Note: You can get full credit for this problem by answering just the last question correctly. The initial questions are meant as hints toward the final answer. They also allow you the opportunity to get partial credit.

Consider the indefinite integral $\displaystyle \int x^{3}\left(4+10 x^{4}\right)^{8}\, dx$

(a) The most appropriate substitution to evaluate this integral is

$u$ =

(b) The substitution above gives $du = f(x)\,dx$ where

$f(x)$ =

(c) After making the substitution we obtain the integral $\displaystyle \int g(u)\,du$ where

$g(u)$ =

(d) Computing the indefinite integral gives us

$\displaystyle \int g(u)\,du$ = $+ C$

(Your answer should be a function of $u$. )

(e) After substituting back for $u$, we obtain the following final form of the answer:

$\displaystyle \int x^{3}\left(4+10 x^{4}\right)^{8}\, dx$ = $+ C$

(Your answer should be a function of $x$. )