∫4x^3 (x^4 + 1)^3 dx =

its
(x^4 + 1)^4 /4

;)

∫ 4x³(x^4 + 1)³ dx

Let u = x^4 + 1. Then,

du/dx = 4x³
du = 4x³ dx

So:

∫ u³ (4x³ dx)
= ∫ u³ du

Finally, by integral power rule ∫ xⁿ dx = x^(n + 1)/(n + 1) + c:

u^(3 + 1)/(3 + 1) + c
= u^4/(4) + c
= (x^4 + 1)^4/(4) + c

I hope this helps!

Use a u substitution. Substitute u for x^4+1. 1/4(x^4+1)^4 is the answer.