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 = 1 p1¡ u2
du
dx (juj < 1) d
dxcos¡1u = ¡1 p1¡ u2
du
dx (juj < 1) d
dxtan¡1u = 1 1 + u2
du dx d
dxcsc¡1u = ¡1 jujp
u2¡ 1 du
dx (juj > 1) d
dxsec¡1u = 1 jujp
u2¡ 1 du
dx (juj > 1) d
dxcot¡1u = ¡1 1 + u2
du dx
Integrals Involving Inverse Trigonometric Functions
Z 1
pa2¡ u2 du = sin¡1³u a
´+ C (Valid for u2< a2)
Z 1
a2+ u2 du = 1
atan¡1³ u a
´+ C (Valid for all u)
Z 1
up
u2¡ a2 du = 1
asec¡1¯¯¯u a
¯¯
¯ + C (Valid for u2> a2)
The Six Basic Hyperbolic Functions
sinh x = ex¡ e¡x 2 cosh x = ex+ e¡x
2 tanh x = sinh x
cosh x= ex¡ e¡x ex+ e¡x
cschx = 1
sinh x= 2 ex¡ e¡x
sechx = 1
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 udu dx d
dxcosh u = sinh udu dx d
dxtanh u = sech2udu dx d
dxcoth u = ¡ csch2udu dx d
dx sechu = ¡ sechu tanh udu dx d
dx cschu = ¡ cschu coth udu dx
Inverse Hyperbolic Identities
sech¡1x = cosh¡1 µ1
x
¶
csch¡1x = sinh¡1 µ1
x
¶
coth¡1x = tanh¡1 µ1
x
¶
Integrals 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 = 1 p1 + u2
du dx d
dxcosh¡1u = 1 pu2¡ 1
du
dx (u > 1) d
dxtanh¡1u = 1 1¡ u2
du
dx (juj < 1) d
dx csch¡1u = ¡1 jujp
1 + u2 du
dx (u6= 0) d
dx sech¡1u = ¡1 up
1¡ u2 du
dx (0 < u < 1) d
dxcoth¡1u = 1 1¡ u2
du
dx (juj > 1)
Integrals Involving Inverse Hyperbolic Functions
Z 1
pa2+ u2 du = sinh¡1³ u a
´+ C (a > 0)
Z 1
pu2¡ a2 du = cosh¡1³ u a
´+ C (u > a > 0)
Z 1
a2¡ u2du = 8>
>>
><
>>
>>
: 1
atanh¡1³ u a
´+ C (if u2< a2)
1
acoth¡1³ u a
´+ C (if u2> a2)
Z 1
up
a2¡ u2 du = ¡1
a sech¡1³u a
´+ C (0 < u < a)
Z 1
up
a2+ u2 du = ¡1
a csch¡1¯¯
¯u a
¯¯¯ + 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
2ln1 + x
1¡ x (jxj < 1) sech¡1x = ln
Ã1 +p 1¡ x2 x
!
(0 < x· 1)
csch¡1x = ln Ã1
x+
p1 + x2 jxj
!
(x6= 0)
coth¡1x = 1
2lnx + 1
x¡ 1 (jxj > 1)
Alternate Form For Integrals Involving Inverse Hyperbolic Functions
Z 1
pu2§ a2 du = ln(u +p
u2§ a2) + C
Z 1
a2¡ u2 du = 1 2aln
¯¯
¯¯ a + u a¡ u
¯¯
¯¯ + C
Z 1
up
a2§ u2du = ¡1 aln
Ãa +p a2§ u2 juj
! + C
