Problem $1.(1$ point$)$

Assume $f(x)$ is a polynomial with values as given in the following table:

$\backslash$table$[[x,-2,-1,0,1,2,3,4,5],[f(x),7,5,6,-2,1,0,3,-4]]$

A certain substitution $u=g(x)$ will convert ${\displaystyle \underset{0}{\overset{4}{}}}72(f(x{)}^2-1{)}^{3}f(x)f^{\text{'}}(x)dx$ into an integral of the form ${\displaystyle \underset{a}{\overset{b}{}}}h(u)du$ where $h(u)$ is a constant times a power of $u.$ You can then use the table of values given above and the substitution $u=g(x)$ to find the exact value of ${\displaystyle \underset{0}{\overset{4}{}}}72(f(x{)}^2-1{)}^{3}f(x)f^{\text{'}}(x)dx.$

$($a$)$ What is the function $g(x)?$

$($b$)$ What is the value of $a?$

$($c$)$ What is the value of $b?$

$($d$)$ What is the function $h(u)?$

$($e$)$ What is the value of ${\displaystyle \underset{0}{\overset{4}{}}}72(f(x{)}^2-1{)}^{3}f(x)f^{\text{'}}(x)dx?$