Difference between revisions of "Antiderivative of tanh"
From specialfunctionswiki
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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. █
