BAB 6. Integral - BAB 6 INTEGRAL

BAB 6. Integral

1 Integral

  Pengertian

Rumus dasar

Sifat Teknik pengintegralan

1 Integral

  

Pengertian

Rumus dasar Sifat

  Turunan Turunan ′

  Y Y Y ” Integral Integral

  Figure: Anti turunan Secara umum _{R R}

  ′ dy ′ ′

  Turunan Turunan ′

  Y Y Y ” Integral Integral

  Figure: Anti turunan Secara umum _{R R}

  ′ dy ′ ′

1 Integral

  

Pengertian

Rumus dasar Sifat

  x

• c (dengan n
• c _R

  2 x dx

  3 ^R sec

  2 ^R sinx dx = − cosx + c

  1 ^R cosx dx = sinx + c

  2. Integral fungsi trigonometri

  lnx

  =

  dx

  1

  −

  = ^R

  1 x

  6= − 1) dan ^R

  1

  n +

  x

• c

  1

  = a n +

  dx

  n

  ax

  1. Integral bentuk aljabar _R

  = tanx

  x

• c (dengan n
• c _R

  2 x dx

  3 ^R sec

  2 ^R sinx dx = − cosx + c

  1 ^R cosx dx = sinx + c

  2. Integral fungsi trigonometri

  lnx

  =

  dx

  1

  −

  = ^R

  1 x

  6= − 1) dan ^R

  1

  n +

  x

• c

  1

  = a n +

  dx

  n

  ax

  1. Integral bentuk aljabar _R

  = tanx

  x

• c (dengan n
• c _R

  2 x dx

  3 ^R sec

  2 ^R sinx dx = − cosx + c

  1 ^R cosx dx = sinx + c

  2. Integral fungsi trigonometri

  lnx

  =

  dx

  1

  −

  = ^R

  1 x

  6= − 1) dan ^R

  1

  n +

  x

• c

  1

  = a n +

  dx

  n

  ax

  1. Integral bentuk aljabar _R

  = tanx

  x

• c (dengan n

  2 x dx

  3 ^R sec

  2 ^R sinx dx = − cosx + c

  cosx dx = sinx + c

  1 ^R

  2. Integral fungsi trigonometri

  lnx

  =

  dx

  1

  −

  = ^R

  1 x

  6= − 1) dan ^R

  1

  n +

  x

• c

  1

  = a n +

  dx

  n

  ax

  1. Integral bentuk aljabar _R

  = tanx

• c _R

  x

• c (dengan n

  2 x dx

  3 ^R sec

  sinx dx = − cosx + c

  2 ^R

  cosx dx = sinx + c

  1 ^R

  2. Integral fungsi trigonometri

  lnx

  =

  dx

  1

  −

  = ^R

  1 x

  6= − 1) dan ^R

  1

  n +

  x

• c

  1

  = a n +

  dx

  n

  ax

  1. Integral bentuk aljabar _R

  = tanx

• c _R

  tanx

  1

  =

  x dx

  2

  sec

  3 ^R

  sinx dx = − cosx + c

  2 ^R

  cosx dx = sinx + c

  1 ^R

  2. Integral fungsi trigonometri

  lnx

  =

  dx

  −

  x

  = ^R

  1 x

  6= − 1) dan ^R

  1

  n +

  x

• c

  1

  = a n +

  dx

  n

  ax

  1. Integral bentuk aljabar _R

• c (dengan n
• c _R

  tanx

  1

  =

  x dx

  2

  sec

  3 ^R

  sinx dx = − cosx + c

  2 ^R

  cosx dx = sinx + c

  1 ^R

  2. Integral fungsi trigonometri

  lnx

  =

  dx

  −

  x

  = ^R

  1 x

  6= − 1) dan ^R

  1

  n +

  x

• c

  1

  = a n +

  dx

  n

  ax

  1. Integral bentuk aljabar _R

• c (dengan n
• c _R

  tanx

  1

  =

  x dx

  2

  sec

  3 ^R

  sinx dx = − cosx + c

  2 ^R

  cosx dx = sinx + c

  1 ^R

  2. Integral fungsi trigonometri

  lnx

  =

  dx

  −

  x

  = ^R

  1 x

  6= − 1) dan ^R

  1

  n +

  x

• c

  1

  = a n +

  dx

  n

  ax

  1. Integral bentuk aljabar _R

• c (dengan n
• c _R

  tanx

  1

  =

  x dx

  2

  sec

  3 ^R

  sinx dx = − cosx + c

  2 ^R

  cosx dx = sinx + c

  1 ^R

  2. Integral fungsi trigonometri

  lnx

  =

  dx

  −

  x

  = ^R

  1 x

  6= − 1) dan ^R

  1

  n +

  x

• c

  1

  = a n +

  dx

  n

  ax

  1. Integral bentuk aljabar _R

• c (dengan n
• c _R

  tanx

  1

  =

  x dx

  2

  sec

  3 ^R

  sinx dx = − cosx + c

  2 ^R

  cosx dx = sinx + c

  1 ^R

  2. Integral fungsi trigonometri

  lnx

  =

  dx

  −

  x

  = ^R

  1 x

  6= − 1) dan ^R

  1

  n +

  x

• c

  1

  = a n +

  dx

  n

  ax

  1. Integral bentuk aljabar _R

• c (dengan n
• c _R

  tanx

  1

  =

  x dx

  2

  sec

  3 ^R

  sinx dx = − cosx + c

  2 ^R

  cosx dx = sinx + c

  1 ^R

  2. Integral fungsi trigonometri

  lnx

  =

  dx

  −

  x

  = ^R

  1 x

  6= − 1) dan ^R

  1

  n +

  x

• c

  1

  = a n +

  dx

  n

  ax

  1. Integral bentuk aljabar _R

• c (dengan n
• c _R

  tanx

  1

  =

  x dx

  2

  sec

  3 ^R

  sinx dx = − cosx + c

  2 ^R

  cosx dx = sinx + c

  1 ^R

  2. Integral fungsi trigonometri

  lnx

  =

  dx

  −

  x

  = ^R

  1 x

  6= − 1) dan ^R

  1

  n +

  x

• c

  1

  = a n +

  dx

  n

  ax

  1. Integral bentuk aljabar _R

• c (dengan n
• c _R

Contoh

Tentukan integral dari soal berikut

_R

  2

  

( 6x − 4x ) dx =

1 R

  3

  2

  2

( 10x 30x − 16x

5 ) dx =

Contoh

Tentukan integral dari soal berikut

_R

  2

  

( 6x − 4x ) dx =

1 R

  3

  2

  2

( 10x 30x − 16x

5 ) dx =

Contoh

Tentukan integral dari soal berikut

_R

  2

  

( 6x − 4x ) dx =

1 R

  3

  2

  2

( 10x 30x − 16x

5 ) dx =

1 Integral

  

