Inverse Hyperbolic Functions

The Inverse Hyperbolic Sine, Inverse Hyperbolic Cosine & Inverse

Hyperbolic Tangent

Rfrangefdom

xxgfunctiontheofiverseThe

hxxf

RfrangeRfdom

xxgfunctiontheofiverseThe

hxxf

frangefdom

xxxgfunctiontheofiverseThe

hxxf

,)1,1(

tanh)(:

arctan)(.3

,

sinh)(:

arcsin)(.2

),0[,),1[

0;cosh)(:

arccos)(.1

The Inverse Hyperbolic Cotangent, Inverse Hyperbolic Secant & Inverse

Hyperbolic Cosecant

}0{,}0{

0csc)(:

csc)(.3

}0{,),1()1,(

0,coth)(:

coth)(.2

),0[,]1,0(

0;sec)(:

sec)(.1

RfrangeRfdom

xhxxgfunctiontheofiverseThe

hxarcxf

RfrangeUfdom

xxxgfunctiontheofiverseThe

xarcxf

frangefdom

xhxxgfunctiontheofiverseThe

hxarcxf

Derivatives of Inverse Hyperbolic Functions

1

1)(arcsin.2

),1(;1

1)(arccos.1

2

2

xh

xx

hx

)1,1(;1

1)(arctan.3

2

x

xhx

)1,0(;1

1)sec(.4

2

x

xxhxarc

),1()1,(;1

1)coth(.5

2

Ux

xxarc

0;1

1

)csc(.6

2

;1

1

;1

1{2

2

xxx

hxarcpositiveisx

xx

positiveisxxx

Proofs

),1(;1

1

sin

1

1sin

1sinh,1;cosh

arccos

:.1

2

2

xxy

y

yy

xysoandxxy

hxy

Let

1

1

cosh

1

1cosh

cosh

1coshsinh

arcsin

:.2

2

2

xyy

yy

negativeneverisybecauserootpositivethechooseWe

xysoandxy

hxy

Let

)1,1(;1

1

sec

1

1sec,)1,1(;tanh

arctan

:.3

22

22

xxhy

y

xhysoandxxy

hxy

Let

)1,0(;1

1

tanhsec

1

1tanhsec

.inttanh),1,0(

sec,

1tanh

1tanh,],1,0(,sec

sec.4

2

2

22

xxxyhy

y

yyhy

ervalthatonpositiveisysoandx

positiveishxarcybecauserootpositivethechooseWe

xy

xysoandxxhy

hxarcy

),1()1,(;1

1

1

1

csc

1

1csc

1csc,),1()1,(;coth

coth

:.5

222

2

22

Uxxxhy

y

yh

xhysoandUxxy

xarcy

Let

0;1

1

tanhsec

1

1cotcsc

coth

1coth,,0,csc

csc.6

2

;1

1

;1

1

;1

;1

22

{

{

2

2

2

2

xxx

yhyy

yyhy

y

xysoandxxy

hxarcy

positiveisxxx

negativeisxxx

positiveisxx

negativeisxx

x

hx

hxy

xhy

hxy

ey

xhy

xhy

casesfollowingtheofeachforyFind

)(arcsin.6

)ln(lnarcsin.5

)ln(arctan.4

.3

)(cosarcsin.2

)3tan5(arccos.1

5

arcsin

25

4

Example (1)

13tan25

sec15

13tan25

3sec5

)3tan5(arccos

2

2

2

2

x

x

x

xy

xhxy

Example (2)

1cos

)(cosarcsin2sin5

1cos

)sin(cos2)(cosarcsin5

)(cosarcsin

4

24

4

24

25

x

xhx

x

xxxhy

xhy

Example (3)

1,1;

1

4

)ln(arctan

1

)ln(arctan

5

10

4

5

5

xlyequivalentorx

x

x

xy

hxy

Integrals Involving Inverse Hyperbolic Functions

}0{;csc1

.6

)1,0(;sec1

.5

),1()1,(;coth1

.4

)1,1(;arctan1

.3

),1(;arccos1

.2

arcsin1

.1

2

2

2

2

2

2

Rxcxharcxx

dx

xcxharcxx

dx

Uxcxarcx

dx

xchxx

dx

xchxx

dx

chxx

dx

Example (1)

ch

dxx

dxx

x

dxx

x

dxx

x

x

x

)(arcsin

1)(

5

1)(

149

23

151

223

423

21

51

32

223

4

21

1049

4

21

10

4

5

5

5

Example (2)

