INTEGRAL SUBTITUSI

Bentuk umum:

∫ k ∙ f'(x) ∙ fⁿ (x) dx
∫ sinⁿ f(x) ∙ cos f(x)dx
∫ sin f(x) ∙ cosⁿ f(x)dx

contoh 1:
∫6x² (x³ - 4)⁷ dx

misalkan U = x³ - 4 ⇒ dU = 3x² dx ⇒ 2dU = 6x² dx
∫6x² (x³ - 4)⁷ dx
= ∫ 2U⁷ dU
=  (2/8)U⁸ + c
= ¼ (x³ - 4)⁸ + c

contoh 2:
∫ [(x³ + 2x) / {(5/4)x⁴ + 5x²}⁶] dx

misalkan U = (5/4)x⁴ + 5x² ⇒ dU = (5x³ + 10x) dx ⇒ ⅕ dU = (x³ + 2x) dx
∫ [(x³ + 2x) / {(5/4)x⁴ + 5x²}⁶] dx
= ∫ (⅕ / U⁶) dU
= ∫ (⅕ Uˉ⁶) dU
= ⅕(-⅕)Uˉ⁵ + c
= (-1/25) U⁵ + c= (-1/25) {(5/4)x⁴ + 5x²}⁵ + c

contoh 3:
∫ (36x³ - 6) ⁷√(2x - 3x⁴)⁵ dx

misalkan U = 2x - 3x⁴ ⇒ dU = (2 - 12x³) dx ⇒ -3dU = (36x³ - 6) dx
∫ (36x³ - 6) ⁷√(2x - 3x⁴)⁵ dx
= ∫ U^(5/7)(-3dU)
= -3 ∫ U^(5/7) dU
= -3(7/12) U^(12/7) + c
= (-7/4) U ∙ U^(5/7) + c
= (-7/4) U ∙ ⁷√U⁵ + c
= (-7/4) (2x - 3x⁴) ∙ ⁷√(2x - 3x⁴)⁵ + c

contoh 4:
∫ {40x / ∛(5x² - 8)²} dx

misalkan U = 5x² - 8 ⇒ dU = 10x dx ⇒ 4dU = 40x dx
∫ {40x / ∛(5x² - 8)²} dx
= ∫ {4dU / ∛U²}
= 4 ∫ {U^(-⅔)} dU
= 4 ∙ 3 ∙ U^(⅓) + c
= 12 ∛U + c
= 12 ∛(5x² - 8) + c

contoh 5:
∫ 10 ∙ sin⁶ 4x ∙ cos 4x dx

misalkan U = sin 4x ⇒ dU = 4 ∙ cos 4x dx ⇒ (5/2)dU = 10 ∙ cos 4x dx
∫ 10 ∙ sin⁶ 4x ∙ cos 4x dx
= ∫ (5/2) ∙ U⁶  dU
= (5/2)(1/7) ∙ U⁷ + c
= (5/14)∙ sin⁷ 4x + c

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