$(18$pts$)$ Evaluate the definite integral.

${\displaystyle \underset{0}{\overset{1}{}}}{x}^3(x^4+1{)}^{3}dx$

a$)$ Identify $u$ :

b$)$ Calculate $du$ :

c$)$ Rewrite the integral in terms of $u$ and evaluate the definite integral. Note you can go back to the original variable if you want to or keep the variable as $u.$