( 2) ( ) Hyperbolic Functions 6E. cosh d (cosh sech ) d sinh tanh. cosh. x x x x x x C. x 1 x. 2 a. x x x = + = + cosh dx sinh C 3sinh C.

Transcription

1 Hyperolic Functions 6E a (sinh+ cosh ) cosh+ sinh+ + cosh cosh (cosh sech ) sinh tanh sinh sinh sechtanh sech+ cosh cosh cosh c a a sinh cosh+ cosh sinh sinh + + ( ) + + ( ) ( ) ( ) ( ) arcosh + ( ) + + ( ) ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) arsinh + a sinh cosh (sinh ) cosh sinh + sinh tanh lncosh+ cosh c coshsinh (cosh ) (sinh ) (cosh ) + ( ) (cosh ) + Pearson Eucation Lt 08. opying permitte for purchasing institution only. This material is not copyright free.

2 5 a sinh sinh + cosh + cosh ln( + cosh ) + + tanh ( tanh )sech cosh + + (sech tanh sech ) tanh + tanh + or tanh sech + c 5cosh+ sinh (5 + tanh ) cosh 5+ lncosh+ 6 sinh cosh cosh cosh sinh+ 9 v u Usingu uv with v v u an sinh 7 a e + e e cosh e (e ) + e + + e e e sinh e (e e 5 ) e e You cannot use integration y parts. You coul use integration y parts twice. c e + e e + e coshcosh orwriteas (cosh+ cosh ) (e e e e ) e e + e e + or sinh + sinh Pearson Eucation Lt 08. opying permitte for purchasing institution only. This material is not copyright free.

3 8 sinh+ cosh (e e ) + (e + e ) e So e e 0 sinh+ cosh 0 0 e 9 a c sinh (cosh ) sinh + sinh cosh sinh cosh 5 sinh cosh + sinh + 8 cosh cosh + ( sinh ) cosh (+ sinh + sinh ) cosh (cosh+ sinh cosh+ sinh cosh ) sinh + sinh + sinh Using cosh u + sinh u 0 ln ln + cosh cosh ln [ + sinh] 0 ln+ ln + [ + ln ] 8 ( + ln6) ln ln e e ln ln ln e, e e Pearson Eucation Lt 08. opying permitte for purchasing institution only. This material is not copyright free.

4 a Let cosh u, so sinhuu sinh u u 9 9cosh u 9 sinh u u cosh u sinhu u sinhu u u+ arcosh + You nee 5sinh u, or 5sinh u, then coshuu 5 cosh u u + 5 5sinh u+ 5 5 coshu u 5 sinh u+ coshu u coshu u u+ arsinh a arsinh Using + a arsinh +. a arcosh + Using a arcosh + a a ( ) arcosh + Pearson Eucation Lt 08. opying permitte for purchasing institution only. This material is not copyright free.

5 { ( 9 )} arsinh + ( ) arsinh arsinh ( ) [ arsinh( ) ] arsinh arsinh (s.f.) [ ] 5 Reminer: The logarithmic form of an inverse hyperolic function is in the Eecel formulae ooklet. a 0 arsinh arsinh arsinh0 ( ) ln + { Using arsinh ln + + } 5 arcosh 5 arcosh arcosh ln + ln ln + ln + 6 ln ln ln { Using arcosh ln + } Using lna ln ln a Pearson Eucation Lt 08. opying permitte for purchasing institution only. This material is not copyright free. 5

6 6 With an u u u u sinh an sin cosh, sinh sinh u u + sinh u+ cosh u sinhu sinhucoshuu + coshu sinh u u (coshu )u sinhu u + sinhu coshu arsinh( ) + + arsinh( ) + sinhu an coshu + sinh u 7 With u an u, 9 u u [ arcoshu] arcosh9 arcosh 0.8 (s.f.) 9 8 Using cosh u, sinhuu cosh sinh u u u sinh u u (coshu )u sinhu u + sinhucoshu u+ arcosh + arcosh + + arcosh coshu an sinhu cosh u 9 a e + e e e e + e cosh sinh So cosh sinh e + e e e + Multiplying numerator an enominator y e. Pearson Eucation Lt 08. opying permitte for purchasing institution only. This material is not copyright free. 6

7 9 Using the sustitution u e, u e an e u + u + u artan + e e arctan + 0 With u sinh, u coshorcosh u u sinh u + 9 9( u + ) cosh sinh So 0 u 0 sinh + 9 u + arsinh( u ) a ( ) 6} { sinh arsinh sinh 0.60 ( s.f.) So ( ) 6 Let u ( ), so u. Then u 6 u arcosh + arcosh ( + ) + } { So ( + ) + Let u ( + ), so u. Then u u + arsinh( u) + arsinh( + ) + 0 Pearson Eucation Lt 08. opying permitte for purchasing institution only. This material is not copyright free. 7

8 c ( + ) + Let u ( + ), so u. Then u u + ( ) u arctan ( + ) arctan So ( 9) ( 9 ) Let u, so u. 9 Then u u ( 9 ) 9u arcosh arcosh + 7 a So ( ) ( ) Let u, so u. Then + 0 u + ( ) u arsinh( u ) + arsinh( ) + Pearson Eucation Lt 08. opying permitte for purchasing institution only. This material is not copyright free. 8

9 5 ( ) + + So + 5 ( ) ( ) Let u, so. u Then u + 5 u ( ) u arcosh + 5 arcosh ( + ) + So ( + ) + Let u ( + ), so u. Then u u + u arsinh arsinh arsinh 0.00 ( s.f.) + ( ) + So + ( ) + [ arsinh( ) ] arsinh ln{+ 5} ar sinh ln{ + + } Pearson Eucation Lt 08. opying permitte for purchasing institution only. This material is not copyright free. 9

10 ( ) + So ( ) + Let u ( ), so u. Then u + ( ) ( ) u arsinh u 0 arsinh ln{ } arsinh ln{ } ln{ + } 6 a In orer to fin the intersection point of the two curves, we solve 5cosh 7 sinh 5 ( e + e ) 7 ( e e ) 5 ( e + ) 7e ( e ) e 7e + 0. Set u e in orer to clearly see the quaratic equation u 7u+ 0 with solutions 7± 9 u 6 7± 5 u 6 i.e. u u. This gives us the values for the intersection points as ln ln ln Pearson Eucation Lt 08. opying permitte for purchasing institution only. This material is not copyright free. 0

11 6 We fin the area of the region R y calculating R ( 7 sinh ) ( 5cosh ) Area ( 7 sinh 5cosh) ln [ 7 cosh 5sinh] ( 7ln 5) ( 7ln 5) + 7ln6 0 ln ln 7 π Volume π sinh (cosh ) 0 0 π sinh π sinh π (e e ) 0 hallenge π e e 8 π (e e ). 8e Using the sustitution + sinh θ, coshθ θ + (sinh θ+ sinhθ+ ) (sinhθ + ) + sinh θ+ cosh θ So cosh θ θ cosh θ ( + ) sech θ θ tanhθ+ + + sinhθ coshθ + sinh θ + Pearson Eucation Lt 08. opying permitte for purchasing institution only. This material is not copyright free.

12 hallenge Using the sustitution u, u, a u (cosh u+ ) u So cosh ( ) cosh u u sinh u+ + 8 sinh( ) cosh ( ) So sech ( ) sech u u tanh u+ tanh( ) + Pearson Eucation Lt 08. opying permitte for purchasing institution only. This material is not copyright free.