Question

∫ 𝑥2(𝑥3 + 1)𝑑𝑥 , using the substitution 𝑢 = 𝑥3 + 1.

Im so confused on this , please help. <3

Answer 1

Hi @AMY! The answer to your question is 2/9(x^3 + 1)^3/2 + C

Answer 2

I don't see any square roots in what @AMY typed, so I'll have to go with

∫ x^2 (x^3+1) dx
u = x^3 + 1
du = 3x^2 dx
so now you have
∫ 1/3 u du = 1/6 u^2 = 1/6 (x^3+1)^2 + C

check:
∫ x^2 (x^3 + 1) dx
= ∫ x^5 + x^2 dx
= 1/6 x^6 + 1/3 x^3
= 1/6 (x^6 + 2x^3 + 1) - 1/6
= 1/6 (x^3 + 1)^2 - 1/6 + C
This is the same as the first solution, differing only in the C involved.