Derivatives, Integrals, and Properties
Of Inverse Trigonometric Functions and Hyperbolic Functions (On this handout, a represents a constant, u and x represent variable quantities)
Derivatives of Inverse Trigonometric Functions
d
dxsin
¡1u = p 1
1¡ u2
du
dx (juj < 1) d
dxcos¡1u =
¡1 p
1¡ u2
du
dx (juj < 1) d
dxtan¡1u =
1
1 + u2
du dx d
dxcsc
jujpu2¡ 1
du
dx (juj > 1) d
dxsec¡1u =
1 jujpu2¡ 1
du
dx (juj > 1) d
dxcot
1 + u2
du dx
Integrals Involving Inverse Trigonometric Functions
p
a2¡ u2 du = sin¡1³u
a
´ + C (Valid for u2< a2)
a2+ u2 du = 1
atan
¡1³ u a
´ + C (Valid for all u)
up
u2¡ a2 du = 1
asec
¡1¯
¯ua¯¯ + C (Valid for u2> a2)
The Six Basic Hyperbolic Functions
sinh x = e
x
¡ e¡x
2 cosh x = e
x+ e¡x 2 tanh x = sinh x
cosh x=
ex¡ e¡x
ex+ e¡x
sinh x=
2
ex¡ e¡x
cosh x=
2
ex+ e¡x
coth x = cosh x
sinh x =
ex+ e¡x
ex¡ e¡x
Identities for Hyperbolic Functions sinh 2x = 2 sinh x cosh x cosh 2x = cosh2x + sinh2x
cosh2x = cosh 2x + 1
2 sinh2x = cosh 2x¡ 1
2 cosh2x¡ sinh2x = 1 tanh2x = 1¡ sech2x coth2x = 1 + csch2x
Derivatives of Hyperbolic Functions d
dxsinh u = cosh u
du dx d
dxcosh u = sinh u
du dx d
dxtanh u = sech
2udu dx d
dxcoth u = ¡ csch2udu
dx d
dx sechu = ¡ sechu tanh ududx d
dx cschu = ¡ cschu coth ududx
Inverse Hyperbolic Identities
sech¡1x = cosh¡1
µ1 x
¶
csch¡1x = sinh¡1
µ1 x
¶
coth¡1x = tanh¡1
µ1 x
¶
Trang 2Integrals of Hyperbolic Functions
Z
sinh u du = cosh u + C
Z
cosh u du = sinh u + C
Z
sech2u du = tanh u + C
Z
csch2u du = ¡ coth u + C
Z
sechu tanh u du = ¡ sechu + C
Z
cschu coth u du = ¡ cschu + C
Derivatives of Inverse Hyperbolic Functions
d
dxsinh
¡1u = p 1
1 + u2
du dx d
dxcosh
¡1u = p 1
u2¡ 1
du
dx (u > 1) d
dxtanh
1¡ u2
du
dx (juj < 1) d
dx csch
jujp1 + u2
du
dx (u6= 0) d
dx sech
up
1¡ u2
du
dx (0 < u < 1) d
dxcoth
1¡ u2
du
dx (juj > 1)
Integrals Involving Inverse Hyperbolic Functions
p
a2+ u2 du = sinh¡1³ u
a
´ + C (a > 0)
p
u2¡ a2 du = cosh¡1³ u
a
´ + C (u > a > 0)
a2¡ u2du =
8
>
>
>
>
1
atanh
¡1³ u a
´ + C (if u2< a2) 1
acoth
¡1³ u a
´ + C (if u2> a2)
up
a2¡ u2 du = ¡1a sech¡1³u
a
´ + C (0 < u < a)
up
a2+ u2 du = ¡1a csch¡1¯¯
¯ua¯¯
¯ + C
Expressing Inverse Hyperbolic Functions As Natural Logarithms sinh¡1x = ln(x +p
x2+ 1) (¡1 < x < 1) cosh¡1x = ln(x +p
x2¡ 1) (x¸ 1)
tanh¡1x = 1
2ln
1 + x
1¡ x (jxj < 1) sech¡1x = ln
Ã
1 +p
1¡ x2
x
! (0 < x· 1)
csch¡1x = ln
à 1
x+
p
1 + x2
jxj
! (x6= 0)
coth¡1x = 1
2ln
x + 1
x¡ 1 (jxj > 1)
Alternate Form For Integrals Involving Inverse Hyperbolic Functions
p
u2§ a2 du = ln(u +p
u2§ a2) + C
a2¡ u2 du = 1
2aln
¯
¯a + u
a¡ u
¯
¯
¯ + C
up
a2§ u2du = ¡1aln
Ã
a +p
a2§ u2
juj
! + C