I tried using substitution on:

1/3 ∫ -1 to 2 x^2(x^3 + 1)^1/2 dx

which ended up to be 2/6 ∫ 0 to 3 u^2 du (after changing limits of integrals to u)

but I didnt get the right answer.

The correct answer was:

1/3 [ 2/9 (x^3 + 1)^3/2] x= -1 to x = 2 = 2

and it doesnt look like they used substitution.