Verified answer

Integrate the original integrand by substitution:

∫ x²(2x³ + 3)⁴ dx

Let u = 2x³ + 3,

du / dx = 6x²

du = 6x² dx

du / 6 = x² dx

∫ x²(2x³ + 3)⁴ dx = ∫ u⁴ du / 6

∫ x²(2x³ + 3)⁴ dx = u⁵ / 30 + C

Since u = 2x³ + 3,

∫ x²(2x³ + 3)⁴ dx = (2x³ + 3)⁵ / 30 + C

Expand the brackets and ask again and Il answer your questain.

(2x^3 + 3)^4

=(2x^3+ 3) multiply by itself 4 times. Expanding wil make it easy 4 you. O u can make substitutions then intergrate it by parts. Expand and il help you.

let (2x^3+4)^4=u

du=6x^2(2x^3+4)^3

dx=du/6x^2(2x^3+4)^3

∫x^2(2x^3+3)^4dx= ∫(1/6)((2x^3+3)du=

= ∫(1/6)u^(1/4)du=

=(1/6)(4/5)u^(5/4)