The definite integral

${\displaystyle \underset{0}{\overset{2}{}}}\frac{x^5}{\sqrt[\phantom{2}]{x^3+2}}dx$

The definite integral

${\displaystyle \underset{0}{\overset{2}{}}}\frac{x^5}{\sqrt[\phantom{2}]{x^3+2}}dx$

can be evaluated by substitution. Student A and Student $B$ are each working on the problem independently.

Student A ran out of time to complete their solution, but wants to make the substitution $u=x^3+2.$

Student B made a different substitution to Studant A$,$ but did not include it as part of their solution. They wrote:

${\displaystyle \underset{0}{\overset{2}{}}}\frac{x^5}{\sqrt[\phantom{2}]{x^3+2}}dx={\displaystyle \underset{0}{\overset{2}{}}}\frac{2(u^2-2)}{3}du=\frac{2}{3}{\displaystyle \underset{0}{\overset{2}{}}}(u^2-2)du=\frac{2}{3}[\frac{u^3}{3}-2u{]}_0^2=-\frac{8}{3}$

$($a$)$ Complete the solution proposed by Student A$.$

$($b$)$ What substitution did Student B make? Are there any errors in their solution? If so$,$ explain what the error$($s$)$ are, and provide a corrected solution.