Primitive of Hyperbolic Cosecant Function/Hyperbolic Tangent Form/Proof 2
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Theorem
- $\ds \int \csch x \rd x = \ln \size {\tanh \frac x 2} + C$
where $\tanh \dfrac x 2 \ne 0$.
Proof
| \(\ds \int \csch x \rd x\) | \(=\) | \(\ds -\ln \size {\csch x + \coth x} + C\) | Primitive of $\csch x$: Logarithm Form | |||||||||||
| \(\ds \) | \(=\) | \(\ds \ln \size {\frac 1 {\csch x + \coth x} } + C\) | Logarithm of Reciprocal | |||||||||||
| \(\ds \) | \(=\) | \(\ds \ln \size {\frac 1 {\frac 1 {\sinh x} + \frac {\cosh x} {\sinh x} } } + C\) | Definition 2 of Hyperbolic Cosecant and Definition 2 of Hyperbolic Cotangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds \ln \size {\frac {\sinh x} {1 + \cosh x} } + C\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \ln \size {\tanh \frac x 2} + C\) | Half Angle Formula for Hyperbolic Tangent: Corollary $1$ |
$\blacksquare$