Primitive of Hyperbolic Tangent Function/Proof 1
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Theorem
- $\ds \int \tanh x \rd x = \map \ln {\cosh x} + C$
Proof
| \(\ds \int \tanh x \rd x\) | \(=\) | \(\ds \int \frac {\sinh x} {\cosh x} \rd x\) | Definition of Hyperbolic Tangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \frac {\paren {\cosh x}'} {\cosh x} \rd x\) | Derivative of Hyperbolic Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \ln \size {\cosh x} + C\) | Primitive of Function under its Derivative | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \ln {\cosh x} + C\) | Graph of Hyperbolic Cosine Function: $\cosh x > 0$ for all $x$ |
$\blacksquare$