Challenging Integral: Solving ∫ [x(8-x^3)^1/3] dx from 0 to 2

[niz73]

Hi all,
Can you guys please help me with the following integration problem

2
∫ [x (8-x3)^1/3 ] dx
0
Thanks in advance.

 

[Pengwuino]

What have you attempted for the problem so far?

 

[niz73]

I have substituted u^3 = 8 - x^3 then 3 u^2 du = - 3 x^2 dx
But now the problem has only x so in order to substitute for dx I have to divide and multiply by x and that means I can not eliminate cube root.