∫x⁵(2-x³)pangkat 1/2 dx=

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Mohon bantuannya dijawab ya segera…
Semoga kebaikan saudara dibalas Allah SWT dengan kebaikan pula….

∫x⁵(2-x³)pangkat 1/2 dx=

Jawaban Terkonfirmasi

displaystyle text{misal :}\x^3=u\3x^2,dx=du\\int x^5sqrt{2-x^3},dx=int sqrt{2-x^3}cdot x^3cdotfrac13(3x^2),dx\int x^5sqrt{2-x^3},dx=int sqrt{2-u}cdot ucdotfrac13,du\int x^5sqrt{2-x^3},dx=frac13intsqrt{2-u}cdot u,du\int x^5sqrt{2-x^3},dx=frac13intsqrt{2-u}(-(2-u)+2),du\int x^5sqrt{2-x^3},dx=frac13int2(2-u)^frac12-(2-u)^frac32,du\int x^5sqrt{2-x^3},dx=frac13left(2cdotfrac23(2-u)^frac32-frac25(2-u)^frac52right)+C
displaystyle int x^5sqrt{2-x^3},dx=frac49(2-u)^frac32-frac2{15}(2-u)^frac52+C\boxed{boxed{int x^5sqrt{2-x^3},dx=frac49(2-x^3)^frac32-frac2{15}(2-x^3)^frac52+C}}

Jawaban Terkonfirmasi

∫ x^5 (2 – x^3)^1/2 dx
= ∫ x^2 . x^3 (1 – x^3)^1/2 dx
Misal
u = x^3
du = 3x^2 dx
dx = du/(3x^2)
= ∫ x^2 . u . (1 – u)^1/2 du/(3x^2)
= ∫ 1/3 u (1 – u)^1/2 du
Gunakan integral parsial dengan cara tabel
Turunan … Integral
(+) 1/3 u ……. (1 – u)^1/2
(-) 1/3 ………. -2/3 (1 – u)^3/2
(+) 0 ………… -2/3 . -2/5 (1 – u)^5/2

= 1/3 u (-2/3 (1 – u)^3/2 – 1/3 (-2/3 . -1/5 (1 – u)^5/2) + C
= -2/9 u (1 – u)^3/2 – 2/45 (1 – u)^5/2 + C
= -2/9 x^3 (1 – x^3)^3/2 – 2/45 (1 – x^3)^5/2 + C

= -2/9 x^3 (1 – x^3) √(1 – x^3) – 2/45 (1 – x^3)^2 √(1 – x^3) + C

^ = pangkat