Primitive of Hyperbolic Tangent Function

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Theorem

$\ds \int \tanh x \rd x = \map \ln {\cosh x} + C$


Proof 1

\(\ds \int \tanh x \rd x\) \(=\) \(\ds \int \frac {\sinh x} {\cosh x} \rd x\) 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 $x$

$\blacksquare$


Proof 2

\(\ds \int \tanh x \rd x\) \(=\) \(\ds -i \int \tan i x \rd x\) Hyperbolic Tangent in terms of Tangent
\(\ds \) \(=\) \(\ds -\int \tan i x \rd \paren {i x}\) Primitive of Function of Constant Multiple
\(\ds \) \(=\) \(\ds \ln \cmod {\cos i x} + C\) Primitive of $\tan x$: Cosine Form
\(\ds \) \(=\) \(\ds \ln \cmod {\cosh x} + C\) Definition of Hyperbolic Cosine
\(\ds \) \(=\) \(\ds \map \ln {\cosh x} + C\) Graph of Hyperbolic Cosine Function: $\cosh x > 0$ for all $x$

$\blacksquare$


Also see


Sources

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