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(1)Hyperbolic Functions
Hyperbolic cosine of x: cosh
2
x x
e e x
−
+ =
Hyperbolic sine of x: sinh
2
x x
e e x
−
− =
Hyperbolic tangent: tanh sinh
cosh
x x
x x
x e e x
x e e
− −
−
= =
+
Hyperbolic cotangent: coth cosh
sinh
x x
x x
x e e x
x e e
− −
+
= =
−
Hyperbolic secant: sech
cosh x x
x
x e e−
= =
+
Hyperbolic cosecant: csch
sinh x x
x
x e e−
= =
−
Identities
2
2
2
sinh cosh
cosh sinh
tanh sech
coth csch
x
x x e
x x
x x
x x
+ =
− =
= − = +
Derivatives
( )
( )
( )
sinh cosh
cosh sinh
tanh sech
d du
u u
dx dx
d du
u u
dx dx
d du
u u
dx dx
= = =
( )
( )
( )
2
coth csch
sech sech
csch csch coth
d du
u u
dx dx
d du
u u u
dx dx
d du
u u u
dx dx
= − = − = −
Integrals
2
sinh cosh
cosh sinh
sech
u du u C
u du u C u du u C
= +
= +
= +
∫ ∫ ∫
2
csch coth
sech sech
csch coth csch
u du u C
u u du u C
u u du u C
= − +
= − +
= − +
∫ ∫ ∫ Useful Identities
1 11
sech x cosh
x
− = − csch 1x sinh 11
x
− = − coth1x tanh 11
x
− = −
Derivatives of Inverse Logarithm Formulas for Evaluating
Hyperbolic Functions Inverse Hyperbolic Functions
( )
( )
( )
( )
( )
( )
1
2
2
2
2
2
2
sinh 1
1
cosh 1
,
1
tanh 1
,
1
coth 1
,
1
sech 1
,
1
csch 1
,
1
d u du
dx u dx
d u du
u
dx u dx
d u du
u
dx u dx
d u du
u
dx u dx
d u du
u
dx u u dx
d u du
u
dx u u dx
−
−
−
−
−
−
= +
= >
−
= <
−
= >
− −
= < <
− −
= ≠
+
( )
( )
1
1
1
2
2
1
sinh ln ,
cosh ln ,
1
tanh ln ,
2
1
sech ln ,
1
csch ln ,
1
coth ln ,
2
x x x x
x x x x
x
x x
x x
x x
x x
x x
x x
x
x x
x
−
−
−
−
−
−
= + + − ∞ < < ∞
= + − ≥
+
= <
−
+ −
= < ≤
+
= + ≠
+
= >
−
Integrals of Inverse Hyperbolic Functions
1
1
1
sinh
cosh ,
1
tanh if
coth if
1
du
u C u
du
u C u u
u C u du
u C u u
−
−
− −
= +
+
= + >
−
+ <
= + >
−
∫ ∫ ∫
1
2
1
2
1
sech cosh
1
1
csch sinh
1
du
u C C
u
u u
du
u C C
u
u u
− −
− −
= − + = − +
−
= − + = − +
+
∫ ∫
2
2
sinh 2 sinh cosh
cosh cosh sinh
cosh
cosh
2
cosh
sinh
2
x x x
x x x
x x
x x
=
= +
+ =