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Integral Fungsi Trigonometri

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Ana Sugiyarti
Ana Sugiyarti

Materi Matematika Peminatan SMA

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Matematika SMA : Integral Page 1 = ( + ) + ( ) = ( + ) ( ) = ( + ) + ( ) = ( ) ( + ) INTEGRAL FUNGSI TRIGONOMETRI Rumus Dasar: sin( + ) = 1 cos( + ) + cos( + ) = 1 sin( + ) + sec2( + ) = 1 tan( + ) + cosec2( + ) = 1 cotan( + ) + sec( + )tan( + ) = 1 sec( + ) + cosec( + ) cotan( + ) = 1 cosec( + ) + Contoh 1. sin(3 + 2) = 1 3 cos(3 + 2) + 2. sec4 tan4 = 1 4 sec 4 + 3. 2cos3 sin = (sin 4 sin 2) = 1 4 cos4 ( 1 2 cos 2) + = 1 2 cos2 1 4 cos4 + 4. sin5 cos Dengan menggunakan teknik substitusi, maka Misal = sin = cos sin5 cos = 5 = 1 6 6 + = 1 6 sin6 + 5. sin 6 0 = [ 1 6 cos6] 0 = ( 1 6 cos6) ( 1 6 cos0) = ( 1 6 1) ( 1 6 1) = 0
Matematika SMA : Integral Page 2 Jika genap, maka : Jika ganjil, maka : Contoh sin3 Jawab Karena = 3, maka sin2 = 1 cos2 dan = cos sin3 = sin sin2 = sin (1 cos2 ) = (sin sin cos2 ) = cos sin cos2 Jika = cos maka = sin sin3 = cos 2 = cos + 1 3 3 + = cos + 1 3 cos3 + Contoh cos4 Jawab Karena = 4, maka cos2 = 1 2 (1 + cos2) cos4 = (cos2 )2 = ( 1 2 (1 + cos2)) 2 = 1 4 (1 + 2 cos2 + cos2 2) sin , cos sin2 = 1 2 (1 cos2) cos2 = 1 2 (1 + cos 2) sin2 = 1 cos2 , = cos cos2 = 1 sin2 , = sin
Matematika SMA : Integral Page 3 = 1 4 (1 + 2cos2 + 1 2 (1 + cos4)) = 1 4 (1 + 2 cos2 + 1 2 + 1 2 cos 4) = 1 4 ( 3 2 + 2 cos2 + 1 2 cos4) = 1 4 ( 3 2 + 2 1 2 sin 2 + 1 2 1 4 cos4) + = 3 8 + 1 4 sin 2 + 1 32 cos4 + Jika , keduanya genap genap, maka : Jika ganjil, maka : Jika ganjil, maka : Contoh sin4 cos5 Jawab Karena = 4 dan = 5 maka gunakan cos2 = 1 sin2 dan = sin = cos sin4 cos5 = sin4 cos4 cos = sin4 (cos2 )2 cos = sin4 (1 sin2 )2 cos = 4(1 2)2 = 4(1 22 + 4) = ( 4 26 + 8) sin cos sin2 = 1 2 (1 cos2) cos2 = 1 2 (1 + cos 2) sin2 = 1 cos2 , = cos cos2 = 1 sin2 , = sin
Matematika SMA : Integral Page 4 = 1 5 5 2 1 7 7 + 1 9 9 + = 1 5 sin5 2 7 sin7 + 1 9 sin9 + Contoh sin3 cos2 Jawab Karena = 3 dan = 2 maka gunakan sin2 = 1 cos2 dan = cos = sin sin3 cos2 = sin2 cos2 sin = (1 cos2 )cos2 sin = (1 2) 2() = (2 + 4) = 1 3 3 + 1 5 5 + = 1 3 sin3 + 1 5 sin5 + Contoh sin4 cos4 Jawab Karena = 4 dan = 4 maka gunakan cos2 = 1 2 (1 + cos2),sin2 = 1 2 (1 cos2) sin4 cos4 = (sin2 cos2 )2 = ( 1 2 (1 cos2) 1 2 (1 + cos2)) 2 = ( 1 4 (1 cos2 2)) 2 = ( 1 4 sin2 2) 2 = 1 16 ( 1 2 (1 cos 4)) 2 = 1 64 (1 2cos4 + cos2 4)
Matematika SMA : Integral Page 5 = 1 64 (1 2 cos4 + 1 2 (1 + cos8)) = 1 64 (1 2 cos4 + 1 2 + 1 2 cos8) = 1 64 ( 3 2 2 cos4 + 1 2 cos8) = 1 64 ( 3 2 2 1 4 sin 4 + 1 2 1 8 sin 8) + = 3 128 1 128 sin 4 + 1 1.024 sin 8 + LATIHAN YUKS! 1. cos2 2. cos3 3. sin5 4. sin7 3 cos2 3 5. sin4 2 cos4 2

