Hasil Dari Integral X (3x - 5) ⁴ dx Adalah??

InTegraL ParsiAL

u = x
(3x - 5)^4 dx --> v = 1/3 . 1/5 (3x - 5)^5 = 1/15 (3x - 5)^5

∫x(3x - 5)^4 dx
= ∫udv
= uv - ∫v du
= x(1/15 (3x - 5)^5) - 1/15 ∫(3x - 5)^5 dx
= 1/15 x(3x - 5)^5 - 1/15 . 1/3 . 1/6 (3x - 5)^6 + C
= 1/15 x (3x - 5)^5 - 1/270 (3x - 5)^6 + C
= (1/270)(3x - 5)^5 [18x - (3x - 5)] + C
= (1/270)(3x - 5)^5 [15x + 5] + C
= (1/54)(3x - 5)^5 [3x + 1] + C

Coba dengan Integral Substitusi

∫ x (3x - 5)⁴ dx
u = 3x - 5  --> du = 3 dx --> dx = 1/3 du
u = 3x - 5 --> 3x = u + 5 --> x = 1/3 (u + 5)
..
∫ x (3x -5)⁴  dx = ∫ 1/3 (u+5)(u)⁴(1/3) du
= 1/9 ∫ (u⁵ + 5u⁴) du
= 1/9 [1/6  u⁶ + u⁵) + c
= 1/9 [ 1/6 u⁵][u + 6] + c
= 1/54 (u+6)(u⁵) + c
= 1/54 (3x-5+ 6)(3x-5)⁵+c
= 1/54 (3x +1)(3x-5)⁵ + c