Pengertian Rumus dasar Sifat Sifat

  1 ^R { f ( x ) ± g ( x ) dx = ^R f ( x ) dx + ^R g ( x ) dx

  2 ^R k f ( x ) dx = k

^R

f ( x ) dx

  3 ^R b

  − a f ( x ) dx = − ^R a b f ( x ) dx

  4 ^R b

  − a f ( x ) dx +

^R

c b f ( x ) dx = ^R c

  − a f ( x ) dx Sifat

  1 ^R { f ( x ) ± g ( x ) dx = ^R

  f ( x ) dx + ^R g ( x ) dx

  2 ^R k f ( x ) dx = k

^R

f ( x ) dx

  3 ^R b

  − a f ( x ) dx = − ^R a b f ( x ) dx

  4 ^R b

  − a f ( x ) dx +

^R

c b f ( x ) dx = ^R c

  − a f ( x ) dx Sifat

  1 ^R { f ( x ) ± g ( x ) dx = ^R

  f ( x ) dx + ^R g ( x ) dx

  2 ^R

  k f ( x ) dx = k ^R f ( x ) dx

  3 ^R b

  − a f ( x ) dx = − ^R a b f ( x ) dx

  4 ^R b

  − a f ( x ) dx +

^R

c b f ( x ) dx = ^R c

  − a f ( x ) dx Sifat

  1 ^R { f ( x ) ± g ( x ) dx = ^R

  f ( x ) dx + ^R g ( x ) dx

  2 ^R

  k f ( x ) dx = k ^R f ( x ) dx

  3 ^R b

  − a

  f ( x ) dx = − ^R

  a b

  f ( x ) dx

  4 ^R b

  − a f ( x ) dx +

^R

c b f ( x ) dx = ^R c

  − a f ( x ) dx Sifat

  1 ^R { f ( x ) ± g ( x ) dx = ^R

  f ( x ) dx + ^R g ( x ) dx

  2 ^R

  k f ( x ) dx = k ^R f ( x ) dx

  3 ^R b

  − a

  f ( x ) dx = − ^R

  a b

  f ( x ) dx

  4 ^R b

  − a

  f ( x ) dx + ^R

  c b

  f ( x ) dx = ^R

  c − a

  f ( x ) dx Sifat

  1 ^R { f ( x ) ± g ( x ) dx = ^R

  f ( x ) dx + ^R g ( x ) dx

  2 ^R

  k f ( x ) dx = k ^R f ( x ) dx

  3 ^R b

  − a

  f ( x ) dx = − ^R

  a b

  f ( x ) dx

  4 ^R b

  − a

  f ( x ) dx + ^R

  c b

  f ( x ) dx = ^R

  c − a

  f ( x ) dx Sifat

  1 ^R { f ( x ) ± g ( x ) dx = ^R

  f ( x ) dx + ^R g ( x ) dx

  2 ^R

  k f ( x ) dx = k ^R f ( x ) dx

  3 ^R b

  − a

  f ( x ) dx = − ^R

  a b

  f ( x ) dx

  4 ^R b

  − a

  f ( x ) dx + ^R

  c b

  f ( x ) dx = ^R

  c − a

  f ( x ) dx Sifat

  1 ^R { f ( x ) ± g ( x ) dx = ^R

  f ( x ) dx + ^R g ( x ) dx

  2 ^R

  k f ( x ) dx = k ^R f ( x ) dx

  3 ^R b

  − a

  f ( x ) dx = − ^R

  a b

  f ( x ) dx

  4 ^R b

  − a

  f ( x ) dx + ^R

  c b

  f ( x ) dx = ^R

  c − a

  f ( x ) dx

1 Integral

  

Pengertian Rumus dasar Sifat a. Cara biasa _R

  1 x 3x 1 dx _R ( − ) =

  2 x 1 3x 5 dx

  ( + )( − ) =

b. Cara subtitusi

• ^R

  n

  

1

1 n

  1 Bentuk linier ax b dx ax b c

( ) = ) + +

+

  

a n

• . .( _R

