Primitive of Hyperbolic Cosecant Function/Hyperbolic Tangent Form

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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 1

Let $u = \tanh \dfrac x 2$.

Then:

\(\ds \int \csch x \rd x\) \(=\) \(\ds \int \dfrac 1 {\sinh x} \rd x\) Definition 2 of Hyperbolic Cosecant
\(\ds \) \(=\) \(\ds \int \dfrac {1 - u^2} {2 u} \dfrac {2 \rd u} {1 - u^2}\) Hyperbolic Tangent Half-Angle Substitution
\(\ds \) \(=\) \(\ds \int \dfrac {\d u} u\) simplifying
\(\ds \) \(=\) \(\ds \ln \size u + C\) Logarithm of Reciprocal
\(\ds \) \(=\) \(\ds \ln \size {\tanh \frac x 2} + C\) substituting back for $u$

$\blacksquare$


Proof 2

\(\ds \int \csch x \rd x\) \(=\) \(\ds -\ln \size {\csch x + \coth x} + C\) 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 $1$

$\blacksquare$


Sources

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