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
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