oh snappp yessss integration by substitution.... okay lets start again from scratch..... so i pick which one to differentiate which is
w = u^3 + 2
dw = 3u^2 + 0 = 3u^2du
1/3dw = u^2du
= √w (1/3du)
= 1/3(2/3u^3/2 + c)
= 2/9u^3/2 + c
Well, I'll give it a shot for you:
So, we have integ((u^2)sqrt((u^3) + 2))du)
By substitution, you can make w = u^3 + 2 and if you get the derivative of that, you get dw = 3u^2 du
By substituting what you have back into the integral, you get integ(((u^2)sqrt(w))du)
Now, since you have a u^2 and your dw is 3u^2, you have multiply the inside of the integral by 3 and multiply the outside by 1/3.
(1/3)integ(((3u^2)sqrt(w)du)) -> (1/3)integ(sqrt(w)dw)
From here, you get the anti derivative -> (1/3)(2/3)(w^3/2) -> (2/9)(w^3/2) + c
c is a constant...
In other words, you got it right