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integral tak tentu

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integral tak tentu
08/11/2015
INTEGRAL TAK TENTU
(ITT)
Anita T. Kurniawati
LOGO
DEFINISI & SIFAT ITT
1
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Beberapa rumus ITT
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Contoh:
2
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Integrasi Parsial
 u dv  uv   v du
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Contoh:
3
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Penerapan 1/D pada ITT
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4
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1
1
1
1
UV  U . V  DU . 2 V  D 2U . 3 V  ...
D
D
D
D
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LOGO
Contoh 1:


I   e 3 x x 2  3x  2 dx 



1 3x 2
1
e x  3x  2  e 3 x
x 2  3x  2
D
D3

1
1
1  D D2
 e3x . .
x 2  3x  2  e 3 x 1  
 ... x 2  3x  2
3  D
3 
3
9

1  
3









1 
1
1 
1 
11
29 
 e 3 x  x 2  3 x  2  2 x  3  .2   C  e 3 x  x 2  x    C
3 
3
9 
3 
3
9 
I


1 3x
e 9 x 2  33 x  29  C
27
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5
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Contoh 2:
x e
3 2 x
dx 
1 3 2 x
1
1
1
1
x e  x 3 . e  2 x  Dx 3 . 2 e  2 x  D 2 x 3 3 e  2 x  D 3 x 3 4 e  2 x
D
D
D
D
D
1
3
3
3
  x 3 e  2 x  x 2 e  2 x  xe 2 x  e  2 x  C
2
4
4
8


1
  e 2 x 4 x 3  6 x 2  6 x  3  C
8
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Rumus Reduksi
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6
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Integrasi Fungsi Pecah Rasional
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7
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Contoh 1:
I 
23  2 x
23  2 x
dx  
dx
2
2 x  1x  5
2x  9x  5
I 
4
3
dx  
dx  2 ln 2 x  1  3 ln x  5  C
2x  1
x5
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LOGO
8
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Contoh 2:
I 
3x  1
3x  1
dx  
dx
x  2x  1
x  12
2
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Contoh 3:
I 
x 1
dx
x  1x 2  1
9
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Contoh 4:
I 
2x 2  3
x
2

1
2
dx
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10
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Integrasi Fungsi Trigonometri
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Bentuk:  sin x cos x dx;  cos x cos x dx;  sin x sin x dx
1 1

1.
 sin 6 x cos 2 xdx   2 sin 8 x  sin 4 xdx  2  8 cos 8 x  4 cos 4 x  C
2.
 cos 6 x cos 3xdx   2 cos 9 x  cos 3xdx  2  9 sin 9 x  3 sin 3x  C
3.
 cos
4.
 sin 3x sin 2 xdx   2 cos x  cos 5xdx  2 sin x  5 sin 5 x  C
1
1
2
2 x dx  
1
1 1

1
1
1  cos 4 xdx  1  x  1 sin 4 x  C
2
2
4

1
1
1

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LOGO
11
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Bentuk:  Rsin x, cos x  dx;
R  fs rasional
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LOGO
Contoh:
dx
 5  4 cos x  

2 dt
dt
2
t
 2
 arctg  C
2
2
3
3
9t
1 t  1 t

5  4
2 
1 t 
1
2
.
2
1 x
arctg  tg   C
3
3 2
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12
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Bentuk:  Rsin x, cos x  dx   R sin x, cos x  dx

dt
1
t 1
 1
 
 dt  ln
 C  ln 1  cot g x  C
t t  1   t  1 t 
t
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Bentuk:  Rtg x  dx
13
08/11/2015
t 
1
1 t
 1
I  

dt   ln 1  t  ln 1  t 2  C   ln
C
2 
2
1  t 1  t 
1 t2
  ln
1  tg x
 C   ln 1  tg x  cos x  C   ln cos x  sin x  C
sec x
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Integrasi dgn substitusi trigonometri
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14
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1.

1
a2  x2
dx  
1
x
.a cos t dt   dt  t  C  arcsin  C
a cos t
a
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LOGO
15
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