Transcript 20-IntegralFUNGSI HIPERBOL INVERS.ppt

ITK-121
KALKULUS I
3 SKS
Dicky Dermawan
www.dickydermawan.890m.com
FUNGSI
HIPERBOL
& INVERS
Fungsi trigonometri
Pas dengan situasi ini:
cos 2   sin 2   1
x 2  y 2  1 (persamaan lingkaran)
FUNGSI HIPERBOL
pas dengan situasi
Fungsi hiperbol
Dimana:
x2  y2  1
cosh 2 t  sinh 2 t  1
cosh 2 t  sinh 2 t
e t  e t
cosh t 
2 t
t
sinh t 
e e
2
Sehingga:
cosh 2 t  sinh 2 t
 e t  e t

 2
2
  e t  e t
  
  2
 e 2t  2  e 2t

4




2
  e 2t  2  e 2t
  
4
 
 4
   1
 4
Fungsi hipernol lain diturunkan dari keduanya:
sinh t
e t e  t
tanh t 

cosh t e t  e t
coth t 
cosh t
sinh t
sech t =
csch t =
1
cosh t
1
sinh t
Turunan Fungsi Hiperbol
d
d  e x  ex
sinh x 

dx
dx 
2
 e x  ex
 
 cosh x
2

d
d  e x  ex
cosh x  
dx
dx  2
d
d  sinh x 
tanh x 


dx
dx  cosh x 
 e x  ex
 
 sinh x
2




d
d  cosh x 
coth x 


dx
dx  sinh x 

cosh x sinh x   sinh x cosh x 
cosh 2 x
'
'
cosh 2 x  sinh 2 x
cosh 2 x
1

2
sec
h
x
2
cosh x
sinh x cosh x   cosh x sinh x 
sinh 2 x
'
'
sinh 2 x  cosh 2 x
sinh 2 x

 csc h 2 x
d
d  1   sinh x
sec h x 
  sec h x tanh x


dx
dx  cosh x  cosh 2 x
d
d  1
 csc h x  
dx
dx  sinh
  cosh x

  csc h x coth x
x  sinh 2 x
Maka,
 cosh x dx  sinh x  C
 sinh x dx  cosh x  C
2
sec
h
x dx   tanh x  C

 csc h
2
x dx   coth x  C
 sec h x tanh x dx   sec h x  C
 csc h x x coth x dx   csc h x  C
INVERS FUNGSI HIPERBOL
Turunannya

1
cosh x  ln x  x  1
2

sinh 1 x  ln x  x 2  1
tanh 1 x 
1 1 x 
ln 

2 1 x 
coth 1 x 
1  x 1
ln 

2  x 1
1 1 x2
sec h x  ln 

x

1





1
1 
csc h 1 x  ln   1  2 
x 
x

1
x 1
1
Domain
x 1
2
x2 1
x
1
1 x2
x 1
1
1 x2
x 1
1
x 1 x2
1
x 1 x2
0  x 1
x0
Contoh:
1
3x 3  2 x 2  x  3
dx

x2 1
2
4x  1
x2 1