We determine a
primitive function
for
, using the hyperbolic functions
and
,
for which the relation
holds. The
substitution
-
yields
-
![{\displaystyle {}\int _{a}^{b}{\sqrt {x^{2}-1}}\,dx=\int _{\,\operatorname {arcosh} \,a\,}^{\,\operatorname {arcosh} \,b\,}{\sqrt {\cosh ^{2}t-1}}\cdot \sinh t\,dt=\int _{\,\operatorname {arcosh} \,a\,}^{\,\operatorname {arcosh} \,b\,}\sinh ^{2}t\,dt\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/484af86b46c8586b2cc038c3d2e042307cbe14b3)
A primitive function of the hyperbolic sine squared follows from
-
![{\displaystyle {}\sinh ^{2}t={\left({\frac {1}{2}}{\left(e^{t}-e^{-t}\right)}\right)}^{2}={\frac {1}{4}}{\left(e^{2t}+e^{-2t}-2\right)}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7dd66e37ab6672969cfe1d0993464bc0434e8683)
Therefore,
-
![{\displaystyle {}\int _{}^{}\sinh ^{2}u\,dt={\frac {1}{4}}{\left({\frac {1}{2}}e^{2u}-{\frac {1}{2}}e^{-2u}-2u\right)}={\frac {1}{4}}\sinh 2u-{\frac {1}{2}}u\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2e8daf77e6501b1de2b8c62c2b9d397d9de89c9)
and hence
-
![{\displaystyle {}\int _{}^{}{\sqrt {x^{2}-1}}\,dx={\frac {1}{4}}\sinh(2\,\operatorname {arcosh} \,x\,)-{\frac {1}{2}}\,\operatorname {arcosh} \,x\,\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8156da1ddee6cd796cad327a983c72d34e847a00)
Due to
the addition theorem
for hyperbolic sine, we have
,
and therefore this primitive function can also be written as
![{\displaystyle {}{\begin{aligned}{\frac {1}{2}}{\left(\sinh {\left(\,\operatorname {arcosh} \,x\,\right)}\cosh {\left(\,\operatorname {arcosh} \,x\,\right)}-\,\operatorname {arcosh} \,x\,\right)}&={\frac {1}{2}}{\left({\sqrt {\cosh {\left(\,\operatorname {arcosh} \,x\,\right)}^{2}-1}}\cdot x-\,\operatorname {arcosh} \,x\,\right)}\\&={\frac {1}{2}}{\left({\sqrt {x^{2}-1}}\cdot x-\,\operatorname {arcosh} \,x\,\right)}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2bb9f28269e6b65dca36b4716ba6f3978af320e)