Integration of Hyperbolic Functions

TOPIC

OBJECTIVES

• identify the different hyperbolic functions;

• find the integral of given hyperbolic functions; •determine the difference between the integrals of hyperbolic functions; and

•evaluate integrals involving hyperbolic functions.

2sinh.1

xx eex

2cosh.2

xx eex

xx

xx

ee

ee

x

xx

cosh

sinhtanh.3

xx

xx

ee

ee

xx

tanh

1coth.4

xx eexhx

2

cosh

1sec.5

xx eexhx

2

sinh

1csc.6

Definitions:

Differentiation Formulas

uduud coshsinh.1

uduud sinhcosh.2

uduhud 2sectanh.3

uduhud 2csccoth.4

uduhuhud tanhsecsec.5

uducothhucschucscd.6

.

Note: The hyperbolic functions are defined in terms of the exponential functions. Its differentials may also be found by differentiating its equivalent exponential form. Similarly, the integrals of the hyperbolic functions can be derived by integrating the exponential form equivalent.

Integration Formulas

Cuudu coshsinh.1

Cuudu sinhcosh.2

Cuuduh tanhsec.3 2

Cuduh cothcsc.4 2

Chuuduhu sectanhsec.5

Chucscuducothhucsc.6

u

uduudu

cosh

sinhtanh.7

cu coshln

u

uduudu

sinh

coshcoth.8

cu sinhln

Cehudu u1tan2sec.9

Cu sinhtan 1

or 1

1lncsc.10 Ce

ehudu

u

u

C1ucosh

1ucoshln

2

1

7.

Hyperbolic Functions Trigonometric Functions

1xsinhxcosh 22

xhsecxtanh1 22

xhcsc1xcoth 22

ysinhxcoshycoshxsinh)yxsinh(

ysinhxsinhycoshxcoshyxcosh

ytanhxtanh1

ytanhxtanhyxtanh

ytanxtan1

ytanxtanyxtan

ysinxsinycosxcosyxcos

ysinxcosycosxsinyxsin

xx 22 sectan1

1sincos 22 xx

xcsc1xcot 22

Identities: Hyperbolic Functions vs. Trigonometric Functions

Hyperbolic Functions Trigonometric Functions

Identities: Hyperbolic Functions vs. Trigonometric Functions

sinh 2x = 2 sinh x cosh x

2/1x2coshxsinh2

2/1x2coshxcosh2 xexsinhxcosh

xexsinhxcosh

2/x2cos1xcos2

2/x2cos1xsin 2

cos 2x = cos2x – sin2x

sin 2x = 2sinx cosx

cosh 2x = cosh2x +sinh2x

Hyperbolic Functions Trigonometric Functions

uducoshusinhd

udusinhucoshd

uduhsecutanhd 2

uduhcscucothd 2

udutanhhusechusecd

uducosusind

udusinucosd udusecutand 2

uducscucotd 2

udutanusecusecd

uducotucscucscd uducothhucschucscd

Differential Formulas

Note: Integration of the hyperbolic is exactly the same as the integration of trigonometric functions, they only differ in signs.

Example: Evaluate the following integrals:

dxx31sinh.1

dxecoshe.2 x2x2

dyycosh

a.3

2

ytanh

3ln

0

2tdthsec.4

xdxcoshxsinh.5 23

dxxcothxhcscx

1.6 24

CLASSWORK

xdx4coshx4sinh.1 dxx2hcscx.2 22

dxelnx2cosh.1 x2sinh

dxhxcscx3.2 2

dxx31hsec.3 4

dx3

xsinh.4 5

dxx3tanhx3hsec.5 36

EXERCISES