Evaluate. $($Be sure to check by differentiating!$)$

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(6t^4+5)t^{3}dt$

Determine a change of variables from $t$ to $u.$ Choose the correct answer below.

A$.u=t^3+5$

B$.u=6t+5$

C$.u=6t^4+5$

D$.u=t^3$

Write the integral in terms of $u.$

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(6t^4+5)t^{3}dt={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}},du$

$($Type an exact answer. Use parentheses to clearly denote the argument of each function.$)$