  1

  4 a. Cara biasa _R

  1

  x 3x

  ( − ) = 1 dx _R

  2 x 1 3x 5 dx

  ( + )( − ) =

b. Cara subtitusi

• ^R

  n

  

1

1 n

  1 Bentuk linier ax b dx ax b c

( ) = ) + +

+

  

a n

• . .( _R

  1

  4 a. Cara biasa _R

  1

  x 3x

  ( − ) = 1 dx _R

  2

  x

  1 3x 5 dx ( + )( − ) =

b. Cara subtitusi

• ^R

  n

  

1

1 n

  1 Bentuk linier ax b dx ax b c

( ) = ) + +

+

  

a n

• . .( _R

  1

  4 a. Cara biasa _R

  1

  x 3x

  ( − ) = 1 dx _R

  2

  x

  1 3x 5 dx ( + )( − ) =

b. Cara subtitusi

• ^R

  n

  

1

1 n

  1 Bentuk linier

  ax b dx ax b c

  

( ) = ) + +

+

  

a n

• . .( _R

  1

  4 a. Cara biasa _R

  1

  x 3x

  ( − ) = 1 dx _R

  2

  x

  1 3x 5 dx ( + )( − ) =

b. Cara subtitusi

• ^R

  n

  

1

1 n

  1 Bentuk linier

  ax b dx ax b c

  

( ) = ) + +

+

  

a n

• . .( _R

  1

  4 a. Cara biasa _R

  1

  x 3x

  ( − ) = 1 dx _R

  2

  x

  1 3x 5 dx ( + )( − ) =

b. Cara subtitusi

• ^R

  n

  

1

1 n

  1 Bentuk linier

  ax b dx ax b c

  

( ) = ) + +

+

  

a n

• . .( _R

  1

  4

c. Integral parsial (pertemuan minggu depan)

  Bentuk umum integral parsial: ^R

  u dv

  =

  uv

  − ^R

  v du. Contoh: _R

  3x .

  cos2x dx =

1 Integral

  

Pengertian

Rumus dasar Sifat

  L1 L2 x = a x = b x = c

  L1 L2 x = a x = b x = c

  L1 L2 x = a x = b x = c

  y = c L1

  L2 y = a y = b

  y = c L1

  L2 y = a y = b

  y = c L1

  L2 y = a y = b

  = g ( x ) Y2

  = f ( x ) Y1 x = a x = b

  Figure: Menghitung luas daerah

  = g ( x ) Y2

  = f ( x ) Y1 x = a x = b

  Figure: Menghitung luas daerah

  Y = f ( x ) x = a x = b

  Y = f ( x ) x = a x = b

  X y = b y = a X = f ( y )

  Y

  X y = b y = a X = f ( y )

  Y Contoh

  2

  1 Hitunglah luas daerah yang dibatasi oleh parabola y x sumbu x, = x 1 dan x 3 !

  = =

  2

  2 Hitunglah luas daerah yang dibatasi oleh parabola y x dan garis = Contoh

  2

  1 Hitunglah luas daerah yang dibatasi oleh parabola y x sumbu x, =

  x 1 dan x 3 !

  = =

  2

  2 Hitunglah luas daerah yang dibatasi oleh parabola y x dan garis = Contoh

  2

  1 Hitunglah luas daerah yang dibatasi oleh parabola y x sumbu x, =

  x 1 dan x 3 !

  = =

  2

  2 Hitunglah luas daerah yang dibatasi oleh parabola y x dan garis =