$(18$pts$)$ Evaluate the indefinite integral and use C to denote the arbitrary constant.

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{x}^4{e}^{x^5}dx$

a$)$ Identify $u$ :

b$)$ Calculate $du$ :

c$)$ Rewrite the integral in terms of $u$ and evaluate the indefinite integral:

d$)$ Go back to the original variable: