Derivatives, Integrals, and Properties Of Inverse Trigonometric

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Derivatives, Integrals, and Properties Of Inverse Trigonometric
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
Identities for Hyperbolic Functions
d
sin¡1 u
dx
sinh 2x = 2 sinh x cosh x
d
cos¡1 u
dx
=
p
=
p
1
du
2
1 ¡ u dx
¡1 du
1 ¡ u2 dx
d
tan¡1 u =
dx
1 du
1 + u2 dx
d
csc¡1 u
dx
=
d
sec¡1 u
dx
¡1
du
p
juj u2 ¡ 1 dx
=
d
cot¡1 u
dx
=
1
du
p
juj u2 ¡ 1 dx
(juj < 1)
cosh 2x = cosh2 x + sinh2 x
(juj < 1)
Z
Z
p
a2
a2
1
du
¡ u2
(juj > 1)
¡1 du
1 + u2 dx
1
du
+ u2
= sin¡1
=
1
p
du =
2
u u ¡ a2
³u´
+C
a
³u´
1
tan¡1
+C
a
a
¯u¯
1
¯ ¯
sec¡1 ¯ ¯ + C
a
a
sinh x
=
¡x
x
¡x
e ¡e
2
sinh2 x =
cosh 2x ¡ 1
2
cosh x
=
e +e
2
tanh x
=
sinh x
ex ¡ e¡x
= x
cosh x
e + e¡x
cschx
=
1
2
= x
sinh x
e ¡ e¡x
sechx
=
1
2
= x
cosh x
e + e¡x
coth x
=
cosh x
ex + e¡x
= x
sinh x
e ¡ e¡x
cosh2 x ¡ sinh2 x = 1
tanh2 x = 1 ¡ sech2 x
coth2 x = 1 + csch2 x
Derivatives of Hyperbolic Functions
(Valid for u2 < a2 )
d
sinh u
dx
= cosh u
(Valid for all u)
d
cosh u
dx
= sinh u
(Valid for u2 > a2 )
d
tanh u =
dx
The Six Basic Hyperbolic Functions
x
cosh 2x + 1
2
(juj > 1)
Integrals Involving Inverse Trigonometric Functions
Z
cosh2 x =
du
dx
du
dx
sech2 u
du
dx
du
dx
d
coth u
dx
= ¡ csch2 u
d
sechu
dx
= ¡ sechu tanh u
d
cschu
dx
= ¡ cschu coth u
du
dx
du
dx
Inverse Hyperbolic Identities
µ ¶
1
sech x = cosh
x
µ ¶
1
¡1
¡1
csch x = sinh
x
µ ¶
1
coth¡1 x = tanh¡1
x
¡1
¡1
Integrals Involving Inverse Hyperbolic Functions
Integrals of Hyperbolic Functions
Z
Z
Z
Z
Z
Z
sinh u du
Z
= cosh u + C
cosh u du
Z
= sinh u + C
2
sech u du
= tanh u + C
csch2 u du
= ¡ coth u + C
1
du
a2 + u2
= sinh¡1
p
1
du
u2 ¡ a2
= cosh¡1
= ¡ cschu + C
d
cosh¡1 u
dx
=
=
d
tanh¡1 u =
dx
d
csch¡1 u
dx
=
d
sech¡1 u
dx
=
d
coth¡1 u
dx
=
p
1
du
1 + u2 dx
1
du
p
2
u ¡ 1 dx
(u > 1)
1 du
1 ¡ u2 dx
(juj < 1)
¡1
du
p
2
juj 1 + u dx
(u 6= 0)
¡1
du
p
2
u 1 ¡ u dx
1 du
1 ¡ u2 dx
p
1
du
2
u § a2
p
x2 + 1)
cosh¡1 x
= ln(x +
p
x2 ¡ 1)
tanh¡1 x
=
1
du
2
a ¡ u2
+C
(a > 0)
+C
(u > a > 0)
sech¡1 x
csch¡1 x
coth¡1 x
= ln(u +
p
(¡1 < x < 1)
(x ¸ 1)
1 1+x
ln
(jxj < 1)
2 1¡x
Ã
!
p
1 + 1 ¡ x2
= ln
(0 < x · 1)
x
Ã
!
p
1
1 + x2
= ln
+
(x 6= 0)
x
jxj
=
1 x+1
ln
2 x¡1
u2 § a2 ) + C
¯
¯
¯a + u¯
1
¯
¯+C
=
ln
2a ¯ a ¡ u ¯
Ã
!
p
Z
1
1
a + a2 § u2
p
du = ¡ ln
+C
a
juj
u a2 § u2
Z
a
= ln(x +
Alternate Form For Integrals Involving
Inverse Hyperbolic Functions
Z
³u´
sinh¡1 x
(0 < u < 1)
(juj > 1)
a
Expressing Inverse Hyperbolic
Functions As Natural Logarithms
Derivatives of Inverse Hyperbolic Functions
d
sinh¡1 u
dx
³u´
8
³ ´
1
¡1 u
>
>
tanh
+ C (if u2 < a2 )
>
>
Z
a
a
<
1
³ ´
du
=
2
1
¡1 u
>
a ¡ u2
>
coth
+ C (if u2 > a2 )
>
>
a
: a
Z
³ ´
1
1
¡1 u
p
du
=
¡
sech
+ C (0 < u < a)
a
a
u a2 ¡ u2
Z
¯u¯
1
1
¯ ¯
p
du = ¡ csch¡1 ¯ ¯ + C
2
2
a
a
u a +u
sechu tanh u du = ¡ sechu + C
cschu coth u du
p
(jxj > 1)

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