Primitive of Hyperbolic Tangent of a x

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Theorem

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


Proof

\(\ds \int \tanh x \rd x\) \(=\) \(\ds \map \ln {\cosh x} + C\) Primitive of $\tanh x$
\(\ds \leadsto \ \ \) \(\ds \int \tanh a x \rd x\) \(=\) \(\ds \frac 1 a \paren {\map \ln {\cosh a x} } + C\) Primitive of Function of Constant Multiple
\(\ds \) \(=\) \(\ds \frac {\map \ln {\cosh a x} } a + C\) simplifying

$\blacksquare$


Also see


Sources