) | \(=\) | \(\ds -\map {\coth^{-1} } {\cosh x} + C\) | Definition of $u$ |
$\blacksquare$
Proof 2
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 $u$ |
$\blacksquare$
Also see
Sources
- 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $5$. Integrals: Integrals of Special Functions: $20$