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Integral Fungsi Trigonometri

  • 1. Matematika SMA : Integral Page 1 = ( + ) + ( ) = ( + ) ( ) = ( + ) + ( ) = ( ) ( + ) INTEGRAL FUNGSI TRIGONOMETRI Rumus Dasar: sin( + ) = 1 cos( + ) + cos( + ) = 1 sin( + ) + sec2( + ) = 1 tan( + ) + cosec2( + ) = 1 cotan( + ) + sec( + )tan( + ) = 1 sec( + ) + cosec( + ) cotan( + ) = 1 cosec( + ) + Contoh 1. sin(3 + 2) = 1 3 cos(3 + 2) + 2. sec4 tan4 = 1 4 sec 4 + 3. 2cos3 sin = (sin 4 sin 2) = 1 4 cos4 ( 1 2 cos 2) + = 1 2 cos2 1 4 cos4 + 4. sin5 cos Dengan menggunakan teknik substitusi, maka Misal = sin = cos sin5 cos = 5 = 1 6 6 + = 1 6 sin6 + 5. sin 6 0 = [ 1 6 cos6] 0 = ( 1 6 cos6) ( 1 6 cos0) = ( 1 6 1) ( 1 6 1) = 0
  • 2. Matematika SMA : Integral Page 2 Jika genap, maka : Jika ganjil, maka : Contoh sin3 Jawab Karena = 3, maka sin2 = 1 cos2 dan = cos sin3 = sin sin2 = sin (1 cos2 ) = (sin sin cos2 ) = cos sin cos2 Jika = cos maka = sin sin3 = cos 2 = cos + 1 3 3 + = cos + 1 3 cos3 + Contoh cos4 Jawab Karena = 4, maka cos2 = 1 2 (1 + cos2) cos4 = (cos2 )2 = ( 1 2 (1 + cos2)) 2 = 1 4 (1 + 2 cos2 + cos2 2) sin , cos sin2 = 1 2 (1 cos2) cos2 = 1 2 (1 + cos 2) sin2 = 1 cos2 , = cos cos2 = 1 sin2 , = sin
  • 3. Matematika SMA : Integral Page 3 = 1 4 (1 + 2cos2 + 1 2 (1 + cos4)) = 1 4 (1 + 2 cos2 + 1 2 + 1 2 cos 4) = 1 4 ( 3 2 + 2 cos2 + 1 2 cos4) = 1 4 ( 3 2 + 2 1 2 sin 2 + 1 2 1 4 cos4) + = 3 8 + 1 4 sin 2 + 1 32 cos4 + Jika , keduanya genap genap, maka : Jika ganjil, maka : Jika ganjil, maka : Contoh sin4 cos5 Jawab Karena = 4 dan = 5 maka gunakan cos2 = 1 sin2 dan = sin = cos sin4 cos5 = sin4 cos4 cos = sin4 (cos2 )2 cos = sin4 (1 sin2 )2 cos = 4(1 2)2 = 4(1 22 + 4) = ( 4 26 + 8) sin cos sin2 = 1 2 (1 cos2) cos2 = 1 2 (1 + cos 2) sin2 = 1 cos2 , = cos cos2 = 1 sin2 , = sin
  • 4. Matematika SMA : Integral Page 4 = 1 5 5 2 1 7 7 + 1 9 9 + = 1 5 sin5 2 7 sin7 + 1 9 sin9 + Contoh sin3 cos2 Jawab Karena = 3 dan = 2 maka gunakan sin2 = 1 cos2 dan = cos = sin sin3 cos2 = sin2 cos2 sin = (1 cos2 )cos2 sin = (1 2) 2() = (2 + 4) = 1 3 3 + 1 5 5 + = 1 3 sin3 + 1 5 sin5 + Contoh sin4 cos4 Jawab Karena = 4 dan = 4 maka gunakan cos2 = 1 2 (1 + cos2),sin2 = 1 2 (1 cos2) sin4 cos4 = (sin2 cos2 )2 = ( 1 2 (1 cos2) 1 2 (1 + cos2)) 2 = ( 1 4 (1 cos2 2)) 2 = ( 1 4 sin2 2) 2 = 1 16 ( 1 2 (1 cos 4)) 2 = 1 64 (1 2cos4 + cos2 4)
  • 5. Matematika SMA : Integral Page 5 = 1 64 (1 2 cos4 + 1 2 (1 + cos8)) = 1 64 (1 2 cos4 + 1 2 + 1 2 cos8) = 1 64 ( 3 2 2 cos4 + 1 2 cos8) = 1 64 ( 3 2 2 1 4 sin 4 + 1 2 1 8 sin 8) + = 3 128 1 128 sin 4 + 1 1.024 sin 8 + LATIHAN YUKS! 1. cos2 2. cos3 3. sin5 4. sin7 3 cos2 3 5. sin4 2 cos4 2