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Bab IVINTEGRAL

IR. Tony hartono bagio, mt, mm

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IV. INTEGRAL

• 4.1 Rumus Dasar• 4.2 Integral dengan Subsitusi• 4.3 Integral Parsial• 4.4 Integral Hasil = ArcTan dan Logaritma• 4.5 Integral Fungsi Pecah Rasional• 4.6 Integral Fungsi Trigonometri• 4.7 Integral Fungsi Irrasional

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4.6 Integral Fungsi Trigonometri

4.6.1 Rumus-rumus Sederhana

∫cos x dx = sin x + C ∫tan x dx = – ln|cos x|+ C

∫sin x dx = – cos x + C ∫cot x dx = ln |sin x|+ C

∫sec2 x dx = tan x + C ∫sec x tan x dx = sec x + C

∫csc2 x dx = – cot x + C ∫csc x cot x dx = – csc x + C

∫sec x dx = ln |sec x + tan x| + C

∫csc x dx = – ln |csc x + cot x| + C

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4.6 Integral Fungsi Trigonometri

4.6.2 Bentuk ∫ R(sin x) cos x dx dan ∫ R(cos x) sin x dx

Jika R fungsi rasional maka

∫ R(sin x) cos x dx = ∫ R(sin x) d(sin x) = ∫ R(y) dy

∫ R(cos x) sin x dx = – ∫ R(cos x) d(cos x) = –∫ R(t) dt

Ingat rumus cos2 x + sin2 x = 1, maka:

∫ R(sin x, cos2 x) cos x dx = ∫ R( y, 1− y2 ) dy

∫ R(cos x, sin2 x) sin x dx = – ∫ R(t, 1− t2 ) dt

Contoh

1. ∫(2cos2 x − sin x + 7) cos x dx

2. ∫sin3 x dx

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4.6 Integral Fungsi Trigonometri

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4.6 Integral Fungsi Trigonometri

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4.6 Integral Fungsi Trigonometri 4.6.3 Integral dengan memperhatikan

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4.6 Integral Fungsi Trigonometri

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4.6 Integral Fungsi Trigonometri

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4.6 Integral Fungsi Trigonometri

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4.6 Integral Fungsi Trigonometri

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4.6 Integral Fungsi Trigonometri

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4.6 Integral Fungsi Trigonometri

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4.6 Integral Fungsi Trigonometri

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4.6.4 Substitusi

Jika R(sin x, cos x) fungsi rasional dalam sin x dan cos x,

maka ∫ R(sin x, cos x) dx dapat dibawa menjadi integral fungsi rasional dalam y dengan menggunakan substitusi

2tan

xy

2tan

xy

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4.6 Integral Fungsi Trigonometri

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4.6 Integral Fungsi Trigonometri

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4.6 Integral Fungsi Trigonometri

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4.6 Integral Fungsi Trigonometri

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