View source for Antiderivative of coth

==Theorem==
The following formula holds:
$$\displaystyle\int \mathrm{coth}(z) \mathrm{d}z=\log(\sinh(z)) + C,$$
for arbitrary constant $C$, where $\mathrm{coth}$ denotes the [[coth|hyperbolic cotangent]], $\log$ denotes the [[logarithm]], and $\sinh$ denotes the [[sinh|hyperbolic sine]].

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

[[Category:Theorem]]
[[Category:Proven]]