),(

),1()(arccos

1)(

5

1)(

149

532

23

23

151

223

423

21

51

32

223

4

21

1049

4

21

10

4

55

5

5

xlyequivalentor

ch

dxx

dxx

x

dxx

x

dxx

xx

x

x

Example (3)

)3

2ln(,1;

)(arctan6

1

)(13

2

4

1

)(14

1

14

1

94

23

23

223

23

223

249

2

xlyequivalentorewhere

ceh

e

dxe

e

dxe

e

dxe

e

dxe

x

x

x

x

x

x

x

x

x

x

Example (4)

{),(;)(arctan

),(),(;)coth(

223

523

41

51

32

223

5

41

1049

5

41

10

5

5325

32

2

53301

5325

32

2

53301

5

5

)(1

5

)(1

194

xch

Uxcarc

x

x

x

x

dxx

dxx

x

dxx

x

dxx

Logarithmic Expressions of inverse hyperbolic Functions

1;)1ln(arccos.1 2 xxxhx

)1ln(arcsin.2 2 xxhx

)1,1(;)1

1ln(2

1arctan.3

xx

xhx

),1()1,(;)1

1ln(2

1coth.5

Uxx

xxarc

]1,0(;)11

ln(sec.42

xx

xhxarc

Proofs

1;)1ln(

1

12

442

012

21

2

2

cosh

01;arccos

:.1

2

2

22

2

2

xxxy

xxechooseWe

xxxx

e

xee

xee

xee

xee

xy

yandxhxy

Let

y

y

yy

yy

yy

yy

)1ln(

1,

12

442

012

21

2

2

sinh

arcsin

:.2

2

2

22

2

2

xxy

xxethenpositiveallwaysiseSince

xxxx

e

xee

xee

xee

xee

xy

hxy

Let

yy

y

yy

yy

yy

yy

)1,1(;)1

1ln(2

1

)1

1ln(2

1

1

1)1(

)1(1

1

1

tanh

)1,1(;arctan

:.3

2

2

222

2

2

xx

xy

x

xy

x

xe

xex

xxeexe

xe

e

xee

ee

xy

xhxy

Let

y

y

yyy

y

y

yy

yy

]1,0(;)11

ln(

)11

ln(

11,

11

2

442

02

)1(2

1

2

2

sec

]1,0(;sec

:.4

2

2

2

22

2

22

2

xx

xy

x

xy

x

xeChoose

x

x

x

xe

xexe

xxeexe

xe

e

xee

xhy

xhxarcy

Let

y

y

yy

yyy

y

y

yy

),1()1,(;)1

1ln(2

1

)1

1ln(2

1

1

)1(1

)1(1

1

1

coth

),1()1,(;coth

:.5

2

2

222

2

2

Uxx

xy

x

xy

x

xe

exx

xxeexe

xe

e

xee

ee

xy

Uxxarcy

Let

y

y

yyy

y

y

yy

yy

Deducing the derivative formulas for inverse hyperbolic functions using their logarithmic

expressions Question:

Use their logarithmic expressions for the inverse hyperbolic sine, the inverse hyperbolic cosine and the inverse hyperbolic tangent to deduce their derivative formulas

1

11

1

1

1

)1

1(1

1

]2)1(1[1

1

])1(ln[

)1ln(arcsin

)1(

2

2

2

2

22

221

2

2

2

21

21

x

x

xx

xx

x

x

xx

xxxx

y

xx

xxhxy

1;1

11

1

1

1

)1

1(1

1

]2)1(1[1

1

])1(ln[

1;)1ln(arccos

)2(

2

2

2

2

22

221

2

2

2

21

21

xx

x

xx

xx

x

x

xx

xxxx

y

xx

xxxhxy

)1,1(;1

11

1.

1

1

)1(

2.

1

1

2

1

)1(

)1)(1()1(.

1

1

2

1

)1,1(;)1

1ln(2

1arctan

)3(

2

2

2

xx

xx

xx

x

x

xx

x

xy

xx

xhxy

Values

)1,1(2,)2(arctan.2

),1[2,)2(arccos.1

fdombecauseexistnotdoesh

fdombecauseexistnotdoesh

)2

3ln(2

1)

1

1ln(2

1)5

1(arctan.5

)103ln()133ln()3(arcsin.4

)83ln()133ln()3(arccos.3

51

51

2

2

h

h

h