Difference between revisions of "Antiderivative of tanh"

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==Theorem==
 
==Theorem==
 
The following formula holds:
 
The following formula holds:
$$\displaystyle\int \tanh(z)\mathrm{d}z = \log(\cosh(z)),$$
+
$$\displaystyle\int \tanh(z)\mathrm{d}z = \log(\cosh(z))+C,$$
 
where $\tanh$ denotes the [[tanh|hyperbolic tangent]], $\log$ denotes the [[logarithm]], and $\cosh$ denotes the [[cosh|hyperbolic cosine]].
 
where $\tanh$ denotes the [[tanh|hyperbolic tangent]], $\log$ denotes the [[logarithm]], and $\cosh$ denotes the [[cosh|hyperbolic cosine]].
  
 
==Proof==
 
==Proof==
 +
By definition,
 +
$$\mathrm{tanh}(z) = \dfrac{\mathrm{sinh}(z)}{\mathrm{cosh}(z)}.$$
 +
Let $u=\mathrm{cosh}(z)$ and use the [[derivative of cosh]], [[u-substitution]], and the definition of the [[logarithm]] to derive
 +
$$\begin{array}{ll}
 +
\displaystyle\int \mathrm{tanh}(z) \mathrm{d}z &= \displaystyle\int \dfrac{1}{u} \mathrm{d} u \\
 +
&= \log \left( \mathrm{cosh}(z) \right) + C,
 +
\end{array}$$
 +
as was to be shown. █
  
 
==References==
 
==References==

Revision as of 22:56, 24 June 2016

Theorem

The following formula holds: $$\displaystyle\int \tanh(z)\mathrm{d}z = \log(\cosh(z))+C,$$ where $\tanh$ denotes the hyperbolic tangent, $\log$ denotes the logarithm, and $\cosh$ denotes the hyperbolic cosine.

Proof

By definition, $$\mathrm{tanh}(z) = \dfrac{\mathrm{sinh}(z)}{\mathrm{cosh}(z)}.$$ Let $u=\mathrm{cosh}(z)$ and use the derivative of cosh, u-substitution, and the definition of the logarithm to derive $$\begin{array}{ll} \displaystyle\int \mathrm{tanh}(z) \mathrm{d}z &= \displaystyle\int \dfrac{1}{u} \mathrm{d} u \\ &= \log \left( \mathrm{cosh}(z) \right) + C, \end{array}$$ as was to be shown. █

References