Integration by part method

[jaychay]

Can you help me with this question ?
I am really struck with this question.
Thank you in advance.

 

[skeeter]

Using integration by parts, I would let

$u = x^3$ and $dv=x^2\sqrt{x^3+1}$


Alternatively, one could use the same setup for the method of substitution letting $t= x^3+1$