Please do these problems in PYTHON code. Thank you! Given $f(x)=x^5(x^3+16{)}^{\frac{1}{2}}$ :

$($a$)$ Determine an appropriate substitution to change ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}f(x)dx$ to a $du-$integral. Include

both $u$ and $du.($use sp$.$Rational$($a$,$b$)$ to input a fraction $\frac{a}{b}$ in the exponent.$)$

$($b$)$ Apply the substitutions to $f($with or without Python$),$ then $($with Python$)$ find the

indefinite $du-$integral. Finally, finish the $u-$substitution integration process.

$($c$)$ Confirm your answer to part $($b$)$ by integrating ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}f(x)dx$ directly $($using integrate on $f$

directly$).$ Show that your answers for parts $($b$)$ and $($c$)$ are the same.

$($d$)$ Use the definite integral from part $($b$)$ and the Fundamental Theorem of Calculus to

evaluate ${\displaystyle \underset{-1}{\overset{1}{}}}f(x)dx.$

$($e$)$ Check your answer to part $($d$)$ by using Python to directly evaluate ${\displaystyle \underset{-1}{\overset{1}{}}}f(x)dx.$