Primitive of Hyperbolic Cosecant Function/Hyperbolic Tangent Form/Proof 1
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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
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 $u$ |
$\blacksquare$