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Anexo
:
Integrales de funciones inversas hiperbólicas
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La siguiente es una lista de
integrales
de las
inversas de las funciones hiperbólicas
:
∫
arsinh
x
c
d
x
=
x
arsinh
x
c
−
x
2
+
c
2
{\displaystyle \int \operatorname {arsinh} {\frac {x}{c}}\,dx=x\operatorname {arsinh} {\frac {x}{c}}-{\sqrt {x^{2}+c^{2}}}}
∫
arcosh
x
c
d
x
=
x
arcosh
x
c
−
x
2
−
c
2
{\displaystyle \int \operatorname {arcosh} {\frac {x}{c}}\,dx=x\operatorname {arcosh} {\frac {x}{c}}-{\sqrt {x^{2}-c^{2}}}}
∫
artanh
x
c
d
x
=
x
artanh
x
c
+
c
2
ln
|
c
2
−
x
2
|
(
para
|
x
|
<
|
c
|
)
{\displaystyle \int \operatorname {artanh} {\frac {x}{c}}\,dx=x\operatorname {artanh} {\frac {x}{c}}+{\frac {c}{2}}\ln \left|c^{2}-x^{2}\right|\qquad ({\text{para }}|x|<|c|)}
∫
arcoth
x
c
d
x
=
x
arcoth
x
c
+
c
2
ln
|
x
2
−
c
2
|
(
para
|
x
|
>
|
c
|
)
{\displaystyle \int \operatorname {arcoth} {\frac {x}{c}}\,dx=x\operatorname {arcoth} {\frac {x}{c}}+{\frac {c}{2}}\ln \left|x^{2}-c^{2}\right|\qquad ({\text{para }}|x|>|c|)}
∫
arsech
x
c
d
x
=
x
arsech
x
c
−
c
arctan
x
c
−
x
c
+
x
x
−
c
(
para
x
∈
(
0
,
c
)
)
{\displaystyle \int \operatorname {arsech} {\frac {x}{c}}\,dx=x\operatorname {arsech} {\frac {x}{c}}-c\operatorname {arctan} {\frac {x\,{\sqrt {\frac {c-x}{c+x}}}}{x-c}}\qquad \left({\text{para }}x\in (0,c)\right)}
∫
arcsch
x
c
d
x
=
x
arcsch
x
c
+
c
ln
x
+
x
2
+
c
2
c
(
para
x
∈
(
0
,
c
)
)
{\displaystyle \int \operatorname {arcsch} {\frac {x}{c}}\,dx=x\operatorname {arcsch} {\frac {x}{c}}+c\,\ln \,{\frac {x+{\sqrt {x^{2}+c^{2}}}}{c}}\qquad \left({\text{para }}x\in (0,c)\right)}
