Note: You can get full credit for this problem by just entering the final answer $($to the last question$)$ correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit.

Consider the definite integral ${\displaystyle \underset{\frac{1}{\sqrt[\phantom{2}]{4}}}{\overset{\sqrt[\phantom{2}]{\frac{2}{4}}}{}}}\frac{x^3}{\sqrt[\phantom{2}]{4x^2-1}}dx$

Then the most appropriate substitution to simplify this integral is $x=g(t)$ where

$g(t)=$

Note: We are using $t$ as variable for angles instead of $,$ since there is no standard way to type $$ on a computer keyboard.

After making this substitution and simplifying $($using trig identities$),$ we obtain the integral ${\displaystyle \underset{a}{\overset{b}{}}}f(t)dt$ where

$f(t)=$

$a=$

$b=$

After evaluating this integral we obtain:

${\displaystyle \underset{\frac{1}{\sqrt[\phantom{2}]{4}}}{\overset{\sqrt[\phantom{2}]{\frac{2}{4}}}{}}}\frac{x^3}{\sqrt[\phantom{2}]{4x^2-1}}